ANALYSIS AND SUPPRESSION OF INSTABILITIES …ANALYSIS AND SUPPRESSION OF INSTABILITIES IN...

ANALYSIS AND SUPPRESSION OF

INSTABILITIES IN VISCOELASTIC FLOWS

By

Karkala Arun Kumar

A DISSERTATION SUBMITTED IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OF

Doctor of Philosophy

(Chemical Engineering)

at the

UNIVERSITY OF WISCONSIN – MADISON

2001

i

Abstract

The viscoelastic character of polymer solutions and melts gives rise to instabilities that are

not seen in the flows of Newtonian liquids. In industrial applications such as coating and

extrusion, these so-called “elastic” instabilities can impose a limitation on the throughput.

Hence, it is important to understand, and if possible, to suppress them.

The first instability we study is the phenomenon of melt fracture, which occurs in the

extrusion of polymer melts and takes the form of gross distortions of the surface of the

extrudate. This instability is linked to the phenomenon of wall-slip, i.e., the velocity of the

polymer at the wall relative to the velocity of the wall itself (also called the slip velocity)

is non-zero. Several slip relations based on microscopic theories for polymers predict

regions in which the slip velocity is multivalued. The expectation is that a multivalued

slip relation will result in a multivalued flow curve, which in turn causes melt fracture.

Using a simple slip relation, we show that when the dependence of the slip velocity on

the pressure is taken into account, this is not necessarily true: a multivalued slip law does

not necessarily imply a multivalued flow curve.

The second instability we study is the “filament stretching instability,” which occurs

in the extension of a polymeric liquid bridge between two parallel plates. This insta-

bility takes the form of a bifurcation to a non-axisymmetric shape near the endplates at

ii

high extensions. Motivated by the idea of stress localization near the free surface, we

model the portion of the filament near the endplates as an elastic membrane enclosing an

incompressible fluid and show this is unstable to non-axisymmetric disturbances.

The third instability that we present results for occurs when polymeric liquids flow

along curved streamlines. Such flows are common in industrial coating operations, and

concentric cylinder geometries where the fluid flows along curved streamlines, such as

circular Couette flow (where the flow is driven by the rotation of one of the cylinders)

and Dean flow (where an azimuthal pressure gradient drives the flow), serve as model

geometries for the more complicated coating flows. In the context of Dean flow, we show

how the addition of a steady or time-periodic axial flow of small magnitude compared

to the primary azimuthal flow, and applied either in Couette or Poiseuille form, can sig-

nificantly delay the onset of the instability. The stabilization mechanism is related to the

generation of axial normal stresses induced by the secondary flow which suppress radial

velocity perturbations.

Recent experimental observations by Groisman and Steinberg (1997) and Baumert

and Muller (1999) in the nonlinear regime of viscoelastic circular Couette flow have

shown the formation of stationary, spatially isolated, axisymmetric patterns, termed “di-

whirls” or “flame patterns.” These patterns are very long wavelength vortex pairs, with a

core region of strong radial inflow, surrounded by a larger region of much weaker radial

outflow. These structures may be connected to localized defects seen in coating flows,

and may also form the building blocks for more complicated patterns seen in viscoelastic

flows. Modeling of these patterns is complicated by the absence of a stationary bifurca-

tion in isothermal circular Couette flow. We show how these solutions can be accessed

by numerical continuation from stationary bifurcations in flows that are geometrically

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similar to circular Couette flow using a simple constitutive equation for the polymer. Al-

though the stationary solutions we compute are unstable, they are very similar, both qual-

itatively and quantitatively, to the experimentally observed diwhirls and flame patterns.

We also use the results from our computations to propose a fully nonlinear self-sustaining

mechanism for these patterns.

iv

Acknowledgments

I wish to express my deep gratitude to my advisor, Professor Michael Graham, without

whose guidance this work would not have been possible. The breadth and depth of his

knowledge, his scientific curiosity, and his attention to detail have been, and will continue

to be, a source of inspiration for me. I would also like to thank Bill Black and Venkat

Ramanan for their patience in answering my questions, and for helping me when I first

started my research.

It was a privilege to have worked with such talented colleagues as John Kasab, Richard

Jendrejack, Gretchen Baier, Philip Stone, Guiyu Bai, and Jun Sato. Along with Bill and

Venkat, they helped create a stimulating atmosphere for research and contributed im-

mensely to my learning experience. Outside of work, Prasad, Shreyas, Rahul, Sirjana,

Ramesh, Kamal, Sanjay, Mukund, Mala, and others too numerous to name made my stay

in Madison enjoyable.

None of what I have accomplished would have been possible without the support of

my parents and my brother. For their love, and for my parents' belief in the value of a

good education, I shall be forever grateful.

v

Contents

Abstract i

Acknowledgments iv

1 Overview 1

1.1 Constitutive equations for polymeric liquids . . . . . . . . . . . . . 3

1.2 Instabilities in polymeric liquids . . . . . . . . . . . . . . . . . . . . 10

2 Pressure dependent slip and flow curve multiplicity 25

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.2 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . 38

2.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3 The filament stretching instability 54

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.2 Planar elongation: a model problem . . . . . . . . . . . . . . . . . 61

3.3 Elongation of a truncated cone . . . . . . . . . . . . . . . . . . . . 64


vi

3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4 Stabilization of Dean flow instability 76

4.1 Instabilities in coating flows . . . . . . . . . . . . . . . . . . . . . . 76

4.2 Elastic instability in Dean flow . . . . . . . . . . . . . . . . . . . . . 81

4.3 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.4 Stability and Numerical Analysis . . . . . . . . . . . . . . . . . . . 91


4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

5 Localized solutions in viscoelastic shear flows 116

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5.2 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

5.3 Discretization and solution methods . . . . . . . . . . . . . . . . . 124

5.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . 138

5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

6 Conclusions and future work 168

A Introduction to finite elasticity 175

A.1 The basic equations of finite elasticity . . . . . . . . . . . . . . . . 175

B Base state solutions and matrix components in Dean flow 182

B.1 Base state solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 182

B.2 Disturbance equations . . . . . . . . . . . . . . . . . . . . . . . . . 184

C Velocity and Stress scalings for Couette-Dean flow 189

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D Time Integration of viscoelastic flows 193

D.1 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . 195

D.2 Numerical Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

D.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . 201

E Branch tracing in three dimensional plane Couette flow 205

E.1 Introduction and Formulation . . . . . . . . . . . . . . . . . . . . . 205

E.2 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

E.3 Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

E.4 Proposals for future work . . . . . . . . . . . . . . . . . . . . . . . 212

Nomenclature 216

Bibliography 226

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List of Figures

1 An illustration of the rod climbing effect. The beaker in (a) contains a

Newtonian liquid, glycerin, which shows a vortex. The beaker in (b)

contains a solution of polyacrylamide in glycerin, which climbs the rod

(figure scanned from Bird et al. (1987a)). . . . . . . . . . . . . . . . . . 2

2 Diagram of simple steady shear flow. As shown here, the two plates are

moved with equal speed in opposite directions. . . . . . . . . . . . . . . 3

3 Diagram of elongational flow with l = exp(ε t). . . . . . . . . . . . . . . 5

4 (a) A random walk calculation showing one of a very large number of

conformations of a polymer molecule (b) A dumbbell model which only

captures the longest relaxation time. . . . . . . . . . . . . . . . . . . . . 5

5 Pictorial representation of the two types of pitchfork bifurcations: (a)

supercritical and (b) subcritical. [y] is some measure of y that captures

the features of the transition. A solid line indicates a stable branch while

a dashed line indicates an unstable branch. . . . . . . . . . . . . . . . . . 13

6 Extrusion related instabilities (a) sharkskin (b) melt fracture. (Agassant

et al., 1991). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

ix

7 A typical flow curve for polymers exhibiting spurt and melt fracture. The

y coordinate 8V/D is proportional to the exit flow rate (Kalika and Denn,

1987). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

8 The filament stretching instability (a) The non-axisymmetric bifurcation

seen from below the bottom plate (b) side view of the instability at a later

stage (Spiegelberg and McKinley, 1996). . . . . . . . . . . . . . . . . . . 18

9 Flow visualization of the purely elastic Taylor-Couette instability (Larson

et al., 1990). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

10 Flow visualization of a blade coating geometry. The figure clearly shows

the presence of an upstream recirculation region with curved stream-

lines (Davard and Dupuis, 2000). . . . . . . . . . . . . . . . . . . . . . . 20

11 Sequence of snapshots showing the transition from non-axisymmetric

disordered flow to solitary vortex structures. On the left in (a), the en-

tire flow geometry is shown, with the box showing the cross section

being visualized. On the right, in (b), the actual transition sequence is

shown (Groisman and Steinberg, 1998). . . . . . . . . . . . . . . . . . . 21

12 Sequence of snapshots showing the transition from non-axisymmetric

flow to the predominantly axisymmetric and localized flame patterns (Baumert

and Muller, 1999). The flow geometry is the same as in figure 11. . . . . 22

13 Plot of equation 30 for different values of pressure (A2 = 4, A3 = 122.24,

β = 0.0102). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

14 Schematic diagram of the constant piston speed experiment. . . . . . . . 36

x

15 Flow curve for a Newtonian Fluid (Λ = 10, A2 = 3, A3 = 122.24,

β = 0.0102, H = 1). The profiles at the points marked `X' are shown in

figures 18 and 19. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

16 Behavior of critical stresses τc2 and τc3 with increasing Λ (A2 = 3, A3 =

122.24): (a) Newtonian model (β = 0.0102, H = 1) (b) shear thinning

model (β = 0.0102, H = 1, n = 0.56) (c) UCM model (β = 0.0102,

H = 1) (d) PTT model (ε = 10−1, β = 0.0102, H = 1). The upper curve

corresponds to τc = τc2 and the lower one to τc = τc3. The curves are

not extended to Λ = 0 because the approximations used are not valid for

small Λ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

17 Flow curve for a Newtonian fluid showing no multiplicity (Λ = 238.62,

A2 = 3, A3 = 122.24, β = 0.0102, H = 1). . . . . . . . . . . . . . . . . 42

18 Profiles of pressure, slip velocity and shear stress on the low flow rate

branch for the Newtonian fluid of figure 15. (∆P =5.22, Λ = 10, A2 =

3, A3 = 122.24, β = 0.0102, H = 1). . . . . . . . . . . . . . . . . . . . 43

19 Profiles of pressure, slip velocity and shear stress on the high flow rate

branch for the Newtonian fluid of figure 15 (∆P =6.39, Λ = 10, A2 = 3,

A3 = 122.24, β = 0.0102, H = 1). . . . . . . . . . . . . . . . . . . . . . 44

20 Flow profiles on the three dimensional slip surface for a shear thinning

fluid: (a) upper and lower limit points (∆P = 6.39, Λ = 10) (b) cusp

point (∆P = 132.24,Λ = 230.46) (c) point in the central portion without

multiplicity (∆P = 140.0, Λ = 240.82). Other parameters are A2 = 3,

A3 = 122.24, β = 0.0102, H = 1, and n = 0.56 for all three cases. . . . . 46

xi

21 Oscillatory flow for a Newtonian fluid with Qp = 0.21:(a) Pressure drop

and exit flow rate vs. time (b) Plot of Qe vs. Pb superimposed on the

steady state flow curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

22 Non-oscillatory flow for a Newtonian fluid with Qp = 0.15: Pressure

drop and exit flow rate vs. time. . . . . . . . . . . . . . . . . . . . . . . 49

23 Non-oscillatory flow of a Newtonian fluid with Λ = 250, Qp = 0.1862

and other parameters as in figure 21: Pressure drop and exit flow rate vs.

time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

24 Oscillatory flow of a UCM fluid (C = 106, Λ = 10, Re = 10−4, κ =

10−5, De = De∗ = 10, λ1 = λ2 = 0.01, Qp = 0.081): (a) Barrel

pressure and exit flow rate vs. time (b) Plot of Qe vs. Pb superimposed

on the steady state flow curve. . . . . . . . . . . . . . . . . . . . . . . . 52

25 Schematic of a filament stretching rheometer. The setup on the left shows

the undeformed state of the liquid bridge. . . . . . . . . . . . . . . . . . 58

26 Schematic of a planar elongation setup and flow field. . . . . . . . . . . . 62

27 Coordinate system for the truncated cone. . . . . . . . . . . . . . . . . . 66

28 Evolution of the shape of a truncated cone under elongation: (a) un-

deformed configuration (b) axisymmetric configuration at l = 0.2 (c)

axisymmetric configuration at l = lc = 0.5 (d) post bifurcation non-

axisymmetric shape at l = lc = 0.5. The figures on the right track the

change in a cross section originally at a distance of 0.75 units from the

left edge. In (d), the perturbation has been exaggerated for clarity. . . . . 72

29 Spatial profile of the hoop stress, τθθ, and the amplitude of the bifurcating

solution ‖ λ1 ‖. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

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30 Some commonly used coating industrial coating processes: (a) Dip coat-

ing and rod coating. (b) Blade coating and air knife coating. (c) Gravure

coating. (d) Reverse roll coating. (e) Extrusion coating. (f) Slide coating

and curtain coating (Cohen, 1992) . . . . . . . . . . . . . . . . . . . . . 78

31 Photograph of ribbing instability in forward roll coating (Coyle et al.,

1990b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

32 Mechanism of the elastic instability in Dean flow. . . . . . . . . . . . . . 84

33 Illustration of the mechanism by which additional axial stresses gener-

ated by a superimposed axial flow stabilize the viscoelastic Dean flow

instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

34 Dean flow geometry, shown with superimposed Poiseuille flow. . . . . . . 87

35 Neutral stability curves for DAC flow (S=0). In each case, the position of

Wpc,min is denoted by a •. . . . . . . . . . . . . . . . . . . . . . . . . . 102

36 Neutral stability curves for DAP flow (S=0). In each case, the position of

Wpc,min is denoted by a •. . . . . . . . . . . . . . . . . . . . . . . . . . 103

37 Plot of Wpc,min vs. Wez for DAC and DAP flows (S=0). Note the linear

scaling at high Wez. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

38 Plots of Wpc,min vs. Wez for S = 0 and S = 10, displaying the stabilizing

influence of solvent viscosity. . . . . . . . . . . . . . . . . . . . . . . . . 104

39 Plot of Wpc,min vs. n for different values of Wez for DAC flow (S = 0). . 104

40 Plot of Wpc,min vs. n for different values of Wez for DAP flow (S = 0). . 105

41 Plot of Wpc,min vs. Wez for DAC flow with n = 1.0 (S = 0). Note the

linear scaling for Wez > 1. . . . . . . . . . . . . . . . . . . . . . . . . . 105

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42 Neutral stability curve of pure Dean flow at high α. The Takens-Bogdanov

bifurcation point is indicated by a •. . . . . . . . . . . . . . . . . . . . . 107

43 Decay of the perturbation hoop stress τθθ when axial flow is imposed.

Parameters are: Wez = 1.0, ω1 = 0.5, Wp = 4.06, S = 0. . . . . . . . . . 110

44 Time sequence of density plots of the perturbation hoop stress τθθ when

axial flow is imposed. The parameters are identical to figure 43, so that

without axial flow, the flow is neutrally stable. Each frame shows a z− r

cross-section of the geometry. . . . . . . . . . . . . . . . . . . . . . . . 111

45 Plot of the magnitude of the Floquet multiplier |β| vs. ω1 for different

values of α. (Wez = 0.5, Wp = 4.06, S = 0). . . . . . . . . . . . . . . . 112

46 Plot of the magnitude of the Floquet multiplier |β| vs. ω1 for different

values of Wez. (Wp = 4.06, α = 6.6, S = 0). |β| asymptotes to 1 at

large ω1 in agreement with the asymptotic prediction. . . . . . . . . . . . 113

47 Plot of the magnitude of the periodic component of the hoop stress τθθ

over a cycle of the forcing for ω1 1 (Wez = 1.0, Wp = 4.06, α = 6.6,

ω1 = 0.01). Note the large increases in magnitude. . . . . . . . . . . . . 114

48 Plot of the magnitude of the periodic component of the hoop stress τθθ

over a cycle of the forcing for ω1 = O(1) (Wez = 1.0, Wp = 4.06,

α = 6.6, ω1 = 1.0). The magnitude remains O(1) over the entire cycle. . 115

49 Geometry of Couette-Dean flow in an annulus . . . . . . . . . . . . . . . 122

50 A spectral element mesh with 16 axial and 16 radial elements with fifth

order polynomials in each direction in each domain. Note the dense con-

centration of points near r = 1 and z = L/2. The high resolution is

necessary to capture the intense stress localization in these regions. . . . . 128

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51 Comparison of ILUT∗ and ILU(0) preconditioners. The test problem was

the calculation of the unit tangent for a point on the nontrivial branch in

Dean flow. The matrix A had a dimension of 21987. . . . . . . . . . . . 137

52 Linear stability curves at δ = 1 (Dean flow) computed using the FENE-P

model. The points marked “TB” are Takens-Bogdanov points. The lines

correspond to points where the base state flow loses stability to stationary

axisymmetric perturbations. . . . . . . . . . . . . . . . . . . . . . . . . 140

53 Linear stability curves at δ = 1 (Dean flow) computed using the FENE-

CR model. As in figure 52, only stationary axisymmetric perturbations

are considered. Note the complete absence of non-stationary bifurcations. 140

54 Continuation in Weθ of a stationary solution in Dean flow. The parameter

values are L = 1.05, b = 700, ε = 0.20, and S = 1.2. At the Hopf point,

a pair of complex conjugate eigenvalues become unstable. These collide

and form two real eigenvalues, one of which re-crosses the imaginary axis

at Weθ = 29.57, where the stationary branch originates. The solution

amplitude used here differs from that used in subsequent figures and is

defined in equation 150. . . . . . . . . . . . . . . . . . . . . . . . . . . 142

55 A path to stationary solutions in circular Couette flow. The parameters

are Weθ = 25.15, L = 3.07, b = 1830, ε = 0.2, and S = 1.2. . . . . . . . 143

56 Results from continuing the stationary circular Couette flow solutions in

L. The parameters are Weθ = 25.15, b = 1830, ε = 0.2, and S = 1.2.

The gaps in the lower branch correspond to places where we changed

the mesh. Note the flatness of the branches as L increases. We have

computed extensions of the upper branch at lower values of Weθ. . . . . 145

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57 Density plot of 〈QQ〉θθ (white is large stretch, black small) and contour

plot of the streamfunction at L = 116.52 (Weθ = 24.29, b = 1830,

S = 1.2, and ε = 0.2). For clarity, most of the flow domain is not

shown. Note the very strong localization of 〈QQ〉θθ near the center. The

maximum value of 〈QQ〉θθ at the core is 1589 which gives τθθ = 12722.

Compared to this, the maximum value of 〈QQ〉θθ in the circular Couette

base state is 706, which gives τθθ = 1150. Away from the core of the

diwhirl, the structure is pure circular Couette flow. The streamlines show

striking similarity to those in figure 10 of Groisman and Steinberg (1998).

This point was generated by stretching the point at the corresponding Weθ

on the upper branch of the curve for L = 9.11 in figure 58. . . . . . . . . 146

58 Diwhirl solution amplitudes as functions of Weθ and L. Note that the

curves at L = 9.11 and L = 4.74 are very close together, while both

curves are well separated from the curve at L = 3.07 (b = 1830, S = 1.2,

and ε = 0.2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

59 Plot of the location of the turning point, Weθ,c, versus L at S = 1.2 and

ε = 0.2. Note the flatness of the curve at large L. . . . . . . . . . . . . . 149

60 Plot of the position of the linear stability limit in circular Couette flow

with respect to axisymmetric disturbances and the turning point in Weθ

for the diwhirls as a function of b. The parameters are S = 1.2 and

ε = 0.2. The computations for the diwhirls were performed at L = 4.74,

which is close to the minimum in figure 59. . . . . . . . . . . . . . . . . 150

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61 (a) The axial variation of vr at r = 0.6 for L = 4.74 and Weθ = 23.50 on

the upper branch. (b) Figure 9 on page 2457 of Groisman and Steinberg

(1998), shown here for purposes of comparison. We have shifted the axial

coordinate so that the symmetry axis of the computed diwhirl in (a) is at

z = 0, to make comparison with (b) easier . . . . . . . . . . . . . . . . . 151

62 Variation of solution amplitudes with ε. Here, Weθ = 25.15, and the

other parameters as in figure 58. . . . . . . . . . . . . . . . . . . . . . . 152

63 Diwhirl solution amplitudes as a function of the parameter cr for two

different wavelengths. Weθ = 25.15 and the other parameters are as in

figure 58. The existence of turning points demonstrates that these solu-

tions cannot be extended to the FENE-CR model. . . . . . . . . . . . . . 153

64 Intensity plot of the dimensionless viscous dissipation for Weθ = 23.73

on the upper branch at L = 4.74. Light areas represent areas of large

viscous dissipation and dark areas represent regions of low viscous dissi-

pation. The horizontal axis is stretched by a factor of two relative to the

vertical axis for clarity. . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

65 (a) Vector plot of v near the outer cylinder at the center of the diwhirl

structure (oblique arrows) and the base state (straight arrows). The length

of the arrows is proportional to the magnitude of the velocity. The axial

velocity is identically zero in the base state, and is zero by symmetry

at the center of the diwhirl. (b) Principal stress directions at the same

location as for (a). The Couette flow stress is not shown because it is

very small in comparison. This figure shows how fluid elements at larger

radii are pulled down and forward sustaining the increase in vθ. . . . . . . 157

xvii

66 Nonlinear self-sustaining mechanism for the diwhirl patterns. . . . . . . . 158

67 Density plot of vr showing the (a) real and (b) imaginary parts of the

destabilizing axisymmetric disturbance, and (c) the streamlines of the

base diwhirl. Note that the core of the diwhirl is entirely unaffected by

the disturbance. The parameters are Weθ = 23.87, S = 1.2, ε = 0.20,

b = 1830, and L = 4.74. . . . . . . . . . . . . . . . . . . . . . . . . . . 161

68 Density plot of the perturbation radial velocity for the three non-axisymmetric

unstable eigenmodes with n = 1. The parameters are identical to those

in figure 67. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

69 Density plot of the perturbation radial velocity for the three non-axisymmetric

unstable eigenmodes with n = 2. The parameters are identical to those

in figure 67. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

70 Bifurcation diagram for Dean flow at b = 1830, S = 1.2, ε = 0.2, and

L = 1.963. We could not continue the branch beyond Weθ = 13.94, for

reasons discussed in section 5.4.6. Radial velocity profiles corresponding

to the points marked by the filled and open circles are shown in figure 71. 164

71 Density plots of the radial velocity at the points circled in figure 70. (a)

Weθ = 15.486 (b) Weθ = 13.930. . . . . . . . . . . . . . . . . . . . . . 164

72 Linear stability curves and existence boundaries for nonlinear solutions

in Dean flow at S = 1.2, ε = 0.2, and b = 1830. The filled triangle shows

the data at L = 1.795 where the solution terminates via collision with the

L/2 branch in a pitchfork bifurcation before the turning point. . . . . . . 165

73 Streamlines for equation 214 with A = 1 and k = π. . . . . . . . . . . . 194

xviii

74 Poincare map of the trajectory of a point in the flow field given by equa-

tion 215. The parameters chosen were k = π, ω = 2π, and ε = 0.25. . . . 195

75 Illustration of the effect of adding a diffusion term to the constitutive

equation on a mesh of 144×72. The parameter values used were We = 1,

Re = 1, ε = 0.2, k = π, ω = 2π, b = 10, and S = 6.67. . . . . . . . . . . 202

76 Effect of mesh refinement on numerical stability. Runs with both meshes

were performed with a = 10−4 and the same parameter values as in

figure 75. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

77 Effect of artificial diffusion when equation 223 in integrated in a known

velocity field. The parameters and mesh are identical to those in figure 75.

Note the earlier blow up when compared to the coupled case. . . . . . . . 203

78 Contours and density plot of the streamwise velocity u in the y− z plane

in the base state. The value of Re = 150, Λ = 0.002, and Fr = 5.08. . . . 212

79 Contours and density plot of the u component of the two unstable eigen-

vectors for the flow shown in figure 78 in the x − z plane at y = 0. The

mode shown in (a) has the higher growth rate of the two unstable modes. . 213

80 Plot of the profile of the u component of the streamwise velocity of the

two unstable eigenvectors for the flow shown in figure 78 at y = 0, and

z = π/γ. The eigenvalue corresponding to eigenvector 1 has the higher

growth rate. Eigenvector 2 is simply a vertically shifted and scaled ver-

sion of eigenvector 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

81 Contours and density plot in the x − z plane of the u component of the

real and imaginary part of the unstable eigenvector at Re = 150, Λ = 0.1,

and Fr = 5.01. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

1

Chapter 1

Overview

Unlike Newtonian fluids, polymeric liquids have some memory of the deformation they

have experienced. Newtonian fluids respond virtually instantaneously to an imposed de-

formation rate, whereas polymeric fluids respond on a macroscopically large time scale,

known as the relaxation time. When subjected to deformation rates much larger than

the inverse relaxation time, their behavior resembles that of elastic solids, whereas their

response to deformation rates much smaller in magnitude than the inverse relaxation

time resembles that of viscous liquids. For this reason, they are known as “viscoelastic”

liquids. The viscoelastic nature of polymer melts and solutions causes them to exhibit

behavior not seen in Newtonian liquids. An interesting example of this behavior is the so

called rod-climbing effect. An illustration of this effect is shown below in figure 1. The

figure shows two liquids being stirred in two different beakers. The liquid in figure 1(a)

is Newtonian, while the liquid in figure 1(b) is a polymer solution. The Newtonian liquid

shows a vortex, with free surface being depressed in the region near the stirrer. We would

expect this intuitively, based on the fact that the centrifugal force pushes liquid near the

2

stirrer outward. In contrast, the polymer solution shows the opposite effect, with the liq-

uid climbing up the stirring rod. This is because the polymer molecules are stretched

along the circular streamlines of the flow. The extra tension in the streamlines exerts an

inward force on the fluid, which acts against the centrifugal force and gravity and pushes

the liquid up the rod. Other examples of viscoelastic behavior include die swell and

elastic recoil (Bird et al., 1987a).

(a) (b)

Figure 1: An illustration of the rod climbing effect. The beaker in (a) contains a New-tonian liquid, glycerin, which shows a vortex. The beaker in (b) contains a solutionof polyacrylamide in glycerin, which climbs the rod (figure scanned from Bird et al.(1987a)).

In industrial applications involving polymeric liquids, instabilities that arise from

their viscoelastic nature can have important consequences, and this document details a

computational investigation of a few of these instabilities. In this chapter, we present an

overview of the topics covered in the rest of the thesis. Before discussing instabilities

in polymerflows however, it is instructive to look at some of the elementary approaches

taken to modeling polymeric liquids, which lead to the constitutive equations that we use

in our analyses.

3

1.1 Constitutive equations for polymeric liquids

Unlike Newtonian liquids, which are very well described by Newton's law of viscosity,

there is no single constitutive equation that describes the entire range of polymeric liq-

uids. Consequently, the approach taken is to use simpleflows to check the predictions of

constitutive equations with the actual behavior of the polymeric liquid being studied. We

will, for the most part, be interested in the behavior of dilute solutions of linear polymers.

In this section, we discuss some of theflows used to characterize these liquids, and then

proceed to describe a few of the simple constitutive equations used to model them.

x

y

z

Figure 2: Diagram of simple steady shearflow. As shown here, the two plates are movedwith equal speed in opposite directions.

The simplestflow used for rheological characterization is the steady shearflow shown

in figure 2. Denoting the velocity components in the three coordinate directions asvx,

vy, andvz, the only component that is not zero isvx. For a Newtonianfluid, the force

required to maintain theflow would only have a component in theflow direction. The

only nonzero components of the stress tensor are the shear stresses,τyx andτxy. Since

the stress tensor is symmetric, these are equal to each other and are given by Newton's

4

law of viscosity as

τxy = τyx = ηγyx, (1)

whereγyx = dvx/dy is the shear rate,γ = |γyx|, andη is the viscosity, which is indepen-

dent of the shear rate. For a polymer solution, the viscosityη, would, in general, depend

on the shear rate, i.e.,η = η(γ). In addition, the stress componentsτxx, τyy, andτzz are

not all zero, unlike in the shearflow of Newtonian liquids. A direct physical consequence

of this is that there are components of thefluid stresses that act to push the plates apart.

Therefore, in addition to the shearing force, a force acting normal to the plates has to be

applied to maintain theflow. The additional nonzero stress components allow us to define

two newfluid properties using the relations

τxx − τyy = Ψ1(γ)γ2yx, (2)

τyy − τzz = Ψ2(γ)γ2yx. (3)

The quantityτxx − τyy is known as thefirst normal stress difference, whileτyy − τzz

is known as the second normal stress difference.Ψ1 andΨ2 are known as thefirst and

second normal stress coefficients respectively, and are functions ofγ.

Another flow used for rheological characterization is elongationalflow (figure 3).

Here, the velocityfield is given by

vx = −1

2εx

vy = −1

2εy

vz = εz, (4)

so thatfluid elements are being stretched exponentially in one direction and compressed

exponentially in the two directions perpendicular to it. The material function of relevance

5

l

1/

z

x

y

l

1/ l

Figure 3: Diagram of elongationalflow with l = exp(ε t).

here is the elongational viscosityη, which is defined by the relation

η =τzz − τxx

ε. (5)

For Newtonian liquids,η = 3η, and the ratioη/η is called the Trouton ratio.

coarsegraining

Q

(a) (b)

Figure 4: (a) A random walk calculation showing one of a very large number of confor-mations of a polymer molecule (b) A dumbbell model which only captures the longestrelaxation time.

6

The steady shear and elongationalflows are used as modelflow fields for testing con-

stitutive equations for polymers. Experimental measurements of the behavior of specific

polymer solutions and melts can be compared with the predictions of the constitutive

equations. The expectation is that constitutive equations that can capture the behavior of

a polymer in these modelflows might reasonably be expected to work in more compli-

catedflow fields as well. Also, the modelflows are used to obtain rheological parameters

needed in constitutive equations. For example, simple shearflow can be used to obtain

the zero shear viscosity and the relaxation time.

We now proceed to discuss simple constitutive equations that are used for modeling

dilute solutions of linear polymers. The constitutive equations that we describe below

are based on molecular models. Figure 4(a) shows a random walk model of a polymer

molecule, which represents one of a large number of conformations that the polymer

molecule can take. These conformations change continually due to thermal motion, with

different time scales associated with motion on different length scales. For example, it is

easy to see that small internal reorientations would take place on a shorter time scale than

a change involving a large number of segments. Thus, the polymer molecule responds on

a wholespectrum of time scales. For the simple models that we will use in this work, we

only consider the upper limit of this spectrum, i.e., the longest relaxation time.

At the level of coarse graining described above, the polymer molecules are modeled

as elastic dumbbells (a pair of beads connected by a spring), as shown infigure 4(b), with

Q representing the end to end vector of the dumbbell. The elastic force here represents

the restoring force due to entropy. A solution of such dumbbells is described by the

probability distribution function forQ, denoted byψ(Q, t). For a dilute solution of these

dumbbells, neglecting inertia and hydrodynamic interaction between the beads allows us

7

to use kinetic theory (Bird et al., 1987b) to derive a diffusion equation forψ(Q, t), which

is given by

∂ψ

∂t+ v ·∇ψ = −

(∂

∂Q·

[κ ·Q]ψ −

2kT

ζ

∂ψ

∂Q−2

ζF cψ

). (6)

Here,v is the velocity vector,k is Boltzmann's constantt represents time,T represents

temperature,F c the spring force,ζ the friction coefficient due to hydrodynamic drag, and

κ is a second order tensor which specifies the local velocityfield.

Equation 6 serves as the starting point for the calculation of the averages of quantities

of physical interest. For the purpose of this work, we are mainly interested in the stress

tensorτ . For elastic dumbbells, this is given by

τ = N 〈QF c〉 − NkT I, (7)

whereN is the number density of dumbbells, and we have once again assumed that the

external forces acting on the two beads of the dumbbell cancel each other out. For simple

models,τ can be approximated from the ensemble average of the second order tensor

QQ. This quantity,〈QQ〉, is given by the equation

〈QQ〉(1) =4kT

ζI −

4

ζ〈QF c〉 . (8)

Here,〈QQ〉(1) is theupper convected derivative of 〈QQ〉 given by

〈QQ〉(1) =∂〈QQ〉

∂t+ v ·∇〈QQ〉−

(∇v)t · 〈QQ〉+ 〈QQ〉 · (∇v)

, (9)

wherev is the velocity vector, and the superscriptt is used to denote the transpose.

In our discussion above, we have not specified a form for the spring force. The sim-

plest case occurs when we assume that the spring is Hookean, i.e.,F c = HQ, where

H is the spring constant. This assumption yields the upper convected Maxwell (UCM)

8

equation. This equation assumes that the beads can stretch to an infinite extent, which

is clearly not a reasonable assumption, especially for so called strongflows, wherefluid

elements undergo exponentially large degrees of stretching. However, the UCM equation

has the virtue of being simple, and for this reason is often used to get qualitative informa-

tion on the effect of viscoelasticity. This simplicity of form is in some respects deceptive,

for it masks the fact that the UCM equation is extremely difficult to use in the numerical

simulation of strongflows. This in turn gives rise to another important use of the UCM

equation: as a test case for numerical methods for solving viscoelasticflow problems.

Since polymers havefinite maximum lengths, a nonlinear spring law makes for a

more realistic approximation. The Finitely Extensible Nonlinear Elastic (FENE) spring

law takes the form

F c = HQ

1−Q2/Q20, Q ≤ Q0, (10)

whereQ2 = tr(QQ) is the square of the dumbbell extension withtr used to denote the

trace of its argument. For small extensions, the spring is nearly Hookean, but increases in

stiffness for larger extensions, up to a maximum extension ofQ0. In the limitQ0 → ∞,

equation 10 becomes Hooke's law. Substituting equation 10 into equation 8 gives

〈QQ〉(1) =4kT

ζI −

4H

ζ

⟨QQ

1−Q2/Q20

⟩. (11)

The second term on the right hand side prevents equation 11 from being an explicit equa-

tion for 〈QQ〉. To obtain a closed form equation for〈QQ〉, it is necessary to make

a closure approximation. One such approximation is the Peterlin closure (Bird et al.,

1987b) which takes the form⟨QQ

1−Q2/Q20

⟩

〈QQ〉1− tr(〈QQ〉)/Q20

. (12)

9

Substituting this relation into equation 11 gives the FENE-P constitutive equation. A

related FENE type equation is the FENE-CR equation (Chilcott and Rallison, 1988),

which is a good model for dilute solutions of some polymers. This equation is

〈QQ〉(1) =1

1− tr(〈QQ〉)/Q20

(4kT

ζI −

4

ζ〈QQ〉

). (13)

In all three equations, knowing〈QQ〉 enables us to get expressions for the stress tensor.

For the UCM equation, this is given by the Kramers expression

τ = NH〈QQ〉−NkT I. (14)

The quantityNkT is equal toG, the relaxation modulus of the polymer. Substituting

equation 14 and the Hookean spring law into equation 8 yields an explicit equation forτ ,

given by

τ + λ τ (1) = ηγ, (15)

which is linear inτ . Here,λ is the relaxation time, given by

λ =ζ

4H, (16)

η is the viscosity, given by

η = λG, (17)

and

γ = I(1) =∇v + (∇v)t (18)

is the shear rate tensor. For the FENE-P equation, the Kramers expression for the stress

tensor is

τ = NH〈QQ〉

1− tr(〈QQ〉)/Q20−NkT I, (19)

10

and for the FENE-CR equation, the stress tensor can be calculated using the relation

τ =1

1− tr(〈QQ〉)/Q20(NH〈QQ〉−NkT I) . (20)

In the latter two cases, it is possible to substitute equations 19 and 20 into the constitutive

equations for〈QQ〉 and get explicit constitutive equations forτ . However, it is much

simpler to work with〈QQ〉, and use equation 19 or 20 to getτ when needed.

Table 1 summarizes the qualitative behavior of the UCM, FENE-P and FENE-CR

equations in shear and elongation. Also presented in table 1 is the behavior of the exact

FENE model in theseflows, obtained from Brownian dynamics simulations. In this work,

we will use the FENE-P and UCM equations for most of our calculations. Occasionally,

we will use the FENE-CR equation when testing for model dependence.

PropertyModel η Ψ1 η

UCM Constant Constant Blows up atfiniteelongation rate

FENE Shear thins Shear thins Saturates at highelongation rate

FENE-P Shear thins Shear thins Saturates at highelongation rate

FENE-CR Constant Shear thins Saturates at highelongation rate

Table 1: Summary of behavior of the UCM, FENE, FENE-P, and FENE-CR constitutiveequations in shear and elongation. All four models predictΨ2 = 0.

1.2 Instabilities in polymeric liquids

Before embarking on a discussion offlow instabilities in polymeric liquids, it is instruc-

tive to clarify what we mean by the term“flow instability.” This is best done with the aid

11

of a simple example. Consider theflow of a Newtonian liquid (such as water) in the gap

between two very long concentric cylinders, driven by the motion of the inner cylinder.

At low rotation rates, we would see aflow where the velocities and pressure only varied

with radial position and are constant along the azimuthal and axial directions. We will

refer to thisflow as the baseflow. As the rotation speed is increased, a transition occurs

to an axisymmetric (i.e. no variation in the azimuthal direction), axially periodic vortex

flow. Thisflow is qualitatively different from the baseflow: in particular, unlike in the

baseflow, the velocities and stresses vary in the axial direction. What is interesting is

that the baseflow is an admissible solution to the governing equations at all values of the

rotation rate. Beyond a critical value of the rotation rate however, perturbations (even in-

finitesimal ones) applied to the baseflow grow in magnitude until a qualitatively different

steady state is reached. We say that the baseflow is unstable beyond the critical rotation

speed, hence the phrase“flow instability.” The qualitative change caused by the variation

of a parameter (in this case, the rotation rate of the inner cylinder) is called abifurcation.

In this work, we will mainly be interested in two types of bifurcations: pitchfork and

Hopf. Pitchfork bifurcations generally occur in systems with symmetry. As a simple

one-dimensional example, consider the differential equation

y = µy − y3. (21)

This equation is invariant with respect to the transformationy → −y, i.e., replacingy by

−y results in the same equation. Steady states are obtained by settingy to zero. They are

solutions to

f(y) = µys − ys3 = 0, (22)

which areys = 0 andys = ±√

µ, with the subscripts being used to denote a steady state.

12

While ys = 0 is a solution for all values ofµ, the solutionsys = ±√µ are valid only for

µ > 0. In this problem,ys =√µ andys = −

√µ bifurcate from the solutionys = 0

at µ = 0. In a small neighborhood of the bifurcation point, the branchesys =√µ and

ys = −√

µ are one sided, i.e., they only exist forµ > 0. This type of bifurcation is called

a pitchfork. Note that the bifurcating branchesys =√µ andys = −

√µ are related by

symmetry.

For the one-dimensional example presented here, the stability of the bifurcating branches

is determined by the sign of the linearization of the steady state equations, evaluated at

the point whose stability is to be determined. If the linearization is positive, the solu-

tion is unstable, because small disturbances grow. If the linearization is negative, small

disturbances decay, and the solution is stable. For equation 21, the linearization is given

by

∂f

∂y= fy = µ− 3y2s . (23)

Substitutingys = 0 givesfy = µ. Thus, the solutionys = 0 is stable for negative values

of µ and unstable for positiveµ. Note that the change in stability occurs at the bifurcation

point,µ = 0. Substitutingys = ±√µ in equation 23 givesfy = −2µ, which is negative

for µ > 0, the only regime where these solutions exist. Therefore, these solutions are

stable. When the bifurcating branch is stable, the bifurcation is said to besupercritical.

The opposite case, asubcritical bifurcation, occurs when the bifurcating branches are

unstable. An example of this is the system

y = y3 + µ y, (24)

which has the steady statesys = 0 andys = ±√−µ. An analysis similar to the one pre-

sented above shows that the steady statesys = ±√−µ bifurcate fromys = 0 at µ = 0,

13

only exist for negative values ofµ and are unstable. Figure 5 shows a pictorial repre-

sentation of supercritical and subcritical bifurcations. For higher dimensional systems,

stability is determined by the eigenvalues of the matrix that results from linearizing the

steady state governing equations, with the linearization being performed about the point

whose stability is being determined. This matrix is called the Jacobian. If all the eigenval-

ues of the Jacobian are negative, the solution is stable. If one or more of the eigenvalues

are positive, the solution is unstable. A bifurcation occurs when the real part of one or

more eigenvalues changes sign.

[y]

µ

[y]

µ

[y]

µ

[y]

µ

(a)

(b)

Figure 5: Pictorial representation of the two types of pitchfork bifurcations: (a) super-critical and (b) subcritical.[y] is some measure ofy that captures the features of thetransition. A solid line indicates a stable branch while a dashed line indicates an unstablebranch.

The other bifurcation that we will present here is the Hopf bifurcation. A Hopf bi-

furcation occurs when a pair of complex conjugate eigenvalues of the Jacobian crosses

14

the imaginary axis. In this case, the bifurcating branches are not steady states, but time-

periodic oscillations (also called limit cycles). As with the pitchfork bifurcation, a super-

critical Hopf bifurcation occurs if the bifurcating branch of periodic solutions is stable,

and a subcritical Hopf bifurcation occurs when they are unstable. Since a Hopf bifurca-

tion requires that a pair of eigenvalues cross the imaginary axis, it follows that a system

must be at least two-dimensional to show a Hopf bifurcation. An example of a system

that undergoes a Hopf bifurcation is

y1 = −y2 + y1(µ− y12 − y2

2),

y2 = y1 + y2(µ− y12 − y2

2), (25)

which can be written in vector form as

y = f(y). (26)

It is trivial to show that(y1s, y2s) = (0, 0) is a steady state. Linearizing about this steady

state, we get the Jacobian matrix

J =∂f

∂y=

µ −1

1 µ

, (27)

which has the complex conjuagte pair of eigenvaluesµ±i. This pair crosses the imaginary

axis atµ = 0, giving rise to a Hopf bifurcation. It can be shown (see Seydel (1994)) that

the bifurcating branch of limit cycles is stable, so this is a supercritical Hopf bifurcation.

With this background, we can now move on to a discussion offlow instabilities in

polymer solutions. Theflow instability that we described at the beginning of this section

as an example is called the Taylor-Couette instability. It is driven by inertial effects,

specifically, the unstable stratification of angular momentum. Another example of an

15

inertia driven instability is the phenomenon of turbulence in pipe and plane Couetteflow

at high Reynolds numbers. In most practical applications, the high viscosity of polymer

melts and solutions results in a low Reynolds number, which means that inertial effects

are of secondary importance. However, polymericfluids display an entirely separate class

of instabilities which have their origin in the elastic nature of thesefluids. In an industrial

setting, these so called“elastic” instabilities, seen in many commercially importantflows,

are detrimental to the quality of thefinal product, and avoiding them involves imposing

limitations on throughput, or modifying theflow apparatus. It is therefore of practical

importance to understand these instabilities, and if possible, come up with methods to

delay their onset. The work presented in this document focuses on four of these: melt

fracture, thefilament stretching instability, the viscoelastic Couette-Dean instability, and

nonlinear pattern formation in Couette-Deanflow. These instabilities are seen either in

industrial processing operations or in simpleflows used to characterize the rheological

properties of viscoelastic liquids.

Thefirst part of our work concentrates on melt fracture, which is an instability seen

in extrusion processes. In an extrusion process, a polymer melt is forced out of a die

by applying a pressure gradient or a constant volumetricflow rate. At lowflow rates

(or pressure drops), the shape of the extrudate is smooth. As theflow rate is increased,

the extrudate, for linear polymers, begins to show surface distortions. These arefirst

seen in the form of small amplitude, small wavelength distortions parallel to the surface

(figure 6(a)). The effect is called sharkskin, and a detailed review of this phenomenon

may be found in Graham (1999). At somewhat higher pressure drops, if theflow is

being driven by a constant pressure gradient, theflow rate shows a sudden large jump

to a higher value. This phenomena is known as spurtflow. If the flow is driven by an

16

imposed constantflow rate, the pressure drop and exit massflow rate show oscillations.

These are accompanied by gross distortions in the shape of the extrudate (figure 6(b)).

This phenomena is known as melt fracture. The dependence of the onsetflow rate (or

pressure drop) for both sharkskin as well as melt fracture (Ramamurthy, 1986; Wang

and Drda, 1997; Piau et al., 1995) suggests that both phenomena have their origin in

the fact thatflows of polymericfluids exhibit slip at the wall, i.e., the velocity of the

polymer at the wall is non-zero relative to the velocity of the wall itself. This is different

from Newtonian liquids which obey at no-slip condition at a solid boundary. Currently,

there exist several slip expressions relating the slip velocity to the wall shear stress which

are based on microscopic theories for polymers (Leonov, 1990; Brochard and de Gennes,

1992; Adjari et al., 1994; Yarin and Graham, 1998; Mhetar and Archer, 1997; Hill, 1998).

These theories predict certain regions in which the slip velocity is multivalued. They do

not, however, consider the effect of pressure on slip velocity.

(a)

(b)

Figure 6: Extrusion related instabilities (a) sharkskin (b) melt fracture. (Agassant et al.,1991).

While sharkskin is a phenomenon related toflow at the exit of the die, melt fracture

is related to theflow profile over the entire length of the die. Aflow curve is generated by

17

plotting theflow rate (or equivalently, the shear rate) versus the pressure drop (figure 7),

and if the slip relation is multivalued (i.e., it has a region where more than one slip

velocity is possible for a given shear rate), theflow curve itself can have a multivalued

region. Spurtflow results from a jump from the lower to the upper branch of theflow

curve when the operation is at constant pressure. Melt fracture arises due to relaxation

oscillations arising from the interaction betweenfluid compressibility and slip, when an

attempt is made to operate on the decreasing part of theflow curve at constant imposed

volumetricflow rate. Since many theoretical models for slip have a multivalued region,

the expectation is that theflow curve will also be multivalued. We show in chapter 2

that this is not always the case. If the slip velocity decreases with increasing pressure

(for which there is experimental evidence in the literature), we get the surprising result

that multivaluedness in the slip relation does not imply multivaluedness in theflow curve.

Thus, there could be certain cases where melt fracture would not be seen even if the slip

relation is multivalued.

Shear Stress (MPa)

8V/D

Figure 7: A typicalflow curve for polymers exhibiting spurt and melt fracture. The ycoordinate8V/D is proportional to the exitflow rate (Kalika and Denn, 1987).

18

The second instability that we present here is related to theflow of polymeric solu-

tions in extension. This so-called“filament stretching instability” wasfirst observed by

Spiegelberg and McKinley (1996) in the extension of a liquid bridge between two parallel

plates. Thisflow has been used to measure the extensional viscosity of polymericfluids.

A visualization of the instability is shown infigure 8. The instability takes the form of

a bifurcation to a non-axisymmetric shape near the endplates, followed by a break-up of

thefilament, and ultimately by complete detachment from one of the endplates. There is

evidence to show that elasticity plays a significant role in this instability. The instability

is never seen in Newtonian liquids, and always occurs when elastic effects are large (ex-

tensional rates on the order of the inverse polymer relaxation time). The mechanism of

this instability is not well understood, and it has been hypothesized that it is related to

the classical Saffman-Taylor instability in Newtonian liquids (Saffman and Taylor, 1958),

which occurs when a less viscous liquid displaces a more viscous one. In chapter 3, we

propose a different mechanism, much more closely related to elastic effects. We model

the region near the endplates (where the instability is seen), as a membrane in the shape

of a thin truncated cone enclosing an incompressiblefluid, and show that this is unstable

to non-axisymmetric disturbances.

(a) (b )

Figure 8: Thefilament stretching instability (a) The non-axisymmetric bifurcation seenfrom below the bottom plate (b) side view of the instability at a later stage (Spiegelbergand McKinley, 1996).

19

The third viscoelastic instability for which we present results is the purely elastic

instability in flows of viscoelasticfluids with curved streamlines. This instability was

first observed in a circular Couette geometry (two concentric cylinders with liquidfilling

the annulus, and theflow driven by one of the cylinders) by Larson et al. (1990), and a

visualization of the instability is shown infigure 9. At a certain criticalflow rate, the

smooth homogeneousflow breaks up into axially and azimuthally periodic cells. The

Reynolds number at the onset of instability in their experiment was close to zero, so

clearly the mechanism is different from the classical inertial Taylor-Couette instability

described at the beginning of this section. The mechanism of this instability is a purely

elastic one, and is related to the coupling of perturbations in the hoop stress with the base

state velocity gradients. Joo and Shaqfeh (1991) showed that a similar instability arises

when theflow is driven by imposing an azimuthal pressure gradient (Deanflow). In this

case, the mechanism is related to the coupling between a radial velocity perturbation and

the base state hoop stress.

Figure 9: Flow visualization of the purely elastic Taylor-Couette instability (Larson et al.,1990).

20

Flows with curved streamlines are common in coating operations. In coating tech-

niques such as forward and reverse roll coating (seefigure 30, and the discussion in sec-

tion 4.1), they arise as a result of the curvature of the geometry. Even in techniques such

as blade coating where the geometry is not curved, curved streamlines can be present in

recirculation regions as shown infigure 10. Instabilities that occur in theseflows impose

a limitation on the operating speed of the processing apparatus and so limit throughput.

Flows in a circular Couette geometry are also used to characterize the properties of poly-

mer solutions in shear. Given the importance of theseflows, any scheme to suppress the

instability may have considerable practical utility. Graham (1998) found that the addition

of a steady axialflow of small magnitude (compared to the azimuthal forcing) either in

Couette or Poiseuille form significantly delays the onset of the elastic instability. The

mechanism is related to the development of an additional axial normal stress induced by

the secondaryflow, which suppresses radial velocity perturbations. In chapter 4 we build

on the work of Graham (1998) by showing that the elastic instability in Deanflow can

be suppressed by the same technique, thus demonstrating its utility for a broader class of

flows. Further, we demonstrate that an oscillatory axial forcing also yields stabilization.

Figure 10: Flow visualization of a blade coating geometry. Thefigure clearly showsthe presence of an upstream recirculation region with curved streamlines (Davard andDupuis, 2000).

21

In chapter 6, we present work on nonlinear pattern formation in viscoelastic circu-

lar Couetteflow. Recent experimental observations by Groisman and Steinberg (1997)

and Baumert and Muller (1999) have shown the formation of stationary, spatially iso-

lated, axisymmetric patterns in circular Couetteflow. These patterns have been termed

“diwhirls” by Groisman and Steinberg, and as“flame patterns” by Baumert and Muller.

These patterns are very long wavelength axisymmetric vortex pairs, with a core region

of strong radial inflow, surrounded by a much larger region of weaker radial outflow. In

the absence of non-isothermal effects, the primary bifurcation in circular Couetteflow is

to a non-axisymmetric, time dependent mode (i.e., a Hopf bifurcartion). Upon further

increasing the strength of theflow, there is a secondary transition to the diwhirl structures

or flame patterns. This transition is shown infigure 11 for the diwhirls and infigure 12

for theflame patterns. If, at this point, theflow strength is reduced, these patterns persist,

until eventually the baseflow is recovered at a shear rate much lower than where thefirst

transition to the non-axisymmetric mode occurred.

t

z

(a) (b)

Figure 11: Sequence of snapshots showing the transition from non-axisymmetric disor-deredflow to solitary vortex structures. On the left in (a), the entireflow geometry isshown, with the box showing the cross section being visualized. On the right, in (b), theactual transition sequence is shown (Groisman and Steinberg, 1998).

Localized structures such as those described above are important for several reasons.

22

Figure 12: Sequence of snapshots showing the transition from non-axisymmetricflowto the predominantly axisymmetric and localizedflame patterns (Baumert and Muller,1999). Theflow geometry is the same as infigure 11.

23

Firstly, they may be connected to localized defects seen in coatingflows. Secondly, they

may form the building blocks of more complex patterns seen in viscoelasticflows, such as

the recently observed phenomenon of elastic turbulence (Groisman and Steinberg, 2000).

Finally, they can serve as a test for the ability of the approximate constitutive equations

described in the previous section to model complexflows. The modeling of these struc-

tures is complicated by the fact that the transition in the space of circular Couetteflow

occurs from a non-axisymmetric, time dependent state. Rather than undertake the ap-

proach of modeling a three-dimensional time dependentflow, we adopt an approach that

is computationally simpler. We start from stationary bifurcations inflows that are geomet-

rically similar to circular Couetteflow, and continue the bifurcating stationary solutions

into the regime of circular Couetteflow. These stationary solutions are very similar to

the experimentally observed diwhirls andflame patterns. We also compute the stability

of these solutions with respect to axisymmetric and non-axisymmetric perturbations.

So far, we have discussed purely elastic instabilities, i.e., instabilities that are caused

by the viscoelastic character of polymers. Polymers can also have significant effects on

instabilities driven by inertia. Of particular interest to us is the transition to turbulence

in plane Couetteflow. It is well known that adding a small quantity of polymer shifts

this transition to higher Reynolds numbers (Giles and Pettit, 1967; White and McEligot,

1970). Since plane Couetteflow is stable to small perturbations at all Reynolds numbers,

non-trivial solutions that may be related to coherent structures seen in turbulence have

to be obtained indirectly. Waleffe (1998) obtained such non-trivial solutions by adding

a forcing term to the Navier-Stokes equations, so that the modifiedflow had a stationary

bifurcation. What is interesting is that the non-trivial solutions emanating from the sta-

tionary bifurcation in the system with forcing persist even when the forcing is removed:

24

they exist as isolated steady states in plane Couetteflow, and show similarities to coher-

ent structures seen in turbulence. Determining the behavior of these solutions when a

polymer is added can help us understand what effect polymers have on turbulence, and

in Appendix E, we present a brief description of some preliminary work done in set-

ting up branch tracing of three dimensional Newtonian plane Couetteflow with Waleffe's

forcing. Future goals would involve coupling this with a polymer constitutive equation.

The main body of this document is divided into four chapters. Chapter 2 describes the

work on melt fracture. Chapter 3 concentrates on the instability infilament stretching.

Chapter 4 describes the stabilization of the elastic instability in Deanflow. Chapter 5

describes the work on the modeling of the diwhirl andflame patterns in circular Couette

flow. In chapter 6, we present a discussion on future directions in the area of elastic insta-

bilities. In Appendix D, we briefly discuss methods to integrate viscoelastic constitutive

equations, andfinally, in Appendix E, we present an application of continuation methods

to track solutions in three dimensional plane Couetteflow.

25

Chapter 2

Pressure dependent slip and flow

curve multiplicity†

The melt fracture instability in the extrusion of polymer melts is often linked to the fact

that the polymer does not obey a no slip condition at the wall of the die. It is thought that

if the slip law, which is a relationship between the velocity of the polymer with respect to

the wall and the shear stress, has a multivalued region, theflow curve will be multivalued

as well. In this chapter, we demonstrate that this is not always the case. In particular, we

show that adding a pressure dependence which preserves the multivaluedness of the slip

law can give rise toflow curve that is not multivalued.

2.1 Introduction

Severalflow instabilities occur during the process of extrusion of melts of linear poly-

mers, and are manifested in the form of distortions in the extrudate. A discussion of

† Most of the material in this chapter has been published in Kumar and Graham (1998a)

26

these may be found in several reviews (Denn, 1990; Larson, 1992; Leonov and Prokunin,

1994). In this work, we consider the phenomena of hysteresis and spurtflow. We can

get a clearer picture of these phenomena by examining theflow curve, which is a plot of

apparent wall shear rate (γA) versus wall shear stress (τw) during capillary extrusion. For

steadyflows in a circular capillary,

γA =4Q

πR3+4usR

, (28)

and

τw =∆pR

2L, (29)

with R andL being the radius and length of the capillary respectively,Q theflow rate,∆p

the pressure drop, andus the slip velocity of thefluid at the wall. Thus,τw is a measure

of the pressure drop, andγA is a measure of theflow rate. The unusual behavior seen

in these curves is usually attributed to the effect of wall slip (i.e., a nonzero value ofus)

because of the dependence of the phenomena on the materials of construction of the die

(Ramamurthy, 1986; Wang and Drda, 1997; Piau et al., 1995). A plot oflog(γA) versus

log(τw) starts as a straight line, indicating a power law regime. At a certain critical stress

τc1, there is a distinct increase in slope. This is followed by a sharp jump in the value

of γA at a second critical shear stressτc2. A rheometer operated in the constant pressure

mode would show a sudden jump to a higherflow rate at this value ofτw. If the pressure

is increased further,γA continues along the highflow rate branch. At this point, if the

pressure is decreased, theflow curve decreases along the highflow rate branch even below

τc2, until there is a jump to the lowflow rate branch at a third critical stressτc3, less than

τc2. Thus, theflow curve exhibitshysteresis (Bagley et al., 1958; Tordella, 1956, 1963;

El Kissi and Piau, 1990). This hysteresis implies that theflow curve displays a region of

27

multiplicity: two stableflow rates are possible at the same pressure drop. If a rheometer is

operated in the constant piston speed mode in the region betweenτc2 andτc3, the pressure

and extrudateflow rate show oscillations (Lupton and Regester, 1965; Kalika and Denn,

1987; Ramamurthy, 1986; Hatzikiriakos and Dealy, 1992a). This phenomenon is known

as spurtflow.

The dependence of the critical shear stressτc2 on the length to diameter ratio (L/D)

of the die can be inferred from several experimental works in the literature. The data

of Vinogradov et al. (1984) for polybutadienes and Wang and Drda (1996) for entangled

linear polyethylenes, which are taken at lowL/D (≤ 25), suggest thatτc2 is virtually

independent ofL/D. The data of El Kissi and Piau (1990) taken at somewhat higher

L/D ratios (L/D = 20 and 40) for polydimethylsiloxanes, corrected for entrance losses

show that there is a small decrease in this value (from 0.61 bar to 0.59 bar) for the constant

pressure experiments where theL/D ratio was varied byfixing D and varyingL. Kalika

and Denn (1987) report data for the constant piston speed experiments using LLDPE

which show thatτc2 decreases asL/D is varied from 33.2 to 66.2, and atL/D = 100.1,

their reported value ofτc3 is greater thanτc2 (in this case, theflow curve shows a large

jump in flow rate in the relatively small region betweenτc2 andτc3) which indicates that

there may not be multiplicity if the rheometer is operated in the constant pressure mode at

highL/D. Finally, Hatzikiriakos and Dealy (1992a) report that they observe an increase

in τc2 with increasingL/D (from L/D = 10 to L/D = 100) for linear high density

polyethylene, and take this as evidence of a pressure dependence of wall slip. Thus, there

seems to be evidence in the literature to support theL/D dependence, but there does not

seem to be an agreement on the trend, and this may differ depending on the polymer used

and other experimental conditions.

28

In addition to experimental evidence for multiplicity in slip behavior, the motivation

for using a multivalued slip model arises from the slip relations obtained by several re-

searchers using microscopic theories for polymers. Several of these theories explicitly

predict multivaluedness in the relation between slip velocity and shear stress, i.e., there

is a region in which three distinct values of slip velocity are possible for the same value

of shear stress. Specific examples of such relations include those proposed by Leonov

(1990); Yarin and Graham (1998); Mhetar and Archer (1997); Hill (1998). In particular,

in the model of Yarin and Graham (1998), the limit point at which the jump from a small

slip to a large slip regime occurs arises from an imbalance between the increasing force

per adsorbed chain and the decrease in the concentration of adsorbed chains, as shear

stress increases. Multivalued slip relations such as these have been used, for example, to

model spurt and oscillations in capillary and Couetteflows of molten polymers (Adewale

and Leonov, 1997).

Georgiou and Crochet (1994) have proposed a computationally convenient phenomeno-

logical slip equation which also shows a maximum and minimum in wall shear stress. Al-

though not derived fromfirst principles, this model has a closed form expression, and can

be viewed as an approximate version of models based on molecular parameters which do

not have closed form expressions. They showed that this slip relation, when taken along

with finite compressibility of the polymer melt, can lead to self sustained oscillations of

the pressure drop and massflow rate at the exit of the die for Newtonianflow in a slit.

However, they do not include a pressure dependence in their slip relation.

Several slip models relating the slip velocity to the wall shear stress and pressure

(or total compressive normal stress on the wall) have been proposed in the literature. In

particular, Hill et al. (1990); Denn (1992); Person and Denn (1997) propose slip relations

29

which show a power law dependence on the wall shear stress and an exponential one

on pressure. Hatzikiriakos and Dealy (1992b) have proposed a similar slip model which

shows saturation at high pressures. They used this model for the lowflow rate branch and

a simple power law dependence on the shear stress for the upper branch of theflow curve

to modelflow oscillations in a capillary rheometer (Hatzikiriakos and Dealy, 1992a).

Thus, their model is in effect a discontinuous multivalued slip relation. The slip model

of Stewart (1993) brings in the pressure dependence through changes in the density. All

these models predict a decrease in slip velocity with increase in pressure. There is also

experimental evidence from White et al. (1991) that this is indeed the case.

In this work, we study the combined effects of adding pressure dependent and mul-

tivalued slip onflow curve multiplicity in capillary extrusion. We do this by modifying

the approximate slip relation proposed by Georgiou and Crochet and adding a pressure

dependence. Consistent with observations, the pressure dependence is such that slip ve-

locity decreases with increasing pressure. As with the model proposed by Georgiou and

Crochet, our relation is not derived fromfirst principles, but contains the same qualita-

tive features found in more complex models derived from molecular considerations. We

apply this equation to the steadyflow of incompressible Newtonian and shear thinning

fluids through a cylindrical die at a constant pressure drop. As also done by Person and

Denn (1997), we simulate the entire axial profile of pressure, stress and slip velocity. The

flow curve obtained shows multiplicity, with the critical shear stressτc2 showing a small

decrease at lowL/D ratios and a more pronounced one at higherL/D ratios. Most im-

portantly, we see that at sufficiently highL/D, theflow curve is no longer multivalued,

despite the fact that the slip relation is multivalued, i.e., it predicts a maximum and mini-

mum in shear stress at all pressures between the entry and exit pressures in the die. Thus

30

we obtain the result that multiplicity in the slip relation does not guarantee multiplicity

in theflow curve.

2.2 Mathematical Model

2.2.1 Slip Model

In terms of dimensionless quantities, our slip model may be written as

τw = usp

(usp +

A21 +A3u2sp

), (30)

where

usp = us(1 + exp(β P )). (31)

Here,τw is the shear stress exerted by thefluid on the wall andus is the slip velocity. The

scaling factors used areu∗ = G/a1, P ∗ = G andτ∗w = G for the velocity, pressure and

shear stress respectively,a1, A2, A3 andβ are parameters of the slip model andG is the

shear modulus of thefluid. For smallusp, this model reduces to the Navier slip condition,

τw = A2usp.

If we arbitrarily chooseA3 = 122.24, we can calculate that the slip model loses

multiplicity for A2 < 2.29. Figure 13 shows a plot of the slip relation for various values

of P for A2 = 3, A3 = 122.24, andβ = 0.0102. Unless otherwise mentioned, we

will work with A2 = 3 andA3 = 122.24 in subsequent calculations. Note that the

pressure dependence is such that the slip velocity decreases with increase in pressure, as

shown infigure 13, but the multiplicity remains. Since the slip model is multivalued by

construction, the natural expectation is that theflow curve will be as well. We shall see

that this expectation is not always fulfiled.

31

P=10

P=50

τw

us

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.00 0.04 0.08 0.12 0.16 0.20

Figure 13: Plot of equation 30 for different values of pressure (A2 = 4, A3 = 122.24,β = 0.0102).

2.2.2 Constant Pressure Formulation

We now develop the governing equations for the constant pressure case using Newto-

nian, shear thinning and UCM constitutive equations. We have already described the

UCM equation in the introduction as a constitutive equation for obtaining the polymer

component of the stress tensor in dilute solutions. It can also be derived using network

models for polymer melts (Bird et al., 1987b). In this case,Q can be thought of as the

length of a polymer segment between two junctions in the network, with the segment

being modeled as a Hookean spring. As shown by Lupton and Regester (1965),fluid

compressibility does not play an important part in the constant pressure case, and we can

simplify our analysis slightly if we assume that thefluid is incompressible. However, for

the constant piston speed case,fluid compressibility plays a crucial role in providing a

mode for storing energy, which is essential for the generation of relaxation oscillations.

32

We first consider the pressure drivenflow of an incompressible Newtonianfluid

through a cylindrical die of radiusR (diameterD) and lengthL. The pressure is de-

fined so that it is zero at the exit of the die. Using the lubrication approximation (Pearson

and Petrie, 1968), the dimensionless volumetricflow rate is given by

Q = us +τwH

, (32)

Here, we assume that the diameter of the die is kept constant and choose to scale the

volumetricflow rate usingQ∗ = πR2u∗. In addition, we scale the lengths byL and

retain the scaling factors for pressure, slip velocity and shear stress discussed above. The

dimensionless numberH = 8ηu∗/GR, whereη is thefluid viscosity. Note that since

γA = 4Q/πR3, QH measures the dimensionless apparent shear rate. The use of the

incompressibility conditiondQ/dz = 0 gives us the following equations for the slip

velocity and pressure:

dusdz

= −dP

dz

∂τw∂P

H + ∂τw∂us

, (33)

dP

dz= −4Λτw, (34)

whereΛ = L/D. Equation 33 is obtained by using equation 32 and differentiating

equation 30 with respect toz. This set of equations is to be solved using the boundary

conditions

P (0) = ∆P, (35)

P (1) = 0. (36)

Since it will be convenient to calculate theflowrate simultaneously with the other

33

quantities, we add the redundant equation and boundary condition

dQ

dz= 0 (37)

Q(0) = us(0) +τw(0)

H. (38)

For a power lawfluid, the constitutive relation between the shear stress exerted by the

fluid on the wall and the radial velocity gradient is given by Bird et al. (1987a) as

τrz = −K

∣∣∣∣∂vz∂r

∣∣∣∣n−1

∂vz∂r

. (39)

Using the lubrication approximation once again, the dimensionless volumetricflow rate

is given by

Q = us +nτ1/nw

H, (40)

whereτw is obtained from equation 30. We define the dimensionless numberH in a

manner similar to the Newtonian case as

H =n(3 + 1/n)K1/n

a1G1/n−1R, (41)

which results in the following set of equations:

dusdz

= −∂τw∂P

∂τw∂us+ H

τ(1/n−1)w

dP

dz, (42)

dP

dz= −4Λτw, (43)

dQ

dz= 0. (44)

These equations have to be solved together with the boundary conditions:

P (0) = ∆P, (45)

P (1) = 0, (46)

Q(0) = us(0) +nτ1/nw (0)

H. (47)

34

The governing equations for the UCM constitutive equation are similar to those for the

Newtonian model. The inclusion of a normal stress only changes the boundary conditions

at the entrance and exit. Neglecting the elastic normal stresses in the barrel, and entrance

and exit effects, these are given by:

P (0) + τzw(0) = Pb, (48)

P (1) + τzw(1) = 0. (49)

Here,τzw is the normal stressτzz evaluated at the wall. At steady state,τzw = −2 τ2w.

We can examine the combined effect of viscoelasticity and shear thinning by using the

Phan-Thien-Tanner (PTT) equation, which can also be derived from network theory (Bird

et al., 1987b). The general form of the PTT equation is given by Bird et al. (1987a) as

Z(tr τ )τ + λ τ (1) +ξ

2λγ · τ + τ · γ = −η γ. (50)

We consider the case of affine motion (ξ = 0) and use the linear form of the functionZ,

i.e,

Z = 1− ε λ tr τ/η.

In this case, the governing equations can be written as

dus

dz= −

dP

dz

∂τw∂P(1 + 4ε/H τ2w)

H + ∂τw∂us(1 + 4ε/H τ2w)

, (51)

dP

dz= 0, (52)

dQ

dz= 0. (53)

35

The boundary conditions are given by

Q(0)− us(0) =τw(0)

H+4ε

3Hτ3w(0), (54)

P (0) + τzw(0) = Pb, (55)

P (1) + τzw(1) = 0. (56)

Our simulations show that the addition of viscoelasticity in the form of either the UCM

or the PTT equation does not change the qualitative behavior of the model.

2.2.3 Constant Piston Speed Formulation

We now consider theflow of a Newtonianfluid through a capillary die as shown infig-

ure 14. Thefluid is driven by a piston moving at constant speed. We assume that the melt

has a constant compressibilityχ, and use a linear relation for the density, i.e.,

ρ = ρ0(1 + χP ), (57)

whereρ0 is the density atP = 0.

Wefirst write the governing equations for the Newtonian case. A mass balance on the

barrel gives

CdPb

dt=1

κ(Qp −Q0) , (58)

wherePb is the dimensionless pressure in the barrel (scaled byG), t is the dimensionless

time (scaled by the residence timet∗ = Qp/πR2L), C = Vb/πR

2u∗t∗ andκ = Gχ

(cf. Lupton and Regester (1965) and Molenaar and Koopmans (1994)). The volumetric

rate of displacement of the piston,Qp is constant during the simulation andQ0 is the

volumetricflow rate at the exit of the barrel. We assume thatC remains constant in the

36

L

diameter = D

die

barrel

plunger

Vb

dVb / dt =Qp

Figure 14: Schematic diagram of the constant piston speed experiment.

time scale of the experiment. A mass balance on the capillary die gives

∂P

∂t= −λ1

(Q

∂P

∂z+1

κ

∂Q

∂z

), (59)

with λ1 = t∗u∗/L. The boundary conditions areQ = Q0, andP = Pb at the entrance of

the die. Finally, integrating the momentum balance overr gives

Re∂Q

∂t= −λ2

(1

4Λ

∂P

∂z+ τw

), (60)

whereλ2 = t∗G/η, Re = ρ0u∗D/4η is the Reynolds number andη is thefluid viscosity.

The momentum equation in this form is also valid for viscoelasticflow if ∂τzz∂z ∂P

∂z.

Numerical results for viscoelasticflow show that this condition is satisfied.

We now need an equation relating the volumetricflow rate to the slip velocity. If the

compressibility is small, we can assume that equation 32 still holds. Hence, the governing

equations for the Newtonian case are equations 58 to 60 together with equation 32 and

the slip relation, equation 30. The same set of equations hold for the shear thinning case,

with the exception that equation 32 is replaced by equation 40.

37

As mentioned before, viscoelasticity does not significantly affect the steady state

equations. However, it is possible that using an evolution equation for the normal and

shear stresses could yield time-dependent results that are qualitatively different from

those for the Newtonian and shear thinning cases. Thus, we felt that it was necessary

to examine the time-dependent case using a simple viscoelastic model. This section dis-

cusses the formulation of the governing equations using the UCM model for viscoelastic-

ity (Bird et al. (1987a)). Assuming that the Reynolds number is small, the UCM model

gives the following evolution equation forτw:

De

2ΛDe∗∂τw∂t=

τwDe∗

− 8(Q− us). (61)

Evaluating the evolution equation forτzz at the wall gives

De

2Λ

∂τzw∂t

+De τwΛ

∂τw∂t= −2τ2w − τzw. (62)

Here,τzw represents the value ofτzz at the wall of the capillary andDe∗ = λu∗/R. The

Deborah number is given byDe = λ〈v〉/R, where〈v〉 = Qp/πR2.

The slip velocity and shear and normal stresses are related by the slip model. The

boundary conditions are:

P (0) + τzw(0) = Pb, (63)

P (1) + τzw(1) = 0. (64)

In the first boundary condition, we assume that the elastic normal stresses in the barrel

are negligible. These boundary conditions neglect entrance and exit effects.

38

2.3 Results and Discussion

2.3.1 Constant Pressure Case

We wish to determine the parameter regimes in whichflow curve multiplicity occurs. A

natural way to do this is through bifurcation analysis. AtfixedΛ, we find the turning

points of theflow curve (i.e. the points where the curve turns back on itself). These

determine the boundariesτc2 andτc3 of the multiplicity region at thatΛ (in the model

we consider, slip occurs at all shear stress levels and henceτc1 does not exist). Then

we track the motion of these points asΛ varies, thus outlining the region in(∆P,Λ)

space whereflow curve multiplicity occurs. The AUTO software package (Doedel, 1981;

Taylor and Kevrekidis, 1990) automatically performs this type of analysis for boundary

value problems like the one we consider here. AUTO uses a domain decomposition

collocation method for spatial discretization and a pseudo-arclength continuation scheme

to trace out steady state solution curves in one parameter or curves of bifurcation points

in two parameters.

Figure 15 shows theflow curve computed by AUTO atΛ = 10,A2 = 3, A3 = 122.24,

β = 0.0102 andH = 1. We see that this curve is multivalued, as expected because the

slip model is multivalued. The turning points occur at∆P = 5.8090 and∆P = 5.6506,

corresponding to critical shear stresses ofτc2 = 0.1452 andτc3 = 0.1413.

The results of the turning point continuation are shown infigure 16a, where the upper

curve corresponds toτc2, and the lower one toτc3. Note that our slip law depends on

pressure, so the slip velocity, and henceτw changes through the length of the die. The

values ofτc2 andτc3 that we report here are computed by dividing the pressure drop∆P

by 4Λ. This is what the (constant) wall shear stress would be in the absence of slip,

39

X

X

Q

∆ P

0.16

0.20

0.24

0.28

0.32

5.0 5.5 6.0 6.5

Figure 15: Flow curve for a Newtonian Fluid (Λ = 10, A2 = 3, A3 = 122.24, β =0.0102, H = 1). The profiles at the points marked `X' are shown infigures 18 and 19.

40

under the same pressure drop. As thefigure shows, the value ofτc2 shows a decrease and

the two critical shear stresses approach one another as the length is increased, and this

effect is more pronounced at higherΛ values. Finally, atΛ = 230.39, we see that the the

two turning points come together in a cusp, indicating that theflow curve beyond thisΛ

value has no multiplicity. The pressure drop at this value is132.19 and the slip relation

still predicts a multiple valued curve for all pressures in this range. Thus, the differential

effect of pressure in the capillary has resulted in the absence of multivaluedness in the

flow curve, although the slip relation itself is multivalued for the entire pressure range

experienced in the die. We also note that at the cusp, the critical stresses have a value of

0.1434, which corresponds to a change of only1.24% from the value ofτc2 atΛ = 10.

Finally, for completeness, we show a plot of theflow curve forΛ = 238.62 in figure 17,

where there is no multiplicity, just as shown onfigure 16a.

Results for the shear thinning case withn = 0.56 and the other parameter values as

for the Newtonian model are similar. The results of the turning point continuation are

shown infigure 16b. The behavior is similar to the Newtonian case where there is a more

pronounced drop inτc2 at higherΛ values. Also, theflow curve loses multiplicity at

Λ = 230.46 and a pressure drop of132.19, which corresponds toτc2 = τc3 = 0.1434.

As with the Newtonian case discussed in the preceding paragraph, the slip relation itself

remains multivalued for all pressures experienced in the die, and the loss of multiplicity

in theflow curve is a result of the differential effect of pressure in the capillary.

As mentioned earlier, wefind that including the effect of viscoelasticity does not

affect the qualitative behavior of the model, in particular, the loss of multiplicity at high

Λ. The result of a turning point continuation using the UCM constitutive equation is

41

τ c

τ c

Three steady states

Three steady states

Three steady states

Three steady states

One steady state

One steady state

One steady state

One steady state

One steady state

One steady state

One steady state

One steady state

Λ

τ c(d)

(c)

(b)

τ c(a)

0.141

0.143

0.145

0.141

0.143

0.145

0.141

0.143

0.145

0.141

0.143

0.145

50 100 150 200

Figure 16: Behavior of critical stressesτc2 andτc3 with increasingΛ (A2 = 3, A3 =122.24): (a) Newtonian model (β = 0.0102, H = 1) (b) shear thinning model (β =0.0102, H = 1, n = 0.56) (c) UCM model (β = 0.0102, H = 1) (d) PTT model(ε = 10−1, β = 0.0102, H = 1). The upper curve corresponds toτc = τc2 and the lowerone toτc = τc3. The curves are not extended toΛ = 0 because the approximations usedare not valid for smallΛ.

42

Q

∆ P

0.16

0.18

0.20

0.22

0.24

125 130 135 140 145 150

Figure 17: Flow curve for a Newtonianfluid showing no multiplicity (Λ = 238.62,A2 = 3, A3 = 122.24, β = 0.0102, H = 1).

shown infigure 16(c). A similar result for the PTT model, withε = 0.1 is shown in

figure 16(d). In both cases, the same parameters were used as for the Newtonian model.

Figures 18 and 19 show the spatial profiles of the pressure, shear stress and slip veloc-

ity at points on the low and highflow rate branches of theflow curve for the Newtonian

fluid shown infigure 15. The profiles at a point on the lowflow rate branch are shown in

figure 18 and those at a point on the highflow rate branch are shown infigure 19. In both

cases, the pressure profiles are nearly linear and the shear stress decreases as we move

towards the exit of the die. The degree of variation in the magnitude of the shear stress is

much smaller than that of the pressure and correspondingly, the slip velocity increases to-

wards the exit where the pressure is lowest. This behavior is expected intuitively and also

seen by Hatzikiriakos and Dealy (1992b) in their simulations at constant piston speed.

43

However, it differs from the results of Person and Denn (1997) where slip velocity has

the smallest magnitude at the exit, which is the region of lowest shear stress. They inter-

pret this result as arising from the strong power law dependence of slip velocity on shear

stress. Finally, the small variation inτw with z provides ana posteriori validation of our

neglect of axial gradients.

τ w

P

us

z

0

2

4

6

0.1302

0.1304

0.1306

0.0 0.2 0.4 0.6 0.8 1.0

0.0304

0.0306

0.0308

Figure 18: Profiles of pressure, slip velocity and shear stress on the lowflow rate branchfor the Newtonianfluid of figure 15. (∆P =5.22,Λ = 10, A2 = 3, A3 = 122.24,β = 0.0102, H = 1).

44

z

τ w

us

P

0

2

4

6

0.139

0.140

0.141

0.142

0.158

0.159

0.160

0.161

0.0 0.2 0.4 0.6 0.8 1.0

Figure 19: Profiles of pressure, slip velocity and shear stress on the highflow rate branchfor the Newtonianfluid of figure 15 (∆P =6.39,Λ = 10, A2 = 3, A3 = 122.24,β = 0.0102, H = 1).

45

It is of interest to observe the location of the profiles of the two turning pointsτc2

andτc3 on the three-dimensional surface defined by the slip model (equation 30). The

pressure profile is almost linear for all cases, and hence, the plots may also be viewed as

the profiles superimposed on the slip curve in(z, us, τw) space, with the higher pressures

corresponding to lowerz values, i.e., points close to the entrance of the die. Such a plot

is shown for the shear thinning case withΛ = 10 in figure 20(a). At this value ofΛ, the

pressure drop is relatively small, and the profiles at the two turning points lie very close

to the location of the maxima and minima of equation 30 at all pressures in the die.

The location of the profile at the cusp is shown infigure 20(b). Here, the high pressure

points (which lie near the entrance of the die) are located to the right of the minimum at

the corresponding pressure. As we move down the die to regions of lower pressures, the

points tend to move toward the right, i.e., closer to the maxima, till at the exit, the points

are located to the left of the maximum. Finally,figure 20(c) shows the profile of a point

in the central region (i.e., the region where there is a sharp increase in the slope of the

flow curve, although there is no multiplicity).

2.3.2 Constant Piston Speed case

For the constant piston speed case, we performed a spatial discretization of the governing

equations using a Chebyshev collocation scheme (Canuto et al., 1988) withN = 30 collo-

cation points for the pressure, volumetricflow rate and slip velocity. The resulting system

of differential-algebraic equations was solved using the integrator DDASAC (Caracotsios

and Stewart, 1985), for imposed values ofQp and zero initial conditions. For the base

case, we chose values ofχ = 10−9m2/N andG = 105N/m2 andRe = 10−10. In all

46

02

4P 0.050.1

0.15

us

0.05

0.15Τw

02

4

050

100P0.05

0.1

us

0.10.20.3

Τw

050

100

050

100P0.05

0.1

us

0.10.20.3

Τw

050

100

(a)

(b)

(c)

Figure 20: Flow profiles on the three dimensional slip surface for a shear thinningfluid:(a) upper and lower limit points (∆P = 6.39, Λ = 10) (b) cusp point (∆P = 132.24,Λ = 230.46) (c) point in the central portion without multiplicity (∆P = 140.0, Λ =240.82). Other parameters areA2 = 3, A3 = 122.24, β = 0.0102, H = 1, andn = 0.56for all three cases.

47

runs,C wasfixed at106 and the parametersλ1 andλ2 werefixed at0.01 and0.1 re-

spectively. All other parameters were chosen to have the same values as in the constant

pressure case.

Results for the Newtonian and shear thinning cases are similar and we only show them

for the Newtonian case. As expected because of the multiplicity in the constant pressure

flow curve, attempting to operate at aflow rate corresponding to the branch betweenτc2

andτc3 of the flow curve (which we call the unstable region) results in oscillations in

the pressure dropPb and the exitflow rateQe. This is shown infigure 21, for a value

of Qp = 0.21 where the steady state solution lies in the unstable region of theflow

curve. Operating on the low or highflow rate branches results in stableflow as shown in

figure 22, forQp = 0.15, where the steady state solution lies on the lowflow rate branch

of theflow curve. These results are similar to those obtained by Georgiou and Crochet

(1994) and Adewale and Leonov (1997), and are to be expected when multivalued slip

models are used.

As long asRe 1, the Reynolds number has virtually no effect on the frequency

of the oscillations. This can be understood by noting that, forRe 1, these are basi-

cally classical relaxation oscillations, as occur in problems with widely separated time

scales (Nayfeh and Mook, 1979). Here, the time scales correspond to the compressibility

(κC) and inertia (Re). Hence,Re must be nonzero for oscillations to be observed, but

is otherwise not important as long as it is small. Decreasing the compressibility has the

effect of increasing the frequency of the oscillations. This can be understood on the same

lines as theRe dependence by noting that the time spent on the high and lowflow rate

branches of theflow curve is directly related to the compressibility. Fluids with lower

48

(b)

(a)

Qe

Pb

Qe

Pb

time

5.6

5.7

5.8

5.9

0.16

0.18

0.20

0.22

0.24

0.26

0.28

0.30

6000 6500 7000 7500 8000 8500 9000 9500

0.16

0.18

0.20

0.22

0.24

0.26

0.28

0.30

5.6 5.7 5.8 5.9

Figure 21: Oscillatoryflow for a Newtonianfluid with Qp = 0.21:(a) Pressure drop andexit flow rate vs. time (b) Plot ofQe vs.Pb superimposed on the steady stateflow curve.

49

time

Qe

Pb

0

1

2

3

4

5

0.00

0.05

0.10

0.15

0 5000 10000

Figure 22: Non-oscillatoryflow for a Newtonianfluid with Qp = 0.15: Pressure dropand exitflow rate vs. time.

50

values ofχ would spend less time on these branches and hence exhibit oscillations of a

higher frequency.

Figure 23 shows the results for a simulation atΛ = 250, where the steady state

flow curve shows no multiplicity. Recall, however, that the slip model itself remains

multivalued, and as in the constant pressure case, the loss of multiplicity is a result of

the differential effect of pressure along the capillary. The value ofQp was chosen to be

0.1862, which lies in the steepest region of theflow curve. As expected, no oscillations

are found.

We conclude with a brief discussion of the results obtained using the UCM consti-

tutive equation, with the same slip model parameters as for the Newtonian case. In this

case, high frequency shock waves are seen at very small values of the Reynolds number

and large values ofκ. To avoid these shock waves, we present results for a parameter set

different from that used for simulations with the Newtonian model. Figure 24 shows the

result of a simulation withRe = 10−4. As before, wefind that Reynolds number does not

affect the frequency of the oscillations provided that it is sufficiently small. We alsofind

that, as for the Newtonian case, decreasing the compressibility increases the frequency of

the oscillations.

2.4 Conclusions

In this chapter, we have studied the behavior of extrusion when a multivalued, pressure

dependent model for wall slip is used. The main features of spurtflow and multiplicity

can be modeled using this relation. The steady stateflow curves using Newtonian, shear

thinning and viscoelastic models predict a critical shear stressτc2 for the onset of spurt

51

time

Qe

Pb

0

50

100

150

0.00

0.05

0.10

0.15

0.20

0 50000 100000 150000 200000 250000 300000

Figure 23: Non-oscillatoryflow of a Newtonianfluid with Λ = 250, Qp = 0.1862 andother parameters as infigure 21: Pressure drop and exitflow rate vs. time.

52

Qe

Pb

Qe

Pb

time

5.6

5.7

5.8

5.9

0.04

0.06

0.08

0.10

0.12

500 600 700 800 900 1000 1100

0.04

0.06

0.08

0.10

0.12

5.6 5.7 5.8 5.9

Figure 24: Oscillatoryflow of a UCMfluid (C = 106, Λ = 10, Re = 10−4, κ = 10−5,De = De∗ = 10, λ1 = λ2 = 0.01, Qp = 0.081): (a) Barrel pressure and exitflow ratevs. time (b) Plot ofQe vs.Pb superimposed on the steady stateflow curve.

53

flow that decreases with increase inΛ. In all cases, we see a loss in multiplicity at highΛ,

although the slip relation itself gives a multivalued slip velocity for a given shear stress

at all pressures in the range involved.

Time-dependent simulations takingfluid compressibility into account show that at-

tempting to operate on the decreasing branch of theflow curve gives rise to oscillations

in the exitflow rate and pressure drop, which is in accord with the results of previous

researchers using multivalued slip models. The frequency of these oscillations is virtu-

ally independent of the Reynolds number, provided that it is small enough. Decreasing

compressibility increases the frequency of the oscillations. No oscillations are observed

if the flow curve does not show multiplicity.

Although there is little experimental data in the literature for experiments conducted

at highΛ values at constant pressure, the behavior we observe is similar to that reported

in Kalika and Denn (1987) for a constant piston speed experiment. The main conclusion

of this work is that multiplicity in the slip model does not guarantee multiplicity in the

flow curve.

54

Chapter 3

The filament stretching instability †

In this chapter, we discuss the instability that occurs during the elongation of a polymer

filament. Here, an initially axisymmetricfilament undergoes a buckling instability pro-

ducing a non-axisymmetric shape. We use the idea of stress localization to model the

process as the stretching of an elastic membrane enclosing a passive, but incompressible

fluid. We show that this simple model exhibits an instability that is similar to that seen in

the stretching of a polymerfilament.

3.1 Introduction

Flows with significant elongational components are common in industrial applications

such asfiber spinning, as well as in rheometry, during the measurement of elongational

viscosity. The most common types of elongationalflow instabilities are draw resonance,

necking, capillary breakup, and elasticfilament breakup, Detailed reviews of which may

be found in Petrie and Denn (1976) and more recently by Larson (1992), so we content

† Most of the material in this chapter has been published in Kumar and Graham (2000)

55

ourselves with a brief overview of these instabilities. We then summarize the observa-

tions in the literature that are related to elasticfilament breakup, and postulate a new

mechanism by which this instability could occur.

Fiber spinning is defined (Petrie and Denn, 1976) as a process whereby afilament or

sheet is extruded from a die and drawn down in cross sectional area by being taken up at

a velocity greater than the extrusion velocity. The draw ratioDr, is defined as the ratio

of the take-up velocity to the extrusion velocity. This also equals the ratio of the initial

to drawnfiber area. Draw resonance is described by Christensen (1962) as a periodic

variation infiber diameter, which occurs at constant take-up speed, when small variations

in the diameter offiber at the take-up spool produce oscillations in thefiber tension. The

analysis of Pearson and Matovich (1969) for a Newtonianfluid with negligible inertia,

gravity, and surface tension showed that the steady state solution is unstable to time-

periodic disturbances whenDr exceeds20.21. A later analysis by Pearson and Shah

(1972) showed that inertia acts to stabilize theflow, while surface tension is destabilizing.

The corresponding stability analysis for viscoelasticfluids has been performed by Ze-

ichner (1973), Fisher and Denn (1975, 1976), and Chang and Denn (1980). An important

parameter in this case is the Deborah number, defined asDe = λ v(0)/L, whereλ is

the relaxation time of thefluid, v(0) is the extrusion velocity, andL is the length of the

fiber. The analysis of Fisher and Denn (1976) for a UCMfluid shows that elasticity has a

stabilizing influence. The analysis also shows that there is no steady solution for certain

values ofDe, an effect related to the unbounded growth of normal stress predicted by the

UCM equation for elongationalflow, at certain Deborah numbers. Nonlinear analyses

carried out for Newtonianfluids (Kase, 1974; Fisher and Denn, 1975) and for power law

56

and UCMfluids (Fisher and Denn, 1976) show thatfinite amplitude effects are stabiliz-

ing. Finally, fluids that thin in elongation are found to be less stable to draw resonance

than Newtonianfluids, while elongation thickeningfluids are more stable (Pearson and

Shah, 1974; Fisher and Denn, 1976).

When thefiber is extruded at a constant tension rather than a constant take-up speed,

Fisher and Denn (1976) found that theflow is stable to draw resonance. However, small

local indentations can grow as they are convected down the length of thefiber. This

phenomenon is known as necking. Necking occurs because the constant force applied

along thefiber length translates to a greater elongational stress in the neck as compared to

adjoining regions in thefiber. Thus, the neck thins faster than the neighboring regions. It

follows that elongation thinning, which speeds up the reduction in area, is destabilizing,

while elongation thickening leads to increased stability. They also showed that if the

residence time of the neck in thefiber is long enough, necking can lead tofiber breakage.

The phenomenon of capillary breakup wasfirst analyzed by Rayleigh (1879). He

showed that a jet of Newtonian liquid with nonzero surface tension is unstable to ax-

isymmetric disturbances whose wavelength is greater than the circumference of the jet.

For fiber spinning, the capillary breakup instability augments necking, but the additional

effect can be shown to be negligible (Larson, 1992). For UCMfluids, the analysis of

Goldin et al. (1969) showed that in the linear limit, disturbances have higher growth rates

than for Newtonianfluids. However, nonlinear effects quickly become significant, and

the lubrication andfinite element analysis of Bousfield et al. (1986) which takes these

into account shows that the growth rates slow down significantly, so that the net effect is

a stabilization compared to Newtonianfluids. Once again, it is the elongation thickening

property of the UCMfluid that leads to increased stability, by resisting the thinning and

57

breakup of threads. While the capillary breakup instability is not of significance infiber

spinning, it has been suggested (Larson, 1992) that the measurement of growth rates of

the disturbances may be used to estimate the surface tension of viscoelasticfluids.

The next instability that we discuss is the breakup of a viscoelastic liquid bridge

in elongation. Unlike the instabilities we have discussed so far, this instability is purely

elastic, in the sense that it is not seen in corresponding experiments with Newtonianfluids

with similar viscosity. The elongation of a liquid bridge confined between two endplates

is used in rheometry to measure the elongational viscosity of afluid (Tirtaatmadja and

Sridhar, 1993; McKinley et al., 1996). A schematic of the setup is shown infigure 25.

In ideal uniaxial elongationalflow, a liquid column of initial lengthL0 is extended at

an exponential rate so that its length at timet is given byLt = L0 exp(ε0 t), whereε0

is the strain rate. The radius of thefilament remains uniform throughout its length. The

elongational viscosity is then defined as

η ≡ (τzz − τrr) /ε0, (65)

whereτzz andτrr are respectively the axial and radial normal stresses. This stress differ-

ence, and henceη, may be calculated from the force required to produce the elongation.

A measure of the displacement is given by the Hencky strainε, given byloge(Lt/L0).

The Deborah number is given byDe = λ ε0, whereλ is the relaxation time of thefluid.

A filament stretching rheometer, such as the one shown infigure 25 attempts to repro-

duce uniaxial elongation by separating the endplates at an exponential rate. The actual

kinematics differ from this due to the presence of the two endplates. While conducting

experiments with polyisobutylene based Bogerfluids in afilament stretching rheometer,

McKinley et al. (1996) found that the liquid bridge partially decohered from one of the

endplates at high strain rates (corresponding toDe > 1). The instability began as a

58

polymer solutionL0

Lt

Figure 25: Schematic of afilament stretching rheometer. The setup on the left shows theundeformed state of the liquid bridge.

non-axisymmetric deformation near the endplates at a certain critical strain, with a well

defined azimuthal wavenumber of 4. Upon further elongation, the column broke into

fibrils, which themselves bifurcated into smallerfibrils at larger elongations. The critical

strain for onset of the instability was found to decrease with increasing Deborah num-

ber. The dependence on the initial aspect ratio (defined asL0/R0, whereR0 is the initial

filament radius) was weaker, with the critical strain increasing slightly at larger aspect

ratios. They report that Newtonian liquid columns of similar viscosity do not display the

instability, and in fact break up due to the capillary instability at comparable strains.

McKinley et al. (1996) propose that the mechanism is similar to the one seen in the

Saffman-Taylor instability (Saffman and Taylor, 1958), which occurs when a less viscous

fluid attempts to displace a more viscous one. In this mechanism, a small perturbation

at the interface between the twofluids causes the less viscousfluid to find itself in a

region of lower pressure, hence amplifying the perturbation. The Bogerfluids used in

59

the experiment of McKinley et al. (1996) are strongly strain hardening. This means that,

at large strains, the central portions of the liquid bridge which are highly stretched resist

further decrease in diameter. This in turn means that a large portion of thefilament is

being extended as a cylinder of almost constant radius, with the additional volume of

fluid being supplied by the rapidly depleting pool of liquid near the two endplates. At

large strains, the thickness of this pool is small compared to its radius, so that theflow of

liquid in this region is radially inward. This indicates that the pressure at the center of the

pool must be lower than at the edge. If the free surface near this region is perturbed, the

surrounding medium (in this case, air at atmospheric pressure)finds itself in a region of

lower pressure, which causes the interface to deform further.

This instability is relatively new, so not much work has been done in terms of model-

ing it. Spiegelberg and McKinley (1998) performed a numerical simulation of the stretch-

ing of afilament of Oldroyd-Bfluid using a commercial software program based on the

finite element method. They reported that the initialflow inhomogeneity caused the for-

mation of stress boundary layers near the free surface of thefilament at large strains.

For Hencky strainsε > 4, they found that it became increasingly difficult to model the

rapidly draining region near the endplates. Significantly, the elastic instability sets in

close toε = 4 for large Deborah numbers.

Rasmussen and Hassager (1999) performed an analysis of the stretching of a UCM

filament. They considered two cases: thefirst was was the elongation of a purely cylin-

drical filament, and the second was the elongation a cylindricalfilament with a small

uniform non-axisymmetric perturbation superimposed on it. For the non-axisymmetric

initial state, they found that while the central region and the region near the endplates

remained almost circular, the deviation from axisymmetry in the region just above the

60

endplates grew as thefilament was extended. They also found that stress boundary lay-

ers were formed near the free surface for the axisymmetric initial condition. For non-

axisymmetric initial conditions, the stress boundary layer was relieved in regions which

bulged out as compared to the axisymmetric state, while it was more pronounced in the

regions that had retracted. Care must be taken when interpreting their results, however,

since the initial condition was non-axisymmetric, in contrast to the axisymmetric base

state in the experiments. Also, numerical difficulties restricted their analysis to Hencky

strains less than2.5 for Deborah numbers larger than1, so their simulations could not

capture the behavior of thefilament at large elongations and high Deborah numbers seen

in the experiments by Spiegelberg and McKinley (1998).

While the mechanism suggested by McKinley et al. (1996) is plausible, the impor-

tance offluid elasticity, as evidenced by the fact that the instability is not observed at small

Deborah numbers or in Newtonianfluids, indicates that a purely elastic mechanism may

be at work. The formation of stress boundary layers near the free surface as seen in the

simulations of Spiegelberg and McKinley (1998) and Rasmussen and Hassager (1999)

suggests that the physics of the instability may be captured by examining a small region

near the interface. Since theflow in the region near the endplates is radially inward, com-

pressive hoop stresses develop in thefluid (Kumar and Graham, 2000). Further, at high

Deborah numbers, the Oldroyd-Bfluid behaves like an elastic solid. Hence, we expect

the region near the free surface to behave like a membrane subjected to a compressive

hoop stress.

We conduct a preliminary investigation of our proposed mechanism by modeling the

region of the liquid bridge near the endplates as an elastic membrane enclosing a passive

incompressiblefluid. The membrane is then stretched with the radii at the two ends kept

61

fixed. The incompressiblefluid then exerts a force of constraint on the membrane, which

acts to keep the enclosed volumefixed. The constraint forces cause compressive hoop

stresses to develop in the membrane, and thus we expect the initial axisymmetric shape

to bifurcate into non-axisymmetric modes at sufficiently large strains.

In the subsequent section, we demonstrate the build up of stress near the surface

by modeling an ideal planar elongation experiment. This serves to justify the use of

our membrane analogy. We next present the formulation of the membrane model for a

reference configuration obtained by chopping away the region near the apex of a cone.

This shape approximates the geometry near near the endplates. We conclude the chapter

with a discussion of the important results and conclusions of our analysis.

3.2 Planar elongation: a model problem

In this section, we examine a model problem that captures the main features of theflow

field near the central region in afilament stretching experiment. Figure 26 shows a por-

tion of the central region in the planar elongation of a UCMfluid. Theflow field is

Hamiltonian, with the stream functionψ = ε x y. Thus, the velocity components are

given by

vx =∂ψ

∂y= ε x

vy = −∂ψ

∂x= −ε y. (66)

The origin is a stagnation point, which means that polymer molecules which are carried

by theflow near the origin will experience large elongations. Thus, on physical grounds,

we would expect high stresses to develop in the neighborhood of the origin.

62

y=- l

y=0

x=0

y

x

Figure 26: Schematic of a planar elongation setup andflow field.

We can get a quantitative picture of the stress buildup by following the motion of a

fluid element initially located at(x0,−l) (which we assume is a point of zero stress) as it

moves along a streamline. The evolution of the stress tensorτ is described by the UCM

equation as

τ + λ

(D τ

D t− τ · ∇v − τ · ∇vt

)= η0 γ, (67)

wherev is the velocity vector,γ = (∇v + (∇v)t), λ is the relaxation time,η is the

viscosity andD/Dt denotes the material derivative∂/∂t+v·∇. We make these equations

dimensionless by scaling the length withl, velocity with l ε , time with ε, and stress with

the shear modulusη/λ. With these scalings, equation 67 is written as

1

Weτ +

(D τ

D t− τ · ∇ v − τ · ∇ vt

)= ˆγ, (68)

with the Weissenberg numberWe is given byλε. Here, the hats above the variables are

used to denote the scaled versions of the respective quantities, and will be dropped in the

rest of this section for convenience.

63

For theflow described above, the UCM equation gives

dτxxdt−

(2−

1

We

)− 2 = 0, (69)

for the evolution ofτxx on a particular streamline. Thefluid element is chosen so that it

is initially located at(x0,−1), wherex0 ∈ (0, 1), with the stressτxx(t = 0) = 0. The

fluid element is carried along with theflow according to

dx

dt= x, (70)

dy

dt= −y. (71)

Equations (69) to (71) may be readily solved to yield

x(t) = x0 exp(t), (72)

y(t) = − exp(−t), (73)

τxx(t) =2We

2We− 1

(−1 + exp

(2−

1

We

)t

). (74)

Equation 73 may be used to eliminatet from equation 74 to yield

τxx(y) =2We

2We− 1

((−1

y

)2−1/We

− 1

). (75)

Equations 74 and 75 yield the well known result that the axial stress in planar elonga-

tion of a UCMfluid blows up atWe = 0.5. It is also evident from equation 75 thatτxx is

independent ofx for the initial condition of zero stress aty = −1. In the limitWe→∞,

equation 75 givesτxx = 1/y2 − 1, which indicates that at high Weissenberg numbers, a

stress boundary layer exists near the free surfacey = 0. This serves as justification for

using the membrane approximation for largeWe, where we assume that the physics of

the instability can be captured by modeling a small region near the interface.

64

At first sight, it seems somewhat surprising that the limiting value ofτxx, or equiv-

alently, the thickness of the boundary layer, is independent ofWe. This is resolved by

observing that, in general, the Weissenberg number governs the lengthl, at which the

normal stresses are close to zero. This in turn means thatWe would affect the length

scale fory, and hence the limiting value ofτxx.

3.3 Elongation of a truncated cone

In this section, we apply the theory of membrane elasticity to determine the evolution of

the shape of a truncated cone under axial elongation. This geometry is chosen because

it approximates the“reservoir” region near the endplates in afilament stretching exper-

iment. The cone encloses an incompressiblefluid, so that its volume remains constant

under elongation. Thus, the liquid exerts a force on the membrane to keep the volume

fixed, but otherwise plays no role in the formulation. This force exerted by the liquid is a

force of constraint which does not contribute to the total energy of the system (Goldstein,

1980). Our goal is to demonstrate that compressive hoop stresses develop in this system,

resulting in a symmetry breaking bifurcation to a non-axisymmetric shape.

3.3.1 Problem Formulation

The mathematical formulation uses aspects of tensor analysis, differential geometry, and

finite elasticity. An overview offinite elasticity, to the extent needed to formulate this

problem, is presented in Appendix A. In keeping with the Einstein notation, we will use

65

superscripts to denote the components of vectors or tensors that transform contravari-

antly, and subscripts for those that transform covariantly. Scalars and constants are in-

variant under transformation, and subscripts used in conjunction with them merely serve

to distinguish between different quantities.

Consider the portion of the cone shown infigure 27. In its undeformed (rest) config-

uration, that the radius varies linearly fromr1 to r2 over a lengthL. Thus, at any axial

locationx3 (with x3 < L), the radius of the cone is given by

r(x3) = r1 − s x3, (76)

wheres = (r1 − r2)/L. In the limit of s = 0, this reduces to a cylindrical membrane.

The volume enclosed by the membrane is given byV0 = π(r21 L− r1 sL2 + s2

3L3)

.

The membrane is subjected to a deformation at constant enclosed volume. This can be

achieved by extending either end or both ends, but it is convenient from our point of view

to think of the left end (the end near the origin) as being heldfixed, while the right end is

extended by an amountl.

We begin by defining the Gaussian surface coordinates for the reference configura-

tion. Letv1 be the distance along a circular cross section of the cone measured from the

intersection of the cone with thex2 = 0 plane. Thus, at any axial location, the azimuthal

angle (measured anti-clockwise) is given byθ = v1/r(x3). The second surface coordi-

nate,v2 is defined to be the distance of a point from the intersection of the cone with

thex3 = 0 plane, measured along its intersection with thex1 x3 plane (seefigure 27).

Thus, thex3 coordinate of a point whose second surface coordinate isv2 is given by

x3 = v2 cos(φ), wheretan(φ) = (r1 − r2)/L. Hence we have0 ≤ v1 < 2 π r(x3),

and0 ≤ v2 ≤ L sec(φ). The position vector in Cartesian coordinates of a point whose

66

x

r1

r2

L

v2

v1

1

x 3

x 2

Figure 27: Coordinate system for the truncated cone.

surface coordinates are(v1, v2) is given by

a = r(v2) cos(θ) e1 + r(v2) sin(θ) e2 + v2 cos(φ) e3. (77)

We scale all lengths byL and write

a∗ = r∗(v2∗) cos(θ) e1 + r(v2∗) sin(θ) e2 + v2∗ cos(φ) e3. (78)

In future, we will drop the asterisks for convenience. We now consider an axisymmetric

deformation such that the position vector of a point whose surface coordinates in the

undeformed configuration are(v1, v2) is given by

A = λ1(v2) r(v2) cos(θ) e1 + λ1(v

2) r(v2) sin(θ) e2 + λ2(v2) e3. (79)

67

The boundary conditions on the stretches are:

λ1(0) = 1,

λ1(sec(φ)) = 1,

λ2(0) = 0,

λ2(sec(φ)) = 1 + l. (80)

Given the position vectorsa andA, we can compute the covariant and contravariant

components of the surface metric tensors in the reference and deformed states, as de-

scribed in section A.1.1. Let us denote byaαβ the covariant components, and byaαβ the

contravariant components of the surface metric tensor in the reference state. The corre-

sponding components in the deformed state are given byAαβ andAαβ respectively. The

strain invariantI1 is then given by

I1 = aαβAαβ + a/A. (81)

Thus, the dimensionless strain energy stored in a neo-Hookean form is given by the equa-

tion

E =

∫ 10

∫ 2π (r1−s z)0

(I1 − 3) dv1 dz, (82)

wherez = v2 cos(φ), and the neo-Hookean coefficientC1, and a factor ofcosφ have been

absorbed intoE to make it dimensionless. Note that the integral is over theundeformed

state of the membrane.

The next step in the formulation is to compute the volume enclosed by the membrane.

One way to do this is to assume that the cone is a solid body, so that the Cartesian

coordinates(x1, x2, x3) of a point enclosed by the membrane are given in terms ofv1 and

68

v2 by

x1 = λ1(v2) r cos(v1/r(v2)),

x2 = λ1(v2) r sin(v1/r(v2)),

x3 = λ2(v2), (83)

where0 < r < r1 − s v2 cos(φ). The Cartesian volume elementdV = dx1dx2dx3 is

related to the differential surface coordinates by

dx1dx2dx3 =| J | dr dv1 dv2, (84)

whereJ is the determinant of the Jacobian matrix of the transformation. Since| J | is

not a function ofr, we can perform the integration overr. The volume enclosed by the

deformed membrane is given by

Vl =

∫ 10

∫ 2π(r1−s z)0

1

2(r1 − s z)λ21 λ2,z dv

1 dz. (85)

Hence, the dimensionless strain energy function is given by

F = E + Λ (Vl − V0), (86)

whereE is given by equation 82 andV by equation 85, andΛ is a Lagrange multiplier.

For a given value ofl, λ1 andλ2 take values so that thefirst variation ofF is zero, i.e.,

λ1 andλ2 are solutions to the Euler-Lagrange equations corresponding to the variational

principleδ F = 0 (Greenberg, 1978).

3.3.2 Method of solution and stability analysis

The standard method of solving variational problems is by means of the Ritz method.

Here, we expandλ1 andλ2 as polynomial series which satisfy the boundary conditions

69

and solve for the coefficients such thatF is extremized. We choose the following func-

tional form:

λ1 = 1 + (z2 − 1)N∑i=0

ai Ti(η),

λ2 = (1 + l) z +N∑i=0

bi Ti(η), (87)

whereη = 2 z − 1 andTi(η) is the Chebyshev polynomial ofith order with argument

η. Substitution of these expressions into equation 86 givesF in terms ofai, bi andΛ.

The variational statementδ F = 0 then corresponds to the requirement that the partial

derivatives ofF with respect to each of theai, bi andΛ vanish. This gives us a set of

2N + 3 nonlinear equations forai, bi andΛ which we can solve by Newton iteration.

Having obtained the axisymmetric solution, we now check for bifurcations to non-

axisymmetric states. To do this, we assume that the position vector in the deformed state

can be written as

A = (λ1(v2) + λ1(v

1, v2)) r(v2) cos(θ + λ3(v1, v2)/r(v2)) e1 +

(λ1(v2) + λ1(v

1, v2)) r(v2) sin(θ + λ3(v1, v2)/r(v2)) e2 +

(λ2(v2) + λ2(v

1, v2)) e3. (88)

The perturbationsλ1, λ2 and λ3 are set to be0 at the two ends of the membrane. In

addition, we impose a periodicity condition in the azimuthal direction.

We can now perform an analysis similar to the one described in the previous section

to determine the constrained strain energy function for non-axisymmetric deformations

Fa = Ea + Λa (Vl,a − V0), (89)

on the same lines as equation 86. Note thatFa reduces toF if we takeλ1, λ2 andλ3 to be

zero (the axisymmetric shape). In the linear limit, the azimuthal modes decouple, so we

70

can examine each mode separately. We therefore choose the following functional form

for the perturbations:

λ1 = (z2 − 1)N∑i=0

ai Ti(η) cos(n θ),

λ2 = (z2 − 1)

(N∑i=0

bi Ti(η) cos(n θ) +N∑i=0

di Ti(η) sin(n θ)

),

λ3 = (z2 − 1)

(N∑i=0

ei Ti(η) cos(n θ) +N∑i=0

fi Ti(η) sin(n θ)

), (90)

whereθ = v2/(r1 − s z), andn is the azimuthal mode number. The form ofλ1 is chosen

so as tofix the phase of the perturbation.

Substituting equations 87 and 90 into equation 89 gives usFa in terms ofai, bi, ai, bi,

di, ei, fi, andΛ. The requirement that the partial derivatives ofFa with respect to these

variables be zero gives us a set of7N + 8 equations in these variables. One solution to

this set of equations corresponds to the axisymmetric solution calculated above, i.e.,ai,

bi andΛ corresponding to the axisymmetric solution, andai, bi, di, ei, fi all equal to zero.

A bifurcation to a different solution occurs when one or more eigenvalues of the Hessian

matrix of Fa, evaluated at the axisymmetric solution, crosses the origin with non-zero

slope.


The axisymmetric shape is obtained using a Newton iteration. In general, 9 Chebyshev

polynomials provide enough accuracy to capture the shape. Since the equations are highly

nonlinear, we need a good initial guess, so we start atl = 0 where the exact solution is

71

given byai = 0, bi = 0, andΛ = 0, and then do a continuation inl, with the initial guess

at each point computed from the tangent at the previous point. We use a Broyden update

method (Broyden, 1965) to avoid having to recompute the Hessian between successive

iterations of the Newton method at a given value ofl. We use two different methods to

check our results. First, fors = 0, the truncated cone is a cylinder, and the evolution

of the shape in the absence of the volume constraint was computed by Yang and Feng

(1970). Our results for the corresponding case are in agreement with theirs. We also

check that equation 179, which is in the nature of a momentum conservation equation

for the membrane, is satisfied both in the base state, and also in linearized form by the

bifurcating solution.

We first consider the specific case ofr1 = 0.5 ands = 0.45, and examine its stability

with respect to a mode3 disturbance. Figures 28(a)–(c) show the development of the

three dimensional structure of the cone as we increasel from 0 to the bifurcation point,

l = 0.5. It is clear from thesefigures that much of the deformation is concentrated in the

central portion of thefilament, and the right end of thefilament stretches into a cylindrical

portion of nearly constant radius. This is what we desire if we wish to match this region

in some way to the constant thickness central region in afilament stretching experiment.

Denoting the elongationl at which the bifurcation occurs bylc, wefind that the cone

undergoes a bifurcation to then = 3 mode atlc = 0.50. The structure of the bifurcat-

ing solution is shown infigure 28(d). This is obtained by adding a small multiple of the

eigenvector corresponding to the bifurcating eigenvalue and adding it to the base eigen-

vector. As mentioned in the introduction, the instability results because portions of the

membrane are in compression. To verify this, we plot the amplitude of the perturbation

radial stretchλ1 and the hoop stressτθθ (calculated from equation 163) infigure 29. We

72

(c)

(d)

(a)

(b)

Figure 28: Evolution of the shape of a truncated cone under elongation: (a) undeformedconfiguration (b) axisymmetric configuration atl = 0.2 (c) axisymmetric configurationat l = lc = 0.5 (d) post bifurcation non-axisymmetric shape atl = lc = 0.5. Thefigureson the right track the change in a cross section originally at a distance of 0.75 units fromthe left edge. In (d), the perturbation has been exaggerated for clarity.

73

0.0 0.5 1.0 1.5z

−0.2

0.0

0.2

0.4

0.6

0.8

τθθ ||

^λ1||

Figure 29: Spatial profile of the hoop stress,τθθ, and the amplitude of the bifurcatingsolution‖ λ1 ‖.

observe that the perturbation has its maximum amplitude close to the point where the

hoop stress has its largest negative value, i.e., near the point of largest compressive stress.

Table 2 shows the values oflc at which then = 3 mode bifurcates for various values of

r1 ands. While the dependence ons is non-monotonic, we see that for afixed value of

s, increasingr1 pusheslc to higher values.

r1 s lc0.5 0.45 0.500.5 0.3 0.310.5 0.2 0.360.7 0.2 0.691.0 0.2 1.08

Table 2: Variation of the bifurcation point for then = 3mode of the truncated cone withr1 ands.

So far, we have confined our discussion to the behavior of then = 3 mode. While

computing values oflc for different modes is a computationally intensive task, we can

74

get an idea of the dependence ofn by looking at the eigenvalue spectrum of different

modes for thelc of then = 3 mode. This computation indicates that modes with higher

wavenumbers bifurcate sooner, an observation consistent with the fact that even very

small increases in length result in compressive hoop stresses in parts of the membrane.

Thus, the elongation of this geometric shape atfixed volume is always unstable to non-

axisymmetric perturbations, with a critical wavenumbern→∞. At first sight, this seems

unphysical, but a similar situation exists in the Saffman-Taylor problem (Saffman and

Taylor, 1958) in the absence of surface tension. What is absent in the present formulation

is a mechanism to damp out large wavenumber disturbances. In a viscoelasticfilament,

two such mechanisms would be surface tension and bending moments arising from non-

zero boundary layer thickness. In fact, an approximate analysis of the radial inflow of a

thin layer of an Oldroyd-Bfluid in a washer shaped domain with free surface boundary

conditions at the top and bottom (Kumar and Graham, 2000), which models theflow the

polymer solution just above the end plates, shows that disturbances of large wavelength

are the most unstable, thus providing evidence for the stabilizing role of surface tension.

Another example of this is in the Saffman-Taylor problem, in which when surface tension

is taken into account, the fastest growing disturbance has afinite wavenumber.

3.5 Conclusions

In this chapter, we have proposed a new mechanism to explain the purely elastic insta-

bility seen infilament stretching at large Weissenberg numbers. In thefirst part of this

chapter, we used the simple model of a planar elongation experiment to show how large

stresses build up in free surfaceflows. Based on this, and upon evidence in the literature

75

(Rasmussen and Hassager, 1999; Spiegelberg and McKinley, 1998) on the formation of

stress boundary layers near the free surface at large Weissenberg numbers, we postulated

that the physics of the instability can be captured by modeling the region near the inter-

face as an elastic membrane enclosing an incompressiblefluid. We showed that such a

membrane is unstable in elongation to non-axisymmetric disturbances. The instability

is related to the formation of compressive hoop stresses in the membrane, and these are

largest close to the point where the perturbation has the greatest magnitude. Our calcu-

lations show that disturbances withn → ∞ are most unstable, but we expect that the

inclusion of selection mechanisms like surface tension and a non-zero bending moment

which impose an energy penalty on large wavenumber disturbances will yield afinite crit-

ical wavenumber. Validation of the mechanism proposed here would require numerical

simulations using a viscoelastic constitutive equation at large Deborah number and up to

large Hencky strains. For the mechanism proposed here to be effective, the simulations

would need to show stress boundary layers with regions of compressive hoop stress close

to the portions of thefilament that have buckled.

76

Chapter 4

Stabilization of Dean flow

instability †

In this chapter, we will discuss the stabilization of the purely elastic instability in the

Deanflow geometry. Instabilities in Deanflow, and the closely related circular Couette

flow geometry are interesting because they serve as models for more complexflows in

polymer coating operations. Therefore, we will motivate the discussion by providing an

overview of instabilities in coatingflows. We will then describe the main features of the

elastic instability in Deanflow and present a means of suppressing this instability.

4.1 Instabilities in coating flows

Coating may be defined as the process of replacing air with a new material on the sub-

strate (Cohen and Gutoff, 1992). Coatings play an important part in everyday life, being

† Most of the material in this chapter has been published in Ramanan et al. (1999)

77

used in the production of common products such as photographicfilm, paper, and mag-

netic media used for data storage. The most visible part of the industry is the manufac-

ture of paints and protectivefilms for automobiles, houses, and other structures. In the

United States, this segment of the industry alone has an annual revenue in of about $17

billion (The Freedonia Group, 2000). When the less visible part of the industry, which

includes such areas as print and publishing and printed circuit boards for electronics for

is taken into account, the value exceeds $300 billion.

There are several coating methods currently in use. The most common single layer

coating methods are rod coating, forward and reverse roll coating, blade coating, air knife

coating, gravure coating, slot coating, and extrusion coating. Multiple layer coatings pro-

cesses include slide and curtain coating. Schematics of some of these processes are shown

in figure 30. The choice of coating method to use for a given application is determined by

several factors: these include the type of substrate (porous or non-porous), coating speed,

the thickness and accuracy desired, the viscosity and viscoelasticity of the coatingfluid,

and the number of layers to be deposited. Booth (Booth, 1958a,b) was thefirst to publish

a guide to choose the right coating method based on the criteria mentioned above. More

recently, a simpler guide has been proposed by Cohen (1992).

Throughput in coating operations is generally limited by interfacial instabilities which

give rise to surface distortions on the coating (Strenger et al., 1997). Perhaps the most

well known of these interfacial instabilities is the so called“ribbing” instability in forward

and reverse roll coating (Saffman and Taylor, 1958; Pearson, 1960; Pitts and Greiller,

1961). A photograph of this instability in forward roll coating is shown infigure 31.

The photograph clearly shows the spatially periodic patterns on the surface of the rollers,

78

(a)

(b)

(e)

(f)

(c)

(d)

Figure 30: Some commonly used coating industrial coating processes: (a) Dip coatingand rod coating. (b) Blade coating and air knife coating. (c) Gravure coating. (d) Reverseroll coating. (e) Extrusion coating. (f) Slide coating and curtain coating (Cohen, 1992)

79

which translate to surface distortions on the coating. Since theflow in the gap in be-

tween the rollers is complicated, most of the earlier analyses of the ribbing instability

used simplified models of theflow (Pitts and Greiller, 1961; Savage, 1977a,b; Gokhale,

1981, 1983b,a; Savage, 1984; Benkreira et al., 1982). With the advent of more pow-

erful computers, it became possible to perform proper asympototic and fully numerical

calculations usingfinite element methods, and to analyze its stability for Newtonian liq-

uids (Ruschak, 1985; Coyle et al., 1986, 1990b). As a result, the critical conditions for

the onset of ribbing for Newtonian liquids are now very well predicted by theory (Coyle,

1992). These results are typically reported in terms of the capillary number, which is a

measure of the ratio of viscous to surface tension effects.

Figure 31: Photograph of ribbing instability in forward roll coating (Coyle et al., 1990b).

Many industrial coating processes use polymericfluids, and this can have a significant

effect on the instability. Bauman et al. (1982) showed that adding as small an amount as

10 ppm of high molecular weight polyacrylamide can reduce the critical capillary number

for instability by a factor of 2 to 5 in forward roll coating. Slot coating (Ning et al., 1996)

and reverse roll coating (Coyle et al., 1990a) are also significantly destabilized when

polymericfluids are used. Strong destabilization has also been observed recently in free

80

surfaceflows in eccentric cylinder geometries (Grillet et al., 1999) with dilute solutions

of polyisobutylene in polybutene/kerosene mixtures. Soules et al. (1988) and Fernando

and Glass (1988) performed a series of experiments which demonstrated that the earlier

onset of ribbing correlates well with the extensional viscosity of the polymer.

It is clear from the observations listed above that the addition of polymer has a marked

effect on the stability of coatingflows in industrial use. Therefore, it is important to

understand the mechanisms by which polymers can causeflow instabilities. Rather than

attempt the complex and computationally demanding task of analyzing coatingflows, we

look instead for simpler geometries where some of these mechanisms are manifested.

Since we are interested in instabilities caused by elastic effects alone, we restrict our

attention toflows where inertial effects are negligible. Examples of these include theflow

of polymer solutions in cone-and-plate and plate-and-plate geometries (Larson, 1988;

Byars et al., 1994), the circular Couette geometry (Larson et al., 1990), and the Dean and

Couette-Dean geometries (Joo and Shaqfeh, 1991, 1994). In the latter three geometries,

thefluid flows in the gap between two concentric cylinders. In circular Couetteflow, the

flow is driven by the motion of one of the cylinders, while in Deanflow, a pressure drop

is applied in the azimuthal direction. In Couette-Deanflow, a combination of the two

methods is used to drive theflow. Apart from these, recirculationflows, such asflow

in a lid-driven cavity (McKinley et al., 1996) also exhibit elastic instabilities. In each

of theseflows, polymer molecules are stretched along the curved streamlines in much

the manner as in the rod climbing effect discussed in chapter 1. This extra tension in

the streamlines, or the so called hoop stress, is what causes the instability. It is obvious

from an examination offigure 30 that curved streamlines exist in many coatingflows. In

addition, recentflow visualizations (figure 10) have shown the presence of recirculation

81

regions in blade coating. Thus, it is plausible that the instabilities seen in theseflows are

driven by mechanisms similar to those that drive the instability in simpler geometries,

and that any stabilization mechanism that suppresses such instabilities in the simpler

geometries such as Couette and Deanflows carries over to the more complex coating

flows as well.

4.2 Elastic instability in Dean flow

Before discussing the elastic instability in Deanflow, we first present an overview of

the instability in the related geometry of circular Couetteflow. Larson et al. (1990)

performed thefirst theoretical and experimental analyses of elaticity-driven instabilities

arising in circular Couetteflow. Their theoretical analysis, using an Oldroyd-Bfluid,

showed that this“viscoelastic Taylor-Couette” (VETC) instability occured even at negli-

gible Reynolds number. The criterion for the instability to occur is thatε1/2Weθ = O(1),

whereWeθ is the azimuthal Weissenberg number (ratio between material andflow time

scales) andε is the gap width, non-dimensionalized with respect to the radius of the cylin-

der. The scaling reflects how largeWeθ must be for the azimuthal normal (hoop) stress

to contribute to the leading order radial momentum balance. For comparison, the well

known inertial Taylor-Couette instability occurs whenε1/2Re is O(1), whereRe is the

Reynolds number. The mechanism of destabilization proposed by Larson et al. (1990)

and later refined by Joo and Shaqfeh (1994) to include non-axisymmetric modes, was

based on the coupling of stress perturbation to the base state velocity gradient to pro-

duce an azimuthal normal stress that drives radial and transverse motions leading to the

formation of cells. The equations governing perturbations to the basic Couetteflow are

82

O(2) symmetric, i.e., they are invariant under both reflections and translations. Theory

(Golubitsky et al., 1985) indicates that bifurcations in such systems are either pitchfork

or degenerate Hopf (i.e., the Jacobian matrix has two pairs of complex conjugate eigen-

values crossing the imaginary axis at the onset of the instability). In agreement with this

prediction, Larson et al. (1990) found that the instability took the form of a degenerate

Hopf bifurcation. While Larson et al. (1990) only considered axisymmetric modes, it

was later found (Joo and Shaqfeh, 1994; Sureshkumar et al., 1994) that the most unsta-

ble disturbance is in fact a non-axisymmetric mode, with a wavenumber between 1 and

3. Further, Renardy et al. (1996) conducted a nonlinear analysis to study mode inter-

actions arising from the introduction of inertia– and thus the classical Taylor-Couette

instability – into the system. Computations (Northey et al., 1992; Avgousti and Beris,

1993; Avgousti et al., 1993) and experiments (Shaqfeh et al., 1992) have shown thatfi-

nite gap effects tend to stabilize theflow. Recently, Al-Mubaiyedh et al. (2000) have

shown that non-isothermal effects can give rise to an entirely new mode of instability

at very low values of the Weissenberg number. In contrast to the isothermal case, this

instability takes the form of a stationary, axisymmetric bifurcation. Al-Mubaiyedh et al.

(2000) also investigated the effect offluid rheology, in particular, accounting for multi-

ple relaxation times, on the viscoelastic circular Couette instability. They found that the

critical Weissenberg number asymptotes to a value that is half that predicted by a single

mode constitutive equation, but that the mechanism of the instability and structure of the

bifurcated modes remain the same. While the analyses discussed above have focused on

the behavior close to the bifurcation point, i.e., the linear and weakly nonlinear regimes,

experiments (Groisman and Steinberg, 1997; Baumert and Muller, 1999) have shown that

a rich variety of dynamical phenomena can occur far away from the bifurcation point. In

83

chapter 5, we describe how a fully nonlinear analysis can capture some aspects of these

observations.

In the context of Deanflow, Joo and Shaqfeh (1991) performed a theoretical investi-

gation for an Oldroyd-Bfluid to show that such an elastic instability did indeed occur. Joo

and Shaqfeh (1992b) generalized this to Couette-Deanflow. In line with the fact that this

instability is purely elastic, they found that the most unstablefluid was one where there

was no contribution to the extra stress tensor from the Newtonian solvent, and that the

onset of the instability was delayed when solvent viscosity was present. Later, Joo and

Shaqfeh (1992a) investigated the effect of inertia on Dean and circular Couetteflows.

They found that Deanflow was destabilized by inertial effects, while circular Couette

flow was stabilized if theflow was driven by the rotation of the outer cylinder, and desta-

bilized if it was driven by the inner cylinder rotation, consistent with the mechanism of

the inertial instability. Joo and Shaqfeh (1994) presented experimental confirmation of

the elastic instability in Deanflow, and also performed an experimental and theoretical in-

vestigation of the effects offinite gap width on the instability. As in circular Couetteflow,

they found both theoretically and experimentally thatfinite gap effects were stabilizing.

The perturbation equations in Deanflow are also O(2) symmetric, and in this case,

Joo and Shaqfeh (1991) found that the axisymmetric mode bifurcates as a pitchfork.

They performed an energy analysis and determined that the mechanism of the instability

involved the coupling of base state hoop stresses with radial velocity perturbations. In

a later publication (Joo and Shaqfeh, 1994), they found experimental confirmation that

the primary bifurcation in Deanflow was a stationary wave, and performed a theoretical

analysis to show that non-axisymmetric modes are always more stable than axisymmetric

ones.

84

The mechanism of the instability may be understood fromfigure 32. Polymer molecules

are modeled as pairs of beads connected by springs. Molecules near the outer cylinder are

more highly stretched than those closer to the center. An inwardly directed radial velocity

perturbation convects highly stretched molecules towards the center, and thus increases

the local hoop stress. This has the effect of reinforcing the radial velocity perturbation,

hence driving the secondaryflow. A simpler way of understanding this mechanism is

to imagine that the azimuthalflow stretches polymer molecules so that they act like a

stretched cylindrical membrane. This“membrane” in turn exerts an inward compressive

force on thefluid, which causesfluid columns to buckle in the axial direction, causing

the instability.

v

ab

dc

Figure 32: Mechanism of the elastic instability in Deanflow.

Knowledge of the mechanism of theflow can be used to develop methods to stabilize

it. A recent study by Graham (1998) has shown that the addition of a relatively weak ax-

ial flow results in significant stabilization of the isothermal VETC instability. The axial

flow increases the critical Weissenberg number for onset of the instability and moves the

most unstable mode to longer wavelengths. The stabilization is due to the additional axial

normal stress generated by the axialflow, which suppresses radial displacements. Scal-

ing analyses and numerical simulation showed that non-axisymmetric disturbances are

85

strongly suppressed. A weakly nonlinear analysis was performed to determine whether

the bifurcation was subcritical or supercritical. In the narrow gap limit, it was found that

the primary bifurcation in circular Couetteflow is subcritical. The bifurcation remains

subcritical if an axial Couetteflow is added, while an axial Poiseuilleflow changes the

bifurcation to a supercritical one.

Given that the effect of an axialflow is to suppress radial velocity perturbations, it

seems reasonable to suppose that the same mechanism would stabilize the Deanflow

instability as well. In the context of the simple instability mechanism stated above, axial

flow has the effect of stretching the polymer“membrane” in the axial direction. This

creates a restoring force that acts against radial buckling, as shown infigure 33. . In

thefirst part of this work we extend the analysis of Graham (1998) to demonstrate that,

as expected, the addition of axialflow stabilizes the viscoelastic Deanflow instability.

In particular, we show that at large axial shear rates, the critical value ofε1/2Weθ scales

linearly with the axial shear rate, thus confirming that a relatively weak axialflow results

in dramatic stabilization. This is similar to the behavior in circular Couetteflow. We also

perform a weakly nonlinear analysis for Deanflow to examine the effect of adding axial

flow on the criticality of the bifurcation.

Having established that the addition of a steady axialflow does stabilize viscoelastic

Deanflow, we next examine the effect of adding a time-periodic axialflow. There are

two reasons to study this. Firstly, it is more practical to superimpose an oscillatory axial

motion in an industrial polymer processing operation than it is to superimpose steady

axial motion. Secondly, viscoelasticfluids have characteristic relaxation time scales,

and these might be expected to interact with the time scale of the parametric forcing to

produce interesting dynamical phenomena. Note that the axial normal stress depends

86

τrz , τzzτrz , τzz

τθθ/r

Figure 33: Illustration of the mechanism by which additional axial stresses generated bya superimposed axialflow stabilize the viscoelastic Deanflow instability

on the square of the axial shear rate in the baseflow. Thus, the average of this stress

over one period of the oscillation is non-zero. Superimposing an oscillatoryflow on the

circular Couetteflow has been shown to stabilize the inertial Taylor-Couette instability

both experimentally (Weisberg et al., 1997) as well as theoretically (Hu and Kelly, 1995;

Marques and Lopez, 1997). No corresponding analysis is available for Newtonian Dean

flow.

The organization of this chapter is as follows. Wefirst begin by formulating the prob-

lem for an Oldroyd-Bfluid in the narrow gap approximation in section 4.2. Section 4.3

describes the details of the linear stability analysis used in the study. To examine the

effect of adding a steady axialflow, we use a standard linear stability analysis. How-

ever, adding an oscillatory axialflow results in time-periodic coefficients in the stability

equations. In this case, the stability analysis is performed using Floquet theory. Rather

than construct the entire monodromy matrix, we use the Arnoldi technique to determine

its dominant eigenvalues. Finally, in section 4.4, we present results from our numerical

analysis and compare them to predictions from the asymptotics.

87

4.3 Formulation

We consider the isothermalflow of an Oldroyd-Bfluid in the annular region between two

infinitely long circular cylinders. Consideration of isothermalflow ensures that only the

purely elastic mechanism of destabilization operates. The annulus is assumed to be open,

i.e., no constraint is imposed on the net axialflow rate. Thefluid has a relaxation time

λ; the polymer and solvent contributions to the viscosity areηp andηs respectively, and

the ratioηs/ηp is denoted byS. The primaryflow is produced by imposing a constant

pressure drop in the azimuthal direction,Kθ = ∂P/∂θ. A schematic of this geometry is

shown infigure 34.

Κθ R 1

R 2

Figure 34: Deanflow geometry, shown with superimposed Poiseuilleflow.

The equations governing theflow are the momentum, constitutive and continuity

equations. In dimensionless form, they are given by

∇ · τ −∇p+WeθS∇2v = 0, (91)

88

τ +Weθ

(∂τ

∂t+ v · ∇τ −

(τ · ∇v + (τ · ∇v)t

))= Weθ

(∇v +∇vt

), (92)

∇ · v = 0, (93)

wherev is the velocity,p is the pressure andτ is the polymer stress tensor. No-slip

boundary conditions are imposed at the two cylinders. The Weissenberg number is de-

fined in terms of the pressure drop as

Weθ = −Kθλε

2 ηp, (94)

whereε is the dimensionless gap width,

ε =R2 −R1

R2. (95)

The Newtonianfluid is recovered in the limitS → ∞, with an appropriate rescaling of

pressure. We measure length in terms of distance from the inner cylinder, scaled by the

gap width, i.e.,

r =r∗ −R1R2 −R1

, (96)

wherer∗ is the dimensional radial position, andr is the dimensionless radial coordinate,

scaled and shifted so thatr = 0 is the inner cylinder andr = 1 is the outer cylinder. Stress

is scaled byηp/λ, and velocity is scaled by−Kθε2R2/2 ηp. Time is made dimensionless

by the ratio of the gap width,εR2, to the velocity scale. In terms of the dimensionless

variables, theflow domain is given by(r, θ, z) : 0 ≤ r ≤ 1, 0 ≤ θ < 2π, −∞ < z <

∞.

Axial Couetteflow is imposed by moving the inner cylinder with the velocityV cos(ωt),

whereV is the amplitude of the modulation andω is the frequency. The scaling used for

89

ω is the inverse of that used for time, so thatωt is O(1). Settingω = 0 is tantamount to

imposing steady axial Couetteflow. Later, we will see thatω → 0 is a singular limit. The

imposition of axialflow introduces an axial Weissenberg number,Wez, defined as

Wez =λV (1− ε)

R2ε. (97)

When the axialflow is imposed by means of a constant axial pressure drop,Pz, we can

similarly define another Weissenberg number

Wez = −PzλεR22 ηp

. (98)

For convenience, we will use the following acronyms:

• DAC: Deanflow with steady Axial Couetteflow superimposed (ω = 0 ).

• DMAC: Deanflow with Modulated Axial Couetteflow (ω = 0).

• DAP: Deanflow with steady Axial Poiseuilleflow.

We only consider the case where the axial pressure drop is steady.

For an Oldroyd-Bfluid, it is possible to obtain exact analytical expressions for the

base state velocitiesv and polymer stressesτ in Deanflow subjected to superimposed

time-periodic or steady axialflow. The narrow gap limit of these expressions (i.e. the

leading term in a Taylor series aboutε = 0) are shown in Appendix B. We are inter-

ested in the stability of these steady and periodic solutions to infinitesimal perturbations.

We define a vector of perturbationsu = (τrr, τrθ, τrz, τθθ, τθz, τzz, vr, vθ, vz, p), where,

for example,τθθ = τθθ − τθθ. It was shown by Joo and Shaqfeh (1991) that the elastic

instability occurs whenWeθ is O(ε−1/2). We enforce this by defining a scaled Weis-

senberg numberWp = ε−1/2Weθ, which isO(1) in the limit ε → 0. We now scale the

90

perturbation stresses and velocities by requiring that all terms in the Oldroyd-B equa-

tions appear in the perturbation equations for steady axialflow. This may be obtained

by imposing the following scalings on the perturbation velocities and stresses:τrr =

O(1), τrθ = O(ε−1/2), τrz = O(1), τθθ = O(ε−1), τθz = O(ε−1/2), τzz = O(1), vr =

O(ε1/2), vθ = O(1), vz = O(ε1/2), p = O(1) andWez = O(1). Terms involving the

azimuthal wavenumbern do not appear in the governing equations at leading order. To

get around this, we must either consider terms ofO(ε1/2) relative to the leading order

terms in the equations (as done by Joo and Shaqfeh (1994)) or equivalently, define a new

azimuthal wavenumbern by n = nε−1/2, wheren is O(1) (as done by Graham (1998)).

We choose the latter approach. Thus, in our formulation,n is not necessarily an integer,

althoughn must be. The scaling on the stresses and velocities is identical to that on the

base state stresses and velocities, and also to the scaling obtained by Graham (1998) for

circular Couetteflow with imposed axialflow. For time-periodic axialflow, a balance of

terms is obtained by rescaling time and frequency so thatω = ω1ε1/2 andt = t1ε

−1/2

whereω1 and t1 areO(1). With this scaling, the dimensionless relaxation time of the

polymer isWp. This is the“low frequency” regime that includes all possible terms in

the narrow gap approximation. Settingω1 = 0 in this regime, we recover the equations

of DAC flow. We also investigate the effect of imposing a“high frequency” oscillation

by scalingω to be anO(1) quantity i.e. we specifyω1 = O(ε−1/2). Dimensionally, this

corresponds to a forcing frequency on the order of the azimuthal shear rate. In this regime

the stability characteristics depend on the order of magnitude of the axial shear rate. In

section 4.5 we show that the relative magnitude ofWez with respect toω1 determines

the appropriate balance of terms that affects the stability of the system. Asymptotic and

numerical results are presented for both regimes.

91

4.4 Stability and Numerical Analysis

4.4.1 Linear Analysis and Floquet Theory

To analyze the stability of the baseflow, we expand the perturbation vector as

u = δφ(r, t)eiαz +O(δ2) + c.c., (99)

wherec.c. stands for“complex conjugate” andα is the axial wavenumber which is a

parameter in the problem. Wefirst consider the imposition of steady axialflow. In this

case, the vectorφ(r, t) is expanded out as

φ = ξ(r)ei(nθ−αct), (100)

whereξ(r) = (τrr(r), τrθ(r), τrz(r), τθθ(r), τθz(r), τzz(r), vr(r), vθ(r), vz(r)), andc =

cr + i ci is the growth rate. This expression is added to the base state stresses and ve-

locities, and the result substituted into equations 91 to 93. The equations may then be

expanded as a series inδ, and atO(δ0) we recover the equations governing the base

state stresses and velocities. AtO(δ), we get the equations that govern the evolution of

perturbations to the base state. We may write these succinctly as

i α cEu = L(α,Weθ,Wez, n)u, (101)

whereL is the linearization of the governing equations about the base state, andE is a

diagonal matrix with1 on the diagonal entries corresponding to the Oldroyd-B equations

and zeros elsewhere. The nonzero components ofE are shown in Appendix B forn = 0.

Equation 101 is in the form of a generalized eigenvalue problem for the growth ratec.

If the other parameters are given, we can numerically solve for the eigenvaluec. For

stability, ci must be less than zero.

92

The numerical technique we use is a Chebyshev collocation method (Canuto et al.,

1988). We use a primitive variable formulation of the conservation, constitutive and con-

tinuity equations, i.e., our unknowns are the six stress components, the three components

of the velocity, and the pressure. Since the conservation equations only contain deriva-

tives of the pressure, it is necessary to define pressure on a staggered grid (Canuto et al.,

1988) to avoid spurious modes. In the formulation we use, the velocity components and

stresses are defined at theN + 1 Gauss-Lobatto points (for anNth order collocation),

which are

rk = cos

(πk

N

), k = 0, . . . , N, (102)

while the pressure is defined at theN Chebyshev-Gauss points, which are given by

r′

k = cos

(π(k + 1/2)

N

), k = 0, . . . , N − 1, (103)

so that pressure is defined at one less point than the other variables. In evaluating the

conservation equations, pressure is interpolated onto the Gauss-Lobatto points from the

staggered grid, while the continuity equation is evaluated on the staggered grid by in-

terpolating the velocity components from the Gauss-Lobatto points. Wefind that it is

sufficient to useN = 32 for accurate calculation ofc and spatial resolution of the eigen-

vectors.

We now move on to the stability analysis for the time-periodic case. In this case, the

vectorφ(r, t) is expanded as

φ(r, t) = (τrr(r, t), τrθ(r, t), τrz(r, t), τθθ(r, t), τθz(r, t), τzz(r, t),

vr(r, t), vθ(r, t), vz(r, t), p(r, t)). (104)

93

We will only consider axisymmetric perturbations because, as will be shown in sec-

tion 4.5, it is the axisymmetric mode which is most unstable whether or not axialflow is

present. The equations governing evolution of the perturbations are given by

−Eu = A(t)u. (105)

Here,A(t) is a matrix with time-periodic coefficients such thatA(t) = A(t+T ), where

T = 2π/ω is the period of the forcing function. The nonzero components ofA are shown

in Appendix B. The problem lends itself to analysis by Floquet theory (Iooss and Joseph,

1989). The solutionu(T ) at t = T , given the initial vectoru(0), is

u(T ) = Φ(T )u(0), (106)

whereΦ(T ) is the monodromy matrix, whose eigenvaluesβ, known as Floquet multipli-

ers, determine the stability of the system. Stable and unstable behavior is indicated by

|β| < 1 and|β| > 1 respectively. The Floquet exponentσ is defined by the relation

β = exp(σ T ). (107)

SupposeΨ is an eigenvector ofΦ(T ) corresponding to the Floquet exponentσ. Then,

it can be shown (Iooss and Joseph, 1989) that the solutionw(t) to E = A(t)w with

w(0) = Ψ has aT periodic componentζ(t) given by

ζ(t) = e−σtw(t), (108)

such that

ζ(t) = ζ(t+ T ). (109)

Thus, theζ(t) corresponding to the dominant Floquet multiplier gives us information on

the spatial structure and time evolution of the disturbance.

94

The standard way to calculateΦ(T ) is to do it column by column, integrating equa-

tion 105 withu(0) = ek, whereek is thekth column of the unit matrix of same dimension

asE or A(t). Given the size of our problem, this is a computationally expensive task.

However, we only need the dominant eigenvalues (those with the largest modulus) of

Φ(T ), and we can conveniently obtain these using the Arnoldi method (Arnoldi, 1951).

We use a primitive variable formulation as in the steady case, with pressure defined on a

staggered grid. We use the public domain code ARPACK (Lehoucq et al., 1997) to carry

out the Arnoldi calculations. The advantage of the Arnoldi method lies in the fact that we

do not need to explicitly constructΦ(T ) in its entirety; we need only determine the action

of the matrix on a vector,q ← Φ(T )p. In our problemq is the solution vector obtained

by integrating the time evolution equations for a given initial vectorp. The choice of the

initial vectorp is arbitrary, with the only requirement being that it satisfy the algebraic

components of equation 105. In our case, this is ensured by choosing an arbitrary stress

profile and solving the linear momentum and continuity equations for the correspond-

ing velocities and pressure. To compute the periodic componentζ(t), we calculate the

disturbance vectorw(t) by integrating the time evolution equation 105 with initial con-

ditions given by the eigenvector of the monodromy matrix obtained from ARPACK. This

choice of starting value ensures that the periodicity condition given by equation 109 is

not violated.

The integration of the time-dependent viscoelastic equations is performed using the

EVSS decomposition (Rajagopalan et al., 1990) of the total stress into elastic and viscous

components, coupled with a fully implicitfirst-order time stepping procedure. The time

integration and eigenvalue routines were benchmarked against known eigenvalues for the

95

steady case (ω = 0) from the DAC results presented later in this chapter. Most calcu-

lations are performed with a time step∆t = T/100, while at very low frequencies we

employ increased temporal resolution of∆t = T/1000. This high resolution is necessary

to ensure the accuracy of the eigenvalue computation, which is checked by monitoring

the periodicity ofζ(t).

4.4.2 Weakly nonlinear analysis

Following the methodology of Graham (1998) which in turn follows Iooss and Joseph

(1989), we conduct a weakly nonlinear analysis to determine the criticality of the bifur-

cation in DAC and DAPflows. In the presence of either type of axialflow, the loss of

stability takes the form of a Hopf bifurcation, while in the absence of axialflow the bifur-

cation is a pitchfork. In either case, the vectors take the form of complex conjugates, and

since we are only interested in real valued solutions, the analysis described below works

whether or not an axialflow is present.

The nonlinear solutions are constructed by expanding all variables as a power series

in the amplitudeδ, substituting in the governing equations, and applying the narrow gap

approximation at each order inδ. We letµ = Wp−Wpc, ω0 = −αcr, ands = ωt. Thus,

the solution takes the form

u(z, s, δ)

µ(δ)

ω(δ)− ω0

=

∞∑k=1

δk

k!

uk(z, s)

µk

ωk

, (110)

whereu is taken to be the solution at the Chebyshev points. The linear operatorL can

96

be written as

L(µ) = L0 + δµ1L′+

δ2

2µ2L

′′+ ... (111)

Further, we define the following inner products:

〈a(z, s), b(z, s)〉 ≡α

4π2

∫ 2π/α0

∫ 2π0

a(z, s) b(z, s) ds dz, (112)

and

[a(z, s), b(z, s)] ≡

9(N+1)+N−1∑l=0

〈al(z, s), bl(z, s)〉. (113)

At O(δ), we recover the linear stability problem, which has the real valued solutionu1 =

z + z∗. Note thatu is time-dependent in the absence of axialflow, and a traveling wave

otherwise. The eigenvectors are normalized so that[z, z] = 1, wherez is the solution to

the adjoint problem. A solvability condition atO(δ2) gives

−2 i ω1 [Eu1, z∗] + 2µ1[L

′u1, z

∗] = −2 [N2(u), z∗]. (114)

The structure of the solutions atfirst order implies that the right hand side of this equation

vanishes. Upon separating the real and imaginary parts, the coefficient matrix of the

resulting2×2 real system has, in general, a nonvanishing determinant, and hence admits

only the trivial solution forµ1 andω1.

Substituting these values into theO(δ2) problem, we see that the particular solution

at that order can be written as

u2 = u20 + u22e2i(α z+s) + u22e

−2i(α z+s). (115)

Here,u20 is given by

L0,0u20 = −2〈N2(u),1〉, (116)

97

whereL0,0 is obtained by replacingα by 0 in L0. The vectoru22 is given by

(−2iω0E +L0,2α)u22 = −2〈N2(u),1e2i(αz+s)〉, (117)

where1 is a column vector with 1 in all its entries, andL0,2α is obtained by replacing

α with 2α in L0. Unlike in Graham (1998), the present formulation retains pressure

in the momentum equations. Hence, neither of the above equations is singular and the

solution is obtained by LU decomposition. A solvability condition atO(δ3) gives the

single complex equation (or the2× 2 real system)

−iω2[Eu1, z∗] + µ2[L

′u1, z

∗] = −2[N3(u), z∗], (118)

from whichµ2 andω2 can be obtained.


4.5.1 Scaling Analysis

As discussed in the introduction, the elastic instability is caused by an unstable stratifica-

tion of the hoop stress,τθθ. Therefore, if this term is absent from the momentum balance

in some asymptotic limit, theflow is stable. It is instructive to perform an asymptotic

analysis to determine how largeWp must be for the hoop stress to enter into the mo-

mentum balance at leading order. This can also be used to check the results from the

numerical analysis.

Wefirst consider the imposition of steady axialflow, withWez 1. The scalings for

both DAC and DAPflows are identical, and the effect ofO(1) solvent viscosity does not

affect them. Wefirst consider the results for axisymmetric perturbations. For these, the

98

scaling regimes and asymptotics are identical to the VETCflow scaling regimes discussed

in Graham (1998), so we merely restate them here. The method of determining these

scalings is to assume thatWp scales asWemz , and then use the constitutive equations to

determine the stress scalings. These are then substituted into the momentum balance, and

the requirement thatτθθ appear at leading order in the radial momentum balance is used

to solve form. Depending on the value ofα, three scaling regimes exist for axisymmetric

disturbances:

αWez 1 : Wp = O(α1/2We3/2z ), (119)

Wez 1, αWez = O(1) : Wp = O(Wez), (120)

Wez 1, αWez 1 : Wp = O(1/α). (121)

Clearly, it is the second scaling which has the lowestWp of the three regimes. This

means that forWez 1, the most unstable wavenumber will beO(1/Wez), and the

critical Wp will scale linearly withWez. For non-axisymmetric modes, wefirst consider

n = O(1), andα = O(1). For this case, the dominant balance reveals thatm = 1, or

Wp = O(Wez). This is the only regime forn = O(1), and indicates that for a givenn,

Wp scales linearly withWez, as in the axisymmetric case. Note that the scaling for non-

axisymmetric modes differs from the corresponding scaling in VETCflow determined

by Graham (1998), whereWp must beO(We2z) for non-axisymmetric disturbances to

become unstable. Disturbances withn 1 are very strongly suppressed. For example,

whenn = O(ε−1), Wp has to beO(ε−1) to keepτθθ in the momentum balance.

The linear scaling ofWp with Wez has an important practical consequence. Recalling

thatWeθ = Wp ε−1/2, we see that the ratioWez/Weθ, which is a measure of the mag-

nitude of axialflow relative to the azimuthalflow, isO(ε1/2). Sinceε 1, this means

99

that a small amount of axialflow provides significant stabilization. Although we have as-

sumed thatWez 1 in the asymptotic analysis above, numerical computations indicate

that these results are valid down to aboutWez = 1, yet another instance of the power of

asymptotics in supplying information about qualitative behavior.

We now present scaling results for the imposition of time-periodic axialflow. Our

aim here is to determine how the imposition of a time-periodic axialflow changes the

stability characteristics of the system. We have already observed that the stabilization

is due to the stressτzz induced by the axialflow. From Appendix B, we see that the

expression forτzz has aWe2z term multiplied by a factor that is the sum of a constant

term and a term that depends on the frequency and time. From our analysis above, we

know that the constant term has a stabilizing influence. We seek to determine the effect

of the frequency and time dependent terms on the stability characteristics. Wefind that,

depending on the frequency, this term can either stabilize or destabilize theflow.

We first consider the high frequency regime,ω1 = O(ε−1/2), with time rescaled ap-

propriately to reflect the shorter time scales. Further, we also retain the scalings on the

axial and azimuthal Weissenberg numbersWez = Wp = O(1) and restrictα to beO(1).

In this regime, the axial strain rate is large, but the strain amplitude is small (O(ε1/2)),

and wefind that the leading order evolution equation forτzz does not contain theWe2z

contribution that has been established to be the source of high normal axial stresses that

contribute to the stabilization in theflow. TheWe2z term comes from the time-averaged

non-zero base state axial stress, which tends to zero with increasingω andfixedWez.

Thus, we expect decreased stabilization in this regime. Further, theω1 = O(ε−1/2) regime

also lets us apply the method of averaging, a rigorous asymptotic technique (Sanders and

100

Verhulst, 1985), to the time dependent equations. Briefly, the averaging method trans-

forms the system

x = εf(x, t, ε) with f(x, t, ε) = f(x, t+ T, ε), (122)

to an autonomous system

y = ε1

T

∫ T

0

f(y, t, 0) dt = ε f(y). (123)

In other words, the averaging procedure replaces the time-dependent coefficients with

their averages over the period of oscillation. This procedure is valid for arbitrary ampli-

tude of oscillation as long asε 1. It can be shown that in our problem the equations

governing the stress perturbations in this regime are of the form shown in equation 122

with ε = ε1/2, ε being the dimensionless gap width defined by equation 95 in section 4.3.

Importantly, we note the absence of the contribution due to the zero frequency terms in

τzz in the leading order equations forτrz andτzz. The remaining coefficients ofWez are

periodic with zero mean and drop out in the averaging procedure, leaving a system that

is independent ofWez andω1. Hence in theω1 = O(ε−1/2) andWez = O(1) regime,

the system is reduced to the steady Deanflow limit with no axial motion. Our numerical

computations also indicate that in the limit of large frequencies, we recover the steady

Deanflow results.

In the second high frequency regime of interest we investigate the effect ofO(1) axial

deformation on the dynamics of the system. In this regime we haveω1 = O(ε−1/2) and

Wez = O(ε−1/2), which corresponds to high frequency and large deformation. By con-

sidering time scales of the order of the relaxation time of thefluid, we discover the pres-

ence of stress boundary layers near the cylinders of sizeO(1/Wez) for O(1) wavenum-

bers and of sizeO(1/αWez) for O(ε−1/2) wavenumbers. The existence of boundary

101

layers at high shear rates is consistent with the analysis of Renardy (1997) and Graham

(1998). The asymptotic balances reveal that the destabilizing azimuthal stress,τθθ, drops

out of the radial momentum balance, thus resulting in stabilization. Again, computational

results confirm this prediction.

4.5.2 Numerical Results

We begin by presenting results for the steady case. As mentioned in the introduction,

the basic Deanflow profile is invariant under translations and reflections, i.e., it has O(2)

symmetry. Symmetry breaking bifurcations in such systems take the form of pitchforks

or degenerate Hopf bifurcations (Golubitsky et al., 1985). In agreement with this pre-

diction, Joo and Shaqfeh (1991) found that the primary bifurcation in Deanflow was a

pitchfork. When axialflow is added, the reflection symmetry is lost, and theflow is SO(2)

symmetric, and theory (Iooss and Joseph, 1989) predicts that symmetry breaking takes

the form of Hopf bifurcations.

In line with the predictions above, wefind that the primary bifurcation in pure Dean

flow is a pitchfork, and when axialflow is added, symmetry breaking takes the form of a

Hopf bifurcation. Figure 35 shows the results of a linear stability analysis for DACflow

whenS = 0. The curves are plots of the critical value ofWp (denoted byWpc) versus

the wavenumberα for givenWez. For each curve, there is a global minimum value of

Wpc, which we denote byWpc,min, with the corresponding wavenumber denoted byαmin.

ThusWpc,min denotes the smallest value ofWp at which the baseflow becomes unstable,

and the destabilizing disturbance has a wavenumberαmin. Thefigure clearly shows that

Wpc,min increases withWez. A similar plot is shown for DAPflow in figure 36. Here too,

the effect is increased stability at highWez. Note also that in both DAC and DAPflows,

102

αmin decreases with increasingWez. Figure 37 shows a plot ofWpc,min versusWez for

both DAC and DAPflows. In accordance with the asymptotic analysis,Wpc,min increases

linearly with Wez. Note also that in both theseflows,Wpc,min increases monotonically

asWez is increased, in contrast to the initial destabilization observed in circular Couette

flow (Graham, 1998). Solvent viscosity has a stabilizing effect in both DAC and DAP

flows, as indicated infigure 38 which plotsWpc,min versusWez for S = 0 andS = 10.

2.0 4.0 6.0 8.0 10.0α

4.0

6.0

8.0

10.0

12.0

14.0

16.0

18.0

20.0

Wp

Wez=0Wez=1Wez=2Wez=3

Figure 35: Neutral stability curves for DACflow (S=0). In each case, the position ofWpc,min is denoted by a•.

As predicted by the asymptotic analysis, non-axisymmetric modes are similarly sta-

bilized. Solvent viscosity has a stabilizing effect on non-axisymmetric modes as well,

and we only show the results forS = 0 . Plots ofWpc,min versusn for differentWez are

shown infigure 39 for DACflow andfigure 40 for DAPflow. We see increased stabi-

lization asn increases and also asWez increases, in accord with the asymptotic analysis.

Figure 41 shows a plot ofWpc,min versusWez for n = 1. The linear scaling predicted by

the asymptotics is evident.

103

2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0α

4.0

5.0

6.0

7.0

8.0

9.0

10.0

11.0

12.0

13.0

Wp c

Wez = 0Wez = 1 Wez = 2Wez = 3

Figure 36: Neutral stability curves for DAPflow (S=0). In each case, the position ofWpc,min is denoted by a•.

0.0 1.0 2.0 3.0 4.0 5.0 Wez

0.0

5.0

10.0

15.0

Wp c,

min

DAP flow DAC flow

Figure 37: Plot ofWpc,min vs. Wez for DAC and DAPflows (S=0). Note the linearscaling at highWez.

104

0.0 1.0 2.0 3.0 4.0 5.0Wez

0.0

10.0

20.0

30.0

Wp c,

min

Axial Couette (S=0)Axial Couette (S=10)Axial Poiseuille (S=0)Axial Poiseuille (S=10)

Figure 38: Plots ofWpc,min vs. Wez for S = 0 andS = 10, displaying the stabilizinginfluence of solvent viscosity.

0.0 0.2 0.4 0.6 0.8 1.0n

2.5

7.5

12.5

Wp c,

min

Wez=0Wez=0.5Wez=1.0Wez=1.5

~

Figure 39: Plot ofWpc,min vs. n for different values ofWez for DAC flow (S = 0).

105

0.0 0.1 0.3 0.4 0.6 0.8 0.9n

3.0

5.0

7.0

9.0

11.0

Wp c,

min

Wez=0Wez=0.5Wez=1.0Wez=2.0

~

Figure 40: Plot ofWpc,min vs. n for different values ofWez for DAP flow (S = 0).

0.0 0.5 1.0 1.5 2.0Wez

0.0

5.0

10.0

15.0

20.0

25.0

30.0

Wp c,

min

Figure 41: Plot ofWpc,min vs. Wez for DAC flow with n = 1.0 (S = 0). Note the linearscaling forWez > 1.

106

Figure 42 shows a portion of the neutral stability curve for pure Deanflow. We point

out that there is a change in the slope of the curve atα = 21.9. At this point, four eigen-

values with zero real parts are simultaneously neutrally stable. This is a codimension-2

Takens-Bogdanov bifurcation point (Takens, 1974; Bogdanov, 1975; Knobloch and Proc-

tor, 1981; Guckenheimer and Knobloch, 1983). Hereafter, we refer to this value ofα as

αtb. For wavenumbers greater thanαtb, the bifurcation is a Hopf, with a four dimensional

center manifold, similar to the one in circular Couetteflow. As we decreaseα towardsαtb,

the period of the Hopf bifurcation increases, reaching infinity atαtb, where the form of the

bifurcation changes to a pitchfork. From a physical point of view, this bifurcation point

is expected, since large wavenumber disturbances are localized in a small region near the

outer cylinder, and hence experience a base velocity profile that is locally linear, similar

to the profile in circular Couetteflow. Thus, we expect that for large wavenumbers, the

destabilization mechanism, and hence the nature of the bifurcation, will be similar to the

one in circular Couetteflow.

Experimental results by Genieser (1997) on theflow of viscoelasticfluids through a

planar contraction display instabilities that occur as transitions from steadyflow to ei-

ther steady or oscillatoryflow, depending on contraction ratio. The oscillatoryflows

have very low frequency, so it may be that these observations are another manifestation

of Takens-Bogdanov bifurcation and concomitant change in destabilization mechanism

found here. The unfolding of a Takens-Bogdanov bifurcation in an O(2) symmetric sys-

tem was performed by Dangelmayr and Knobloch (1987). They showed that, depending

on the value of the bifurcation parameters, observable patterns include a nontrivial steady

state, traveling waves, standing waves and modulated waves.

The results of the weakly nonlinear analysis atWpc,min andαmin are summarized in

107

20.0 20.5 21.0 21.5 22.0 22.5 23.0 23.5 24.0α

6.5

7.0

7.5

8.0

Wp c pitchfork

(steady)

Hopf

(oscillatory)

Takens−Bogdanov point

Figure 42: Neutral stability curve of pure Dean flow at highα. The Takens-Bogdanovbifurcation point is indicated by a•.

Tables 3 and 4 for DAC flow and Tables 5 and 6 for DAP flow. Negative values ofµ2

indicate subcritical behavior while positive values imply supercriticality. We note that

in the absence of solvent viscosity (S = 0), the bifurcation changes from subcritical

to supercritical atWez ≈ 0.05 while at finite solvent viscosity the change occurs at

Wez ≈ 1. Results for higherWez show that forS = 0 the bifurcations revert back to

being subcritical. Clearly, by varying the extent of axial motion we may not only change

the position of the bifurcation point but also ensure that we stay in a regime that admits

stable solutions. Similar behavior arises for the DAP flow, where there is a change in the

sign ofµ2 at atWez ≈ 0.25 for S = 0, andWez ≈ 1.0 whenS = 1. Here, however,

nonzeroS does not change the criticality of the dominant unstable mode at higher values

of Wez. We note that a change in the criticality of bifurcation has also been observed in

108

Wez αmin Wpc,min µ2 ω20 6.6 4.06 −11.21 0

0.05 6.6 4.07 −6.91 −0.740.1 6.5 4.07 0.27 −0.581.0 4.4 5.25 7.05 −0.482.0 2.9 7.39 26.92 −0.423.0 2.2 9.73 46.21 0.244.0 1.8 12.18 52.65 0.575.0 1.5 14.68 54.15 0.66

Table 3:µ2 andω2 for DAC flow (S = 0).

Wez αmin Wpc,min µ2 ω20 6.3 5.88 −484.67 01.0 4.8 7.03 −38.17 −2.582.0 3.6 9.36 144.25 0.403.0 3.0 11.81 123.18 4.154.0 2.6 14.28 −145.25 4.775.0 2.3 16.76 −528.80 3.69

Table 4:µ2 andω2 for DAC flow (S = 1).

the weakly nonlinear analysis of cone and plate flow of an Oldroyd-B fluid (Olagunju,

1997).

In the final portion of this section, we present results for the imposition of time-

periodic axial flow. As in the steady case, solvent viscosity has no qualitative effect on

the nature of the results, and we confine our discussion to the fluid with zero solvent

viscosity. Addition of solvent viscosity simply results in increased stabilization relative

to S = 0, but does not otherwise change the nature of the results. Also, our analysis

for the imposition of steady axial flow indicates that non-axisymmetric modes are always

more stable than axisymmetric ones, so we will only consider axisymmetric perturbations

here.

The addition of time-periodic axial flow introduces a new parameterω1, which in-

creases the dimension of the parameter space. Scanning the entire regime ofWez, α and

109

Wez αmin Wpc,min µ2 ω20 6.6 4.06 −11.21 00.2 6.5 4.08 −3.95 −0.420.3 6.5 4.10 2.20 −0.141.0 5.6 4.54 7.73 −0.382.0 4.1 5.91 22.55 −2.053.0 3.2 7.60 40.10 −3.654.0 2.6 9.63 47.64 −2.835.0 2.1 11.70 20.18 0.46

Table 5:µ2 andω2 for DAP flow (S = 0).

Wez αmin Wpc,min µ2 ω20 6.3 5.88 −484.67 01.0 5.7 6.42 −95.63 −1.352.0 4.5 7.92 151.33 −1.203.0 3.7 9.89 459.44 7.994.0 3.2 11.99 717.29 19.515.0 2.8 14.24 813.98 25.62

Table 6:µ2 andω2 for DAP flow (S = 1).

ω1 is a computationally demanding task. Since our primary goal is to determine the ef-

fect of the periodic axial flow on the stability of pure Dean flow, we limit ourselves to a

smaller parameter space. For the most part, we chooseWp = 4.06, which corresponds to

Wpc,min for pure Dean flow.

For all the values ofα, Wez andω1 that we considered, we always found increased

stabilization relative to pure Dean flow. Thus, the effect of time-periodic axial flow is

always stabilizing. This can be seen quite dramatically in figure 43, which shows the

decay of the Euclidean norm of the destabilizing perturbation hoop stress when time-

periodic axial flow is imposed. The forcing chosen wasWez = 1 andω1 = 0.5. The

initial condition was chosen to be the eigenvector for pure Dean flow withWp = 4.06

andα = 6.6, which corresponds to the most unstable disturbance in pure Dean flow. A

time sequence of density plots of the perturbation hoop stress,τθθ, corresponding to these

110

parameters is shows in figure 44. Byt = 12T , the perturbation has essentially decayed

to the point where the base flow is recovered. The complete stabilization found in Dean

flow is in contrast to the results reported by Ramanan et al. (1999) for the imposition

of time-periodic axial flow in the circular Couette case. While they find that axial flow

results in increased stability for most cases, they also find that the system shows instability

(synchronous resonance) for certain values ofω1 andWez.

0.0 2.0 4.0 6.0 8.0 10.0t/T

0.0

2.0

t/T

0.0 2.0 4.0 6.0 8.0 10.0t/T

0.0

2.0

t/T

Figure 43: Decay of the perturbation hoop stress τθθ when axial flow is imposed. Param-eters are: Wez = 1.0, ω1 = 0.5, Wp = 4.06, S = 0.

As mentioned in section 4.4, stability requires that the magnitude of the Floquet mul-

tiplier be less than one. We show a plot of the magnitude of the Floquet multiplier, |β|

versus the frequency ω1 for different wavenumbers in figure 45. The other initial condi-

tions were Wez = 0.5, and Wp = 4.06. In each case, the magnitude of the Floquet mul-

tiplier is less than 1, indicating stabilization relative to pure Dean flow. Figure 46 shows

the effect of increasing Wez. The parameters chosen were Wp = 4.06 and α = 6.6. As

in the steady case, increasing Wez increases stabilization, reflected here in the smaller

111

t=0

t=T/4

t=3T/4

t=T t=3T

t=6T

t=9T

t=12T

Figure 44: Time sequence of density plots of the perturbation hoop stress τθθ when axialflow is imposed. The parameters are identical to figure 43, so that without axial flow, theflow is neutrally stable. Each frame shows a z − r cross-section of the geometry.

112

magnitude of the Floquet multipliers for larger axial forcing. The figure also shows that

for large ω1, the Floquet multiplier tends asymptotically to unity. This is in agreement

with the scaling analysis for high frequency discussed in the preceding section, which

indicates that for ω1 1, the O(We2z) contribution to τzz in the base flow has a zero

average over a single period of the forcing, thus reducing the problem to zero axial forc-

ing. For the values of Wp and α we chose in this simulation, the base flow is marginally

stable, and hence |β| → 1.

0.0 0.5 1.0 1.5ω1

0.0

0.2

0.4

0.6

0.8

1.0

|β|

α=2.0α=4.0α=6.6α=8.0

Figure 45: Plot of the magnitude of the Floquet multiplier |β| vs. ω1 for different valuesof α. (Wez = 0.5, Wp = 4.06, S = 0).

In the limit ω1 → 0, we might expect the results from our simulations with time-

periodic forcing to reduce to those with steady axial forcing. It turns out however, that

this is not the case, i.e., the limit ω1 → 0 is singular. This has also been observed in the

stabilization of Newtonian circular Couette flow (Marques and Lopez, 1997), as well as in

the linear stability analysis of other time-dependent flows (Rosenblat, 1968; von Kerczek

and Davis, 1974). Davis and Rosenblat (1977) analyzed a damped Mathieu-Hill equation

113

0.0 0.5 1.0 1.5 2.0ω1

0.0

0.2

0.4

0.6

0.8

1.0

|β|

Wez=0.5Wez=1.0Wez=2.0

Figure 46: Plot of the magnitude of the Floquet multiplier |β| vs. ω1 for different valuesof Wez. (Wp = 4.06, α = 6.6, S = 0). |β| asymptotes to 1 at large ω1 in agreement withthe asymptotic prediction.

with externally imposed modulation and showed that at low frequencies the eigenfunc-

tions have large temporal increases in magnitude within the period of oscillation; these

become unbounded as ω1 → 0. Thus, while the Floquet analysis may predict overall

stable behavior for a suitable choice of parameters at low frequencies, the long periods

of oscillation allow for large transient increases in the response. Since the perturbations

in the system are now no longer infinitesimally small as is required for linear stability

analysis, one would have to resort to a nonlinear theory to be able to adequately represent

the solution of the system. This type of behavior is seen in DMAC flow at small ω1, and

is shown in figure 47, where the Euclidean norm of the periodic component of the hoop

stress is plotted against time (normalized by the period of the oscillation) for ω1 = 0.01.

The hoop stress is normalized with respect to its minimum value. We see in the figure

that there are periods where the stress increases close to 500 times its minimum value in

114

the cycle. In contrast, a similar plot for ω1 = 1.0 (figure 48) does not show this behavior,

with the stress remaining O(1) relative to its minimum value in the cycle. Similar be-

havior has been observed in the context of periodic axial flow applied to circular Couette

flow (Ramanan et al., 1999).

0.0 0.2 0.4 0.6 0.8 1.0t/T

0.0

100.0

200.0

300.0

400.0

500.0

||~ τ θθ||

Figure 47: Plot of the magnitude of the periodic component of the hoop stress τθθ over acycle of the forcing for ω1 1 (Wez = 1.0, Wp = 4.06, α = 6.6, ω1 = 0.01). Note thelarge increases in magnitude.

The results presented above for periodic axial forcing reflect the stability properties of

the forced system relative to the critical Wp of pure Dean flow. We conclude this section

with a sample calculation showing how the minimum critical value of Wp is shifted by the

presence of axial oscillations. For DMAC flow, simulations for S = 0, ω1 = 0.8,Wez =

2.0 yield Wpc,min = 4.37 and αmin = 6.0. This is greater than Wpc,min for pure Dean

flow (4.06), thus indicating increased stabilization. For comparison, the value of Wpc,min

for Wez = 2 in steady axial flow would be 7.39 for DAC flow.

115

0.0 0.2 0.4 0.6 0.8 1.0ω1

1.0

1.0

1.0

1.1

||~ τ θθ||

Figure 48: Plot of the magnitude of the periodic component of the hoop stress τθθ over acycle of the forcing for ω1 = O(1) (Wez = 1.0, Wp = 4.06, α = 6.6, ω1 = 1.0). Themagnitude remains O(1) over the entire cycle.

4.6 Conclusions

In this chapter, we showed that the elastic instability in isothermal Dean flow could be

delayed by the addition of steady axial flow either in Couette or Poiseuille form. Both

axisymmetric and non-axisymmetric disturbances are suppressed. The stabilization is a

result of the additional axial normal stress resulting from the axial flow. We conducted

a weakly nonlinear analysis to determine the criticality of the bifurcation in Dean flow

with and without axial flow. Our results indicate that the bifurcation is subcritical for pure

Dean flow, and the subsequent nature of the bifurcation depends on Wez and S. We also

showed that time-periodic axial Couette flow can be used to stabilize Dean flow. Finally,

we report a codimension-2 Takens-Bogdanov bifurcation point at an axial wavenumber

of 21.9 for Dean flow without axial forcing. This bifurcation point represents a change in

the mechanism of the instability.

116

Chapter 5

Localized solutions in viscoelastic

shear flows†

In the previous chapter, we discussed how the primary instability in Dean flow can be

suppressed by the addition of an axial flow that is small in magnitude when compared to

the primary flow. The tools that we used were asymptotic analysis, linear stability analy-

sis, and weakly nonlinear analysis. These tools are useful in giving us information about

the dynamics close to the point where the flow first loses stability. As recent experimental

work (Groisman and Steinberg, 1997, 1998; Baumert and Muller, 1999; Groisman and

Steinberg, 2000) shows, the dynamics far from the bifurcation point can be very complex.

Of particular interest to us are the observations of stationary, long wavelength structures

in circular Couette flow by Groisman and Steinberg (1997), Groisman and Steinberg

(1998), and Baumert and Muller (1999). This chapter describes how a fully nonlinear

† Most of the material in this chapter has been published in Kumar and Graham (2000a) and submittedfor publication in J. Fluid Mech.

117

analysis can capture the main features of these localized solutions in circular Couette

flow.

5.1 Introduction

Spatially localized structures are common in pattern forming physical systems (Cross

and Hohenberg, 1993). Such patterns are interesting and important because they are an

indication of significant nonlinear effects, and their interaction with other patterns may

give information on spatiotemporal behavior. Examples of oscillatory localized struc-

tures can be found in binary liquid mixtures (Moses et al., 1987; Heinrichs et al., 1987;

Kolodner et al., 1988), parametrically excited surface waves (Wu et al., 1984), elas-

tic media (Wu et al., 1987), granular media (Umbanhowar et al., 1996; Lioubashevski

et al., 1996; Fineberg and Lioubashevski, 1998), and colloidal suspensions (Liouba-

shevski et al., 1999). Recently, stationary, two dimensional finite amplitude localized

states have been computed in Newtonian plane Couette flow (Cherhabili and Ehrenstein,

1995, 1997). These solutions are isolated from the base Couette flow branch and were

computed by numerical continuation of traveling wave solutions in plane Poiseuille flow.

Although unstable, these may be related to coherent structures observed in turbulent plane

Couette flow.

In flows of viscoelastic liquids, long wavelength structures were first observed by

Beavers and Joseph (1974) in a circular Couette device. These structures, termed “ tall

Taylor cells” , are primarily inertia driven patterns (Taylor vortices) modified by elastic-

ity. Similar patterns have been computed by Lange and Eckhardt (2000). In contrast,

the structures seen by Groisman and Steinberg (1997), Groisman and Steinberg (1998),

118

and Baumert and Muller (1999) are driven purely by elasticity, since the Reynolds num-

ber is negligibly small in their experiments. There are three interesting aspects to their

observations: (1) isothermal linear stability analysis in this geometry never predicts sta-

tionary bifurcations, (2) these vortex pairs, dubbed “diwhirls” by Groisman and Steinberg

(1997), and “fl ame patterns” by Baumert and Muller (1999) and are very localized, i.e.,

there does not seem to be a selected axial wavelength for these patterns, and (3) the tran-

sition back to the base Couette flow is hysteretic, i.e., the shear rate at which the Couette

flow base state is recovered is much lower than the onset point at which it loses stability.

Here, we seek answers to the following questions, motivated by these observations: (1)

Do isolated branches of stationary solutions exist in a simple model for a viscoelastic

fluid? (2) Are such solutions, if they exist, localized in space? (3) Can the results from

the computations be used to postulate a self sustaining mechanism for these structures?

We address these questions by fully nonlinear computations of the branching behav-

ior of an isothermal inertialess Finitely Extensible Nonlinear Elastic (FENE) dumbbell

fluid in the circular Couette geometry. Our computations show that an isolated branch of

stationary solutions does indeed exist in the circular Couette geometry. In common with

the experimentally observed patterns (which, adopting the nomenclature of Groisman

and Steinberg (1997), we term diwhirls), they are long wavelength solutions, exhibit sig-

nificant asymmetry between radial inflow and outflow, and show hysteresis. In addition,

these solutions persist at arbitrarily large wavelengths: some of the solutions we have

computed have an axial wavelength that is more than a hundred times larger than the gap

width. We also use the results from our computations to propose a self-sustaining mech-

anism for these patterns. Along with the circular Couette flow base state, these structures

119

may form the building blocks for complex spatiotemporal dynamics in the flow of elas-

tic liquids, such as the recently observed phenomenon of elastic turbulence (Groisman

and Steinberg, 2000). In addition, they may be linked to localized defects seen in poly-

mer processing operations and possibly to the strongly nonlinear and long-wave features

observed in free surface flows (Grillet et al., 1999).

In the previous chapter, we discussed the mechanisms by which elastic instabilities

arise in Dean and circular Couette flows due to the inward radial force associated with ten-

sile stresses along curved streamlines. As mentioned earlier, non-isothermal effects can

give rise to stationary, axisymmetric bifurcations. However, the shear rates at which these

bifurcations occur are an order of magnitude lower that those at which the diwhirls are ob-

served. Therefore, the non-isothermal mechanism appears to have limited relevance for

the diwhirls, but seems to explain the very weak stationary vortices seen experimentally

by Baumert and Muller (1995, 1997). Hereafter we consider isothermal flow.

The absence of a stationary bifurcation from the circular Couette flow base state

means that any branch of stationary solutions that exists in this flow must be isolated

from the base state flow, i.e., there can be no direct path from the base state flow to this

branch of solutions. One way of accessing such an isolated branch is to use a technique

known as “homotopy.” The idea in this technique is to start with a problem different from

the original one, but whose solution has the desired properties. For example, the modified

problem may be easier to solve than the original one. After the solution to the modified

problem is computed, it is tracked as the problem is morphed to the original one. Re-

cently, homotopy has been used to find isolated solutions in plane Couette flow (Waleffe,

1998). For our problem, we seek a flow whose base state has a stationary bifurcation, and

which can be easily morphed to circular Couette flow. Clearly, Dean flow is a very good

120

candidate to satisfy this criterion, since the flow geometry is identical to that of circular

Couette flow, and there is a stationary bifurcation from the base state for a wide range of

parameters. Starting from Dean flow, we can approach circular Couette flow in a smooth

way by progressively decreasing the pressure drop, while simultaneously increasing the

rotation speed of one of the cylinders. The linear stability characteristics of viscoelastic

Couette-Dean flow were studied by Joo and Shaqfeh (1992b). As we might expect, this

flow is unstable to a stationary axisymmetric mode when the pressure gradient is the dom-

inant driving force, whereas a non-axisymmetric oscillatory mode is the most dangerous

when cylinder rotation dominates.

Work on nonlinear analysis in viscoelastic circular Couette and Dean flows has con-

centrated on regimes close to the bifurcation point - there have been no extensive com-

putational studies of fully nonlinear behavior in these flows. Renardy et al. (1996) con-

ducted a nonlinear analysis to study mode interactions arising from the introduction of

inertia into the system. Graham (1998) performed a weakly nonlinear analysis to de-

termine the criticality of the bifurcation in circular Couette flow in the narrow gap limit

upon addition of axial flow. Later, Ramanan et al. (1999) extended this analysis to Dean

flow. Khayat (1999) used a low dimensional model in an attempt to determine the dy-

namical behavior in purely elastic and inertio-elastic circular Couette flow. It should be

noted, however, that stress localization (a striking example of which will be seen below)

is common in viscoelastic flows, and it is questionable whether a simple low dimensional

model, which is essentially a low-resolution Galerkin projection, can adequately capture

such behavior.

The strategy we adopt to search for isolated branches of stationary solutions in cir-

cular Couette flow is a fully nonlinear analysis of the governing equations. We use a

121

numerical continuation procedure (Seydel, 1994) to trace out stationary nontrivial solu-

tions bifurcating from the trivial branch in Dean or Couette-Dean flow and see if these

solutions persist as a parameter is varied smoothly to change the flow from Dean or

Couette-Dean to pure circular Couette flow. Any such stationary solutions that persist in

the limit of circular Couette flow have to be part of an isolated branch since there is no

stationary bifurcation from the base state isothermal circular Couette flow. In the remain-

der of the chapter, we report our procedure and results as follows. Section 5.2 contains a

discussion of the geometry, governing equations, and scalings that are used in the com-

putations. In section 5.3, we present a discussion of the discretization scheme and the

numerical method that we use to solve the sparse linear systems arising in the Newton

iterations during the continuation process. This section includes discussion on a precon-

ditioner that we have found to be especially useful. In section 5.4, we discuss the results

of continuation in the various parameters, mechanism of the diwhirl solutions we com-

pute, their stability with respect to time dependent axisymmetric and non-axisymmetric

disturbances, and present a quantitative comparison of our computed diwhirls with exper-

imental data (Groisman and Steinberg, 1998). Finally, we conclude in section 5.5 with a

summary of our main findings.

5.2 Formulation

We consider the flow of an inertialess polymer solution between two concentric cylinders

(figure 49). The inner cylinder has radius R1 and the outer cylinder has radius R2.The

fluid has a relaxation time λ; the polymer and solvent contributions to the viscosity are

122

ΩR 1

R 2Kθ

Figure 49: Geometry of Couette-Dean flow in an annulus

denoted respectively by ηp and ηs, with the ratio ηs/ηp denoted by S. The solution vis-

cosity ηt, is given by the sum of the solvent and polymer viscosities, ηs + ηp. The flow

is created by a combination of the motion of the inner cylinder at a velocity ΩR1 and by

the application of an azimuthal pressure gradient Kθ = ∂P/∂θ.

The equations governing the flow are the dimensionless momentum and continuity

equations

∇ · τ −∇p+WeθS∇2v = 0, (124)

∇ · v = 0, (125)

where v is the velocity, p is the pressure and τ is the polymer stress tensor. The polymer

molecules are modeled as dumbbells connected by finitely extensible springs. Approx-

imate constitutive equations for this model include the FENE-P equation (Bird et al.,

1987b) and the FENE-CR equation (Chilcott and Rallison, 1988), which were described

123

in chapter 1. In dimensionless form, they are

Weθ

(∂〈QQ〉

∂t+ v · ∇〈QQ〉− 〈QQ〉 · ∇vt − 〈QQ〉 · ∇v

)

+

(〈QQ〉

(1− tr(〈QQ〉)/b)−

I

(1− crtr(〈QQ〉)/b)

)= 0, (126)

where 〈QQ〉 is the ensemble average of the polymer conformation tensor,√b is a dimen-

sionless measure of the maximum extensibility of the dumbbells (the dimensionless form

of Q0 in chapter 1), Weθ is the Weissenberg number, which is the product of the polymer

relaxation time and a characteristic shear rate, and cr is a parameter which takes the value

1 for the FENE-CR model and 0 for the FENE-P model. Values of cr between 0 and 1 do

not correspond to any standard constitutive equation; this parameter merely serves as a

convenient way of performing numerical continuation between the FENE-P and FENE-

CR equations. We have already discussed the behavior of the FENE-P and FENE-CR

models in shear and extension in chapter 1. Here, we simply mention that the FENE-

P model has been found to better approximate the behavior of the kinetic theory based

FENE model in steady shear and elongational flows than the FENE-CR model (Herrchen

and Ottinger, 1997). Given the differences between the FENE-P and FENE-CR models

even in simple flows, we would expect them to exhibit different behavior in complex

flows as well, and our computations confirm this.

For both models, Q20 and the components of 〈QQ〉 are scaled by kT/H , where k

is Boltzmann's constant, T is the temperature, and H is the spring constant. Distance

is scaled by the gap width, and time by the inverse of a characteristic shear rate. Since

the FENE-P model does not yield an analytical solution for the base state in Couette-

Dean flow, we take the characteristic shear rate to be the shear rate at the outer cylin-

der for an Oldroyd-B fluid i.e., 1/b = 0) flowing through the same geometry. The

124

velocity scale is chosen to be the product of the time scale and the gap width. The

polymer stress, scaled by the shear modulus, is obtained from 〈QQ〉 using the relation

τ = 〈QQ〉/(1−tr(〈QQ〉)/b)−I/(1−crtr(〈QQ〉)/b). Other parameters of importance

are the dimensionless gap width ε = (R2 − R1)/R2, and δ, which measures the relative

importance of the pressure gradient as the driving force for the flow, given by

δ =−Kθε

2R2/(2 ηt)

(1− ε)R2Ω−Kθε2R2/(2 ηt), (127)

so that δ = 0 is circular Couette flow and δ = 1 is Dean flow. Explicit forms for

the scalings used are presented in Appendix C. The velocity satisfies no slip boundary

conditions on the walls of the cylinder.

5.3 Discretization and solution methods

Equations 124, 125 and 126 form a set of partial differential equations for the three com-

ponents of the velocity, the pressure, and the six components of 〈QQ〉. We look for

steady, axisymmetric solutions that are periodic in the axial direction with a dimension-

less period (scaled by the gap width) of L, so each variable only depends on two spatial

directions, the radial direction r (shifted and scaled so that r = 0 is the inner cylinder

and r = 1 the outer cylinder), and the axial direction z. In performing the numerical

discretization, we can take advantage of certain symmetry properties of the solutions

we seek. In particular, we take the radial and azimuthal velocities to be reflection sym-

metric about the plane z = L/2, and the axial velocity to be reflection anti-symmetric.

This implies that 〈QQ〉rr, 〈QQ〉rθ, 〈QQ〉θθ, 〈QQ〉zz, and p are reflection symmetric, while

〈QQ〉rz and 〈QQ〉θz are reflection anti-symmetric. Thus, the computational domain is

Γ = 0 ≤ r ≤ 1, L/2 ≤ z ≤ L, which is half the size of the physical domain.

125

Considerable care needs to be exercised when choosing a discretization scheme.

Since we are looking for localized solutions, our primary consideration is to choose a

discretization scheme that can place a high concentration of points in regions of strong

velocity and stress localization. In our work, we experimented with two discretization

schemes. The first scheme that we used was a global spectral method. Spectral meth-

ods enjoy the very desirable property of exponential convergence as long as all the fea-

tures of the solution are captured. For our problem, we used Chebyshev discretization

in both directions. In the axial direction, we implemented the symmetry conditions by

choosing odd Chebyshev interpolants for the reflection anti-symmetric components, and

even Chebyshev interpolants for the reflection symmetric components. Chebyshev in-

terpolants work better than Fourier interpolants in the periodic direction because they

have an uneven distribution of collocation points. Regions at the two ends of the domain

have a higher concentration of collocation points, and we can get an improvement in con-

vergence compared to Fourier method by making these regions coincide with the areas

of stress localization. The early computations were performed with this global method.

However, we found that as the degree of localization of the solutions increased, very high

order polynomials were needed, which resulted in unacceptable increases in memory re-

quirements, and presented difficulties in solving the linear systems associated with the

continuation scheme outlined below due to the poor condition number of the matrices.

Our experience with the spectral method indicated that the appropriate discretization

scheme would permit efficient local concentration of points. To this end, we attempted a

domain decomposition Chebyshev collocation scheme (Canuto et al., 1988) in which we

split the computational domain into four conforming rectangular sub-domains and used

Chebyshev collocation in both directions in each of the domains. The interface conditions

126

for each variable were continuity of the value of the variable and its normal derivative be-

tween the domains. Since the governing equation for 〈QQ〉 is hyperbolic, continituity of

the normal derivative is not technically required for its components. However, we found

that the solutions on the non-trivial branch would not converge unless this condition was

imposed. Even when these conditions were imposed, the performance of this scheme was

worse than the global spectral method. One reason could be that the interface conditions

on the components of 〈QQ〉 impose stronger continuity requirements than needed, thus

inhibiting convergence. Also, there is a degree of arbitrariness in the direction of the

normal derivative at the corner point common to all four domains.

The next scheme that we tried was the spectral element method (Patera, 1984). This

method, which we describe below, allows efficient local refinement by subdividing the

domain into several sub-domains or elements. Within each element, the solution com-

ponents are approximated by tensor products of high order orthogonal polynomials. As

long as the solution components within each element are well resolved, this scheme pre-

serves the exponential convergence properties of the global spectral method. As will

be evident from the description below, interface and corner points are treated naturally

within this formulation: the governing equations at these points are the sum of contri-

butions from all elements that border them. We found that this method was superior to

both the global spectral method as well as the domain decomposition collocation scheme

in terms of resolving the stress and velocity localization. In addition, we found that the

linear systems associated with the continuation scheme presented later in this section are

also better conditioned and more sparse for a spectral element scheme than for a global

spectral method. Finally, the computational domain is rectangular in shape, which makes

this method relatively easy to implement.

127

In the spectral element formulation, we apply Galerkin weighting on the conser-

vation and continuity equations (i.e., the test and weight functions are the same), and

streamline-upwind/Petrov-Galerkin weighting (Brooks and Hughes, 1982; Marchal and

Crochet, 1987) on the constitutive equations. The formulation, which is based on the

weak form of the governing equations, is given by∫Γ

(−p I + τ +WeθS∇v) : ∇u dΓ =

∫∂Γ

u · (−pI + τ +WeθS∇v) · nds, ∀u ∈ U,

(128)

∫Γ

q∇ · vdΓ = 0, ∀ q ∈ Q, (129)

∫Γ

(Weθ

(∂〈QQ〉

∂t+ v · ∇〈QQ〉− 〈QQ〉 · ∇vt − 〈QQ〉 · ∇v

)+

〈QQ〉(1− tr(〈QQ〉)/b)

−I

(1− crtr(〈QQ〉)/b)

)(w + c

′v · ∇w

)dΓ = 0, ∀w ∈W,

(130)

where U ∈ H1(Γ), the space of functions whose first derivatives are square integrable

over Γ, W,Q ∈ L2(Γ), the space of functions which are square integrable overΓ, and c′=

h′/V

′where h

′is a characteristic length scale of an element, and V

′is a characteristic

velocity. We take h′

to be the square root of the area of the element, and V′

to be the

average of the magnitude of the velocity at the four corners of each element. We take

u, q, and w to be the same as the interpolating functions for the velocity, pressure and

polymer stress components. These functions must be chosen to satisfy the symmetry

conditions discussed above.

In each element, the variables are approximated by tensor products of Lagrange poly-

nomials defined on the Gauss-Lobatto-Legendre (GLL) grid. We take the symmetry prop-

erties into account by treating the elements bordering the axial edges in a different way

128

z

r

2.5 3 3.5 4 4.50

0.2

0.4

0.6

0.8

1

Figure 50: A spectral element mesh with 16 axial and 16 radial elements with fifth orderpolynomials in each direction in each domain. Note the dense concentration of pointsnear r = 1 and z = L/2. The high resolution is necessary to capture the intense stresslocalization in these regions.

than the interior elements. Since the GLL grid is defined in the domain −1 ≤ ξ ≤ 1,

we map each interior element to −1 ≤ ξr ≤ 1,−1 ≤ ξz ≤ 1. We map the elements

bordering the left edge (z = L/2) to the range −1 ≤ ξr ≤ 1, 0 ≤ ξz ≤ 1, and use

even axial interpolants for the reflection symmetric components and odd interpolants for

the reflection anti-symmetric components. Similarly, we map the elements bordering the

right edge (z = L) to −1 ≤ ξr ≤ 1,−1 ≤ ξz ≤ 0 and use even or odd interpolants as

appropriate. A sample spectral element mesh is shown in figure 50.

In order to avoid spurious pressure modes, the relative approximation orders of ve-

locity and pressure need to satisfy the Ladyzheskaya, Babuska and Brezzi condition. We

ensure this by using interpolants for the pressure that are based on a grid that has two

129

fewer points in each direction than the velocity grid in each element. We choose the ap-

proximation orders for the components of 〈QQ〉 to be the same as for the velocity. We

perform the integrations in equations 128 to 130 using Gauss-Legendre quadrature on the

GLL grid in each element and construct the final system by direct stiffness summation.

This procedure reduces the system of nonlinear partial differential equations to a system

of nonlinear algebraic equations for the nodal values of the variables on the appropriate

GLL grid in each element. These equations can be written in compact form as

E∂y

∂t= f(y,Weθ, S, ε, b, L, δ). (131)

The matrixE has zeros in the rows corresponding to the momentum and continuity equa-

tions. Steady states correspond to solving

f(y,Weθ, S, ε, b, L, δ) = 0. (132)

Solutions to equation 132 are tracked using a numerical continuation procedure, the

starting point for which is the base state Oldroyd-B solution. This solution is used as an

initial guess for the FENE-P or FENE-CR base state solution and refined using a Newton

iteration. We calculate subsequent points along the branch using a pseudo-arclength con-

tinuation algorithm (Seydel, 1994), which we briefly describe here. Let us denote the set

of the values of the variables at the collocation points by the vector y, and the continua-

tion parameter by µ. Here, µ could be Weθ, b, or L. In pseudo-arclength continuation, we

consider both y and µ to be functions of a step length parameter s. Thus, we can write

the set of discretized equations in the compact form

f(y(s), µ(s)) = 0. (133)

130

Given a point (y0, µ0) on the solution branch, the idea is to find the next point (y1, µ1)

such that, apart from satisfying the governing equations, it obeys an additional constraint

N(y, µ) = y0 · (y − y0) + µ0(µ− µ0)−∆s = 0, (134)

where (y0, µ0) is the unit tangent at the point (y0, µ0) , and∆s is a specified step length.

This equation requires that the next computed solution point lie a distance ∆s from the

current point, in the direction of the tangent to the solution curve. At each step, we use a

Newton iteration to solve the augmented set of equations

f(y(s), µ(s)) = 0, (135)

N(y(s), µ(s)) = 0.

The Jacobian matrix of this system is given by

J =

fy fµ

y0 µ0

(136)

and is not singular at a turning point.

While tracing a solution branch, we check for stationary bifurcations using a test

function method (Seydel, 1994). The test function is a scalar function that changes sign

at a stationary bifurcation point, and is relatively inexpensive to compute. Suppose that

(y0, µ0) is a stationary bifurcation point. Then, it follows that

fy(y0, µ0)h = 0, (137)

where h is the eigenvector corresponding to the zero eigenvalue of fy(y0, µ0). Suppose

now that we are at a point (y, µ) different from the bifurcation point. Then equation 137

with (y0, µ0) replaced by (y, µ) has no nontrivial solution for h. However, we can get

131

a solution to an equation that resembles this closely. We arbitrarily choose two indices l

and k, and require that hk = 1. We do this by replacing the lth row in equation 137 by the

equation etkh = 1, where ek is the column vector with a 1 in the kth position and zeros

elsewhere. After this substitution, equation 137 becomes

J lkh = el, (138)

where J lk is the matrix obtained after performing the indicated substitutions in fy. If we

are exactly at the bifurcation point, then h is simply the eigenvector corresponding to the

zero eigenvalue, normalized so that its kth component is 1. If equation 138 is solved close

to a bifurcation point, then h is a good approximation to the eigenvector corresponding

to the zero eigenvalue. In particular, the scalar function

tlk = etlfy(y, µ)h (139)

is zero at a bifurcation point, and changes sign as a bifurcation point is crossed. We use

tlk with l = k to check for stationary bifurcations.

If a stationary bifurcation is detected, as for example when the trivial branch in Dean

flow becomes unstable, we need to begin tracking the new branch. To compute a first

approximation to a point on the new branch, we use the fact that h closely approximates

the eigenvector corresponding to the zero eigenvalue, and write

z = y + δ0h, (140)

for some small value of δ0, as an approximation to a point on the new branch. However, if

we perform a Newton iteration starting with z as the initial guess, we will likely converge

back to the old branch. Instead, we perform a Newton iteration on the augmented system

132

of equations f(z, µ)

zk − z

= 0 (141)

to solve for a point (z∗, µ∗) on the new branch. In equation 141, we are simply specifying

the value of a solution component on the new branch and solving for the point (z∗, µ∗)

where this holds. The method is especially effective when switching from the base solu-

tion to the nontrivial branch because we can choose k to be a component of vr or vz, both

of which are zero in the base state.

The question of determining stability with respect to oscillatory disturbances can be

divided into two parts. For axisymmetric disturbances, we need to find the eigenvalues ω

of the generalized eigenvalue problem

fy q = ωE q. (142)

If any of the eigenvalues have positive real parts, the solution is unstable with respect to

disturbances that have the same symmetry properties that y does and have wavelengths L∗

such that L/L∗ is an integer, otherwise it is stable with respect to such disturbances. We

do not attempt here the much more demanding task of determining stability with respect

to disturbances of arbitrary wavelength. Since we are only interested in determining

stability, we do not need to find the entire spectrum of eigenvalues. It is only necessary

to check if any of the ω have positive real parts. To do this, we use an Arnoldi scheme,

as implemented in the public domain software package ARPACK (Lehoucq et al., 1997),

to calculate a few such eigenvalues. Since ARPACK does not have a built in option

to calculate a specified number of eigenvalues with positive real parts, we use a spectral

transformation suggested by Christodolou and Scriven (1988). Using this transformation,

133

we find the eigenvalues κ of the matrix P = (E − fy)−1(E + fy). These eigenvalues

are related to the the eigenvalues ω in equation 142 by means of the transformation

κi =1 + ωi

1− ωi

. (143)

The eigenvectors of P are identical to those of equation 142. This transformation maps

the eigenvalues in the left half of the complex plane to the interior of the unit circle.

Thus, the eigenvalues of equation 142 with positive real parts map on to the eigenvalues

of P with the largest magnitude, and are easily found by ARPACK. We should mention

here that the Arnoldi scheme constructs a Krylov subspace by the successive action of P

on a vector. As is evident from the definition of P , the construction of each such vector

requires the solution of a linear system. Thus, the eigenvalue computation is an expensive

process, and we only perform it for a few points.

The process for determining stability with respect to non-axisymmetric modes is

somewhat more complicated. We write the solution vector φ as

φ(r, z, θ) = φ(r, z) + ε∗φ(r, z) exp(c t+ i n θ), (144)

where ε∗φ is a small perturbation, c is the growth rate, and n is the azimuthal wavenumber

of the perturbation, assumed to be an integer. Substituting this in the governing equations,

and retaining terms at O(ε∗) gives a complex generalized problem for the growth rate c:

J u = cE u. (145)

As in the axisymmetric case, we can reduce this to a regular eigenvalue problem using

the spectral transformationK = (E − J)−1(E + J).

At this point, it is clear that every step of the procedure involves the solution of a

134

system of linear equations. Let us denote the generic equation we solve as

Ax = b. (146)

It is well worth expending effort to make the solution process efficient. Firstly, A is a

sparse matrix, so considerable savings in memory result from just storing the nonzero

entries together with integer pointer arrays that store information about the coordinates

of each stored entry. The sparse nature of A also means that a properly implemented

iterative method could be more efficient at solving equation 146 than a direct scheme.

The iterative scheme we use is GMRES (Saad and Schultz, 1986). However, GMRES

will converge only if A is well conditioned, and this is generally far from being true for

the systems we solve. Therefore, we solve a preconditioned system

MAx =Mb, (147)

where M is an approximation to A−1. In the remainder of this section, we present a

discussion on some of the preconditioners that we have used.

The first preconditioner that we describe is the incomplete LU decomposition precon-

ditioner with zero fill level, or ILU(0) for short. This preconditioner is a sparse version of

the full LU decomposition algorithm, with sparsity being preserved by only keeping those

entries in theL andU matrices where the corresponding positions inA are nonzero. This

technique gives no consideration at all to the size of the entries that are dropped, so we

would not expect this to be a good preconditioner. However, our experience has been

that this preconditioner is surprisingly robust, and works reasonably well up to b = 1500,

where b is the finite extensibility parameter. At this value of b, we need about twenty

five thousand degrees of freedom to resolve the components of 〈QQ〉, and solving these

linear systems takes about 450 GMRES iterations. At larger values of b, however, ILU(0)

135

fails due to numerical instability: the dropping of entries causes the buildup of very large

or very small numbers in the LU decomposition, which makes it impossible to complete

it. Pivoting actually makes the situation worse by giving rise to zero pivots. Although

we can complete the decomposition by replacing these with small numbers, the resulting

matrix is useless as a preconditioner.

The discussion above suggests that a preconditioner that drops entries based on nu-

merical size would be more effective than ILU(0). A variant of ILU which takes this into

account is threshold ILU, or ILUT (Saad, 1996). In this method, entries in the incomplete

LU decomposition are dropped based on tolerances applied at two different stages. The

algorithm is shown below, where we denote the dimension of the matrix by N and use

the symbols l and u to represent the entries in the L and U matrices respectively. In each

entry, the first subscript represents the row, and the second represents the column. The

algorithm used a one-dimensional work array of length N , which we denote by w below.

1. Do i=1, N

2. w:=ai,∗ (Copy the nonzero entries of row i into w)

3. Do k=1, i− 1 and when wk = 0

4. wk := wk/ukk

5. Apply a dropping rule to wk

6. If wk = 0 then

7. wj = wj − wk ∗ ukj

8. EndIf

136

9. EndDo

10. Apply a dropping rule to row w

11. lij = wj for j = 1, . . . , i− 1

12. uij = wj for j = i, . . . ,N

13. w := 0

14. EndDo

The decomposition is sparse because of the dropping rules applied in steps 5 and

10. The usual choice in step 5 is to keep entries which are larger than some fraction

(tol) of the norm of the row in the original matrix. In step 10, it is usual to keep the

p entries which have the largest magnitude in order to control the amount of fill in. In

using ILUT, we found that the algorithm would generate zero pivots unless p was made

unacceptably large. Again, we can complete the decomposition by replacing these pivots

with small numbers, but the resulting preconditioner is completely ineffective. We found

that this problem could be overcome by performing ILUT on a modified version of A.

We construct this matrix, denoted by A, by setting to zero inA the entries corresponding

to the velocities and pressure in the rows corresponding to the constitutive equation, and

the entries corresponding to the components of 〈QQ〉 in the rows corresponding to the

momentum and continuity equations. We then construct an ILUT decomposition of A

and apply this as a preconditioner for A. We find that this is a very effective technique,

and the largest linear systems that we have solved (O(60000) unknowns) converge to a

relative accuracy of 10−6 in about 350 GMRES iterations, with the preconditioner having

about three times the number of nonzero entries that A does. Smaller systems converge

137

faster and fewer nonzero entries can be kept in the preconditioner. Henceforth, we shall

refer to this preconditioner as ILUT∗. Figure 51 shows a comparison between ILU(0) and

ILUT∗ for a sample problem, which clearly demonstrates the superiority of the ILUT∗

preconditioner over ILU(0).

0.0 50.0 100.0 150.0 200.0 iterations

10−8

10−6

10−4

10−2

100

Log

10(r

esid

ual)

ILUT*

ILU(0)

Figure 51: Comparison of ILUT∗ and ILU(0) preconditioners. The test problem was thecalculation of the unit tangent for a point on the nontrivial branch in Dean flow. Thematrix A had a dimension of 21987.

In using ILUT∗ to solve the complex generalized eigenvalue problem indicated by

equation 145, we need to modify the problem so that it only involves real numbers. Recall

that the spectral transformation requires the solution of the linear system

(E − J)x = b (148)

for each iteration of ARPACK. Instead, we rewrite equation 145 as a real valued problem

c

E 0

0 E

ur

ui

=

Jr J i

J i Jr

ur

ui

, (149)

138

where the subscripts r and i denote respectively the real and imaginary parts of the vec-

tor or matrix. The spectral transformation now requires the solution of linear systems

involving the matrix

S =

(E − J)r (E − J)i

−(E − J)i (E − J)r

,

which is real. To precondition this matrix, we note that the matrix (E−J)i has far fewer

entries than (E−J)r. Therefore, as a first approximation, we can neglect it in computing

the preconditioner. We construct the preconditioner by performing ILUT∗ on (E − J)r.

If M∗ represents this decomposition, we precondition S using the matrix M ∗−1 0

0 M ∗−1

.

5.4 Results and discussion

5.4.1 Stationary bifurcations from the Dean and Couette-Dean

flows

Rather than explore large volumes of parameter space, we restrict our attention to values

close those used in the experiments by Groisman and Steinberg (1997). Specifically, we

fix the value of S at 1.2, and except when we examine the effect of varying the gap width,

set ε = 0.2. For most of our work, we use the FENE-P model which has been shown

to be a better approximation to the exact FENE model (Herrchen and Ottinger, 1997), at

least in simple flows.

In order to track stationary nontrivial branches, it is first necessary to find out from

where they bifurcate. Therefore, the logical starting point of our investigation is the

139

linear stability diagram for Dean flow. Figure 52 shows such a diagram computed using

the FENE-P model. Takens-Bogdanov points, where the bifurcation switches from a

stationary mode to an oscillatory mode, are marked as “TB” . Unlike in the Oldroyd-B

model, where one such point is seen only at very small wavelengths (Ramanan et al.,

1999), we see that, as the polymer becomes stiffer (i.e., b decreases), these points are

shifted to larger wavelengths. We also see that for a sufficiently small value of b, there is

no stationary bifurcation from the base state at all. From the point of view of numerical

simulation, we would prefer to work with as small a value of b as possible, since we would

expect stress boundary layers to be less sharp for smaller values of b, which in turn makes

the computations easier. However, decreasing b tends to lower the elastic character of the

fluid and suppresses elastic instabilities. Before moving on to the nontrivial solutions, we

present some linear stability results for Dean flow of the FENE-CR model in figure 53.

Note that, unlike in the FENE-P model, stationary bifurcations are seen even at low values

of b. Thus, the FENE-CR model predicts linear stability behavior that is qualitatively

different from that predicted by the FENE-P model. We will have more to say on the

differences between the two models in section 5.4.3. For now, we simply concentrate on

the FENE-P model.

5.4.2 The branch structure of viscoelastic Couette-Dean flows

We begin the nonlinear analysis by tracking the bifurcating branch of stationary solu-

tions in Dean flow at b = 700 and L = 1.05. Since this value of b is small, numerical

continuation is not difficult and a crude numerical scheme suffices. We performed these

calculations using a global Chebyshev collocation scheme in both the radial and axial

directions. At L = 1.05, a pair of complex conjugate eigenvalues crosses the imaginary

140

3.5 4.5 5.5α

15.0

20.0

25.0

30.0

35.0

40.0

We θ

b=600b=700b=800b=900

TBTB

TB

Unstable

Stable

Figure 52: Linear stability curves at δ = 1 (Dean flow) computed using the FENE-Pmodel. The points marked “TB” are Takens-Bogdanov points. The lines correspond topoints where the base state flow loses stability to stationary axisymmetric perturbations.

4.0 4.5 5.0 5.5 6.02π/L

15.0

19.0

23.0

27.0

31.0

We θ

b=500b=600b=700

Figure 53: Linear stability curves at δ = 1 (Dean flow) computed using the FENE-CRmodel. As in figure 52, only stationary axisymmetric perturbations are considered. Notethe complete absence of non-stationary bifurcations.

141

axis at Weθ = 28.73. Upon further increasing Weθ, the two unstable eigenvalues coalesce

and form a pair of unstable real eigenvalues which then move in opposite directions. The

smaller one of these re-crosses the imaginary axis at Weθ = 29.57. We track in Weθ the

stationary branch bifurcating as result of this crossing. The result is shown in figure 54,

where we plot the solution amplitude, measured by the quantity

‖ vr ‖=

(Nr∑i=0

Nz∑j=0

|vr,ij|2

)1/2, (150)

as a function of Weθ, where Nr +1 and Nz +1 are the number of Chebyshev collocation

points used in the radial and axial directions respectively. We see that the bifurcation

is mildly supercritical, but quickly turns back and shows a marked hysteretic character.

The turning point at Weθ = 22.34 is much lower than the value of 28.73 where the base

solution loses stability. When we pick a point on this branch and continue it down in

δ, we find however, that it does not extend all the way to δ = 0. For instance, picking

Weθ = 23.35 on the upper branch in figure 54 and tracking it in δ, we find that the branch

turns back at δ = 0.69. We have tried this for other points as well, but in all cases, they

turn back well before reaching δ = 0. Therefore, at least for this value of b and L, there is

no direct path from nontrivial solutions in Dean flow to those (if any) in circular Couette

flow.

Given the apparent absence of a direct route from δ = 1 to δ = 0, we focussed

on smaller values of δ and larger values of b and L. As b increases, the solutions be-

come more localized, and the global scheme we originally used is inefficient. Hence, we

switched to the spectral element/SUPG method described in section 5.3.

At δ = 0.576, b = 1830, and L = 2.71, a stationary bifurcation occurs at Weθ =

25.15. We tracked the bifurcating branch at this value of Weθ up to L = 3.07 and then

142

22.0 24.0 26.0 28.0 30.0Weθ

0.00

0.02

0.04

0.06

0.08

||vr||

Hopf point

Figure 54: Continuation in Weθ of a stationary solution in Dean flow. The parametervalues are L = 1.05, b = 700, ε = 0.20, and S = 1.2. At the Hopf point, a pairof complex conjugate eigenvalues become unstable. These collide and form two realeigenvalues, one of which re-crosses the imaginary axis at Weθ = 29.57, where thestationary branch originates. The solution amplitude used here differs from that used insubsequent figures and is defined in equation 150.

143

−0.10 0.10 0.30 0.50δ

0.000

0.002

0.004

0.006

0.008

0.010

||vr||

Multiple steady states incircular Couette flow

Continuationto L=2.71.

δ=0

Figure 55: A path to stationary solutions in circular Couette flow. The parameters areWeθ = 25.15, L = 3.07, b = 1830, ε = 0.2, and S = 1.2.

down in δ. This path is shown in figure 55. The velocity norm used in this and all

subsequent figures is defined as

‖ vr ‖=

(∫Γ

v2r dΓ

)1/2. (151)

As figure 55 shows, we found that this branch persists all the way to δ = 0. This compu-

tation demonstrates that an isolated branch of nontrivial solutions does indeed exist in the

circular Couette geometry. This is the first time that stationary nontrivial solutions have

been computed in zero Reynolds number circular Couette flow.

5.4.3 Nontrivial stationary solutions in Couette-Dean flow -

Diwhirls

We now proceed to discuss the effect of changing parameters on the stationary solution

at δ = 0. The first parameter we focus on is the wavelength. The results of continuing

144

our solution at Weθ = 25.15 in L are shown in figure 56. The upper branch, being a

stronger flow, is much harder to track than the lower branch, and the end point of this

branch represents the largest value of L at which we could obtain converged solutions

on the upper branch for this value of Weθ. The lower branch presents fewer problems

and we were able to track it with relative ease. The key observation from figure 56 is

that as L increases, both the lower and upper branches become flat, suggesting that the

spatial patterns are becoming independent of the size of the computational domain, i.e.,

they are becoming localized. Examination of the solution components confirmed that this

was indeed the case, with the localization occurring in the region near z = L/2. Since

the components of the solution show little or no axial variation far away from z = L/2,

we can simply use their values at the collocation points for a lower value of L as an

initial guess for the solution at a larger value of L, while increasing the axial extent of

the domains bordering the edges (i.e., z = L). This method of remeshing captures the

localization effectively and avoids the necessity of computing solutions at intermediate

values of L. Using this technique, we were able to get converged solutions on the lower

branch for wavelengths that are in excess of 100 times the width of the gap between the

cylinders. We have also used this technique to compute such long wavelength solutions

on the upper branch at lower values of Weθ. We show the results of one such computation

in figure 57. This figure shows the streamfunction contours and a density plot of 〈QQ〉θθ

at L = 116.52 and Weθ = 24.29. For clarity, we only show the center and edges of

the domain. The streamfunction contours are strongly localized near the center of the

flow cell, which is a region of very strong inflow. Away from the core is a region of

weak outflow, and even further away, the solution is pure circular Couette flow. The

〈QQ〉θθ field shows an even stronger localization. It is the necessity of capturing this

145

2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0L

0.000

0.002

0.004

0.006

0.008

0.010

0.012

||vr||

Figure 56: Results from continuing the stationary circular Couette flow solutions in L.The parameters are Weθ = 25.15, b = 1830, ε = 0.2, and S = 1.2. The gaps in thelower branch correspond to places where we changed the mesh. Note the flatness of thebranches as L increases. We have computed extensions of the upper branch at lowervalues of Weθ.

strong localization that requires the use of a numerical method that permits efficient local

refinement. The streamlines at the core are remarkably similar to those in figure 10 on

page 2457 of Groisman and Steinberg (1998). Henceforth, we will call our solutions

diwhirls as well.

Figure 58 shows the results of continuation in Weθ for solutions at three different

values of L. We see that all three curves show turning points in Weθ, i.e., there is a lower

limit in Weθ below which the diwhirls are not seen. Note that the curves at L = 9.11

and 4.74 are close together, and are both well separated from the curve at L = 3.07.

This further highlights the independence of the solutions on L for large enough L. At

L = 3.07, the critical Weθ at which the base circular Couette flow solution loses stability

to an axisymmetric time-dependent mode is 29.65, which is significantly higher than the

146

L=116.52

2.8

1

1

Outer cylinder

Inner cylinder

Outer cylinder

Inner cylinder

Figure 57: Density plot of 〈QQ〉θθ (white is large stretch, black small) and contour plotof the streamfunction at L = 116.52 (Weθ = 24.29, b = 1830, S = 1.2, and ε = 0.2).For clarity, most of the flow domain is not shown. Note the very strong localization of〈QQ〉θθ near the center. The maximum value of 〈QQ〉θθ at the core is 1589 which givesτθθ = 12722. Compared to this, the maximum value of 〈QQ〉θθ in the circular Couettebase state is 706, which gives τθθ = 1150. Away from the core of the diwhirl, the structureis pure circular Couette flow. The streamlines show striking similarity to those in figure10 of Groisman and Steinberg (1998). This point was generated by stretching the pointat the corresponding Weθ on the upper branch of the curve for L = 9.11 in figure 58.

147

23.0 23.5 24.0 24.5 25.0Weθ

0.000

0.002

0.004

0.006

0.008

0.010

||vr||

L=9.11L=4.74L=3.07

Figure 58: Diwhirl solution amplitudes as functions of Weθ and L. Note that the curvesat L = 9.11 and L = 4.74 are very close together, while both curves are well separatedfrom the curve at L = 3.07 (b = 1830, S = 1.2, and ε = 0.2).

location of the turning point, located at 24.98. The critical Weθ for the other values of L

are even higher. Therefore, the overall bifurcation structure shows hysteresis.

In figure 59, we show a plot of the location of the turning point in Weissenberg num-

ber (Weθ,c) as a function of the wavelength. The most interesting feature in figure 59 is

the flatness of the curve at large L, indicating yet again that for large L, the character-

istics of the solution are independent of the wavelength. Another interesting feature in

figure 59 is that the curve shows a minimum, i.e., the diwhirl patterns exhibit wavelength

selection. This minimum, which occurs at a Weissenberg number of approximately 23.3,

is therefore the lowest Weissenberg number at which the FENE-P model with the cho-

sen parameters predicts diwhirls to occur. More important than the absolute value is the

relative position of the turning point and linear stability limits. The base state circular

Couette flow is unstable with respect to axisymmetric disturbances above Weθ = 20.37.

148

This means that all the solutions that we compute lie above the linear stability limit of

circular Couette flow. In contrast, Groisman and Steinberg (1998) observe diwhirls at

Weissenberg numbers as low as 10, well below the linear stability limit of the base flow.

One reason for this discrepancy could be the approximate nature of the FENE-P model,

which does not take into account the internal degrees of freedom of a real long chain

polymer. Yet another could be that our numerical simulations have not been able to ac-

cess a sufficiently high value of b. In figure 60, we plot the position of the turning point

in Weθ as a function for b for L = 4.74, which is close to the minimum in figure 59. Also

plotted in the figure is the minimum critical value of Weθ at which the base state circular

Couette flow loses stability with respect to axisymmetric perturbations. This figure shows

that as b increases, the position of the turning point shifts to lower Weissenberg numbers

at a faster rate than the shift in the minimum of the linear stability curve. This result is

not unexpected, because the polymer molecules are much more highly stretched in the

core of the diwhirl than in the base state. Therefore, we would expect the nonlinearity

of the FENE-P spring law to have a greater effect on the diwhirls than on the base state

Couette flow. It is conceivable to suppose, based on the results shown in figure 59, that

the two curves would cross at larger values of b (which we are not able to access due to

limitations in the numerical scheme) and that the diwhirls would come into existence be-

low the linear stability limit of the base flow. We point out here that Baumert and Muller

(1999) and Groisman and Steinberg (1997) performed their experiments with very high

molecular weight polymers, for which the values of b are likely to be much higher than

we have been able to access in our simulations.

We now present a rough quantitative comparison of our patterns with those from

the experimental observations of Groisman and Steinberg (1998). In figure 9 of their

149

2.0 4.0 6.0 8.0 10.0 12.0 14.0L

23.0

23.5

24.0

24.5

25.0

We θ,

c

diwhirls

no diwhirls

Figure 59: Plot of the location of the turning point, Weθ,c, versus L at S = 1.2 andε = 0.2. Note the flatness of the curve at large L.

paper, Groisman and Steinberg present the radial velocity profile as a function of z at a

constant radial position near the middle of the gap, where vr has maximum amplitude.

To compare our results with this figure, we chose a point on the upper branch of the

curve for L = 4.74 with Weθ = 23.50 in figure 58 of this work. We then converted

our radial velocity into dimensional units by using values for the physical parameters

from Groisman and Steinberg (1998). Specifically, the values we used were λ = 1 s

and R2 = 41 mm. For ε = 0.2 used in our computations, this gives a gap width of

8.2 mm, slightly higher than the 7 mm gap used in the experiments. In figure 61(a), we

present a profile of the radial velocity as a function of z for r = 0.6, where the radial

velocity is maximum. The peak inflow velocity we find is 5.9 mm/sec, which should

be compared to the value of 3.8 mm/sec that Groisman and Steinberg (1998) show in

their figure 9, which we reproduce here as figure 61(b). The qualitative and quantitative

150

1750 1770 1790 1810 1830 b

19.0

20.0

21.0

22.0

23.0

24.0

25.0

26.0

27.0

We θ,

c

diwhirl onset

linear onset

Figure 60: Plot of the position of the linear stability limit in circular Couette flow withrespect to axisymmetric disturbances and the turning point in Weθ for the diwhirls as afunction of b. The parameters are S = 1.2 and ε = 0.2. The computations for the diwhirlswere performed at L = 4.74, which is close to the minimum in figure 59.

151

similarity between these two values is remarkable, more so when we consider how simple

the FENE-P model is, and that the Weissenberg number used in figure 61(a) is roughly

twice that at which Groisman and Steinberg (1998) report their results, which means a

larger radial velocity should be expected. Furthermore, the wiggles at the shoulders of

the peak in figure 61(a) are not numerical artifacts; similar features in figure 61(b).

0 1 2-1-2

1

0

-1

-2

-3

-4

-5

-6

Vr

, mm

/s

z , d

(a) (b)

Figure 61: (a) The axial variation of vr at r = 0.6 for L = 4.74 and Weθ = 23.50 on theupper branch. (b) Figure 9 on page 2457 of Groisman and Steinberg (1998), shown herefor purposes of comparison. We have shifted the axial coordinate so that the symmetryaxis of the computed diwhirl in (a) is at z = 0, to make comparison with (b) easier

The dimensionless gap width or curvature ε plays a critical role in generating elastic

instabilities. Based on the generic mechanism of elastic instabilities, we expect a decrease

in curvature to have a stabilizing effect, i.e., keeping other parameters fixed, we would

expect the diwhirl pattern to vanish at small enough values of ε. In figure 62, we show

the dependence of the diwhirl solution amplitudes on ε. In agreement with expectations,

we observe turning points as ε decreases. The role of streamline curvature will become

clear in section 5.4.4 where we propose a mechanism for the diwhirls.

As mentioned in section 5.1, an important reason for attempting to numerically simu-

late experimentally observed flow patterns is to determine whether a constitutive equation

152

0.192 0.194 0.196 0.198 0.200ε

0.000

0.002

0.004

0.006

0.008

0.010

||vr||

L=4.74L=9.11

Figure 62: Variation of solution amplitudes with ε. Here, Weθ = 25.15, and the otherparameters as in figure 58.

can model complex flows of viscoelastic liquids. Both the FENE-P and the FENE-CR

equations are derived by applying closures to the evolution equation for 〈QQ〉 for a dilute

solution of noninteracting dumbbells connected by nonlinear springs. We have already

seen that the linear stability curves predicted by the FENE-P and FENE-CR models show

significant differences, and we have seen that the FENE-P model equation has station-

ary solutions in circular Couette flow, which indicates that it is able to capture, at least

qualitatively, the mechanism behind the diwhirls. A natural question to ask is whether

the FENE-CR model can do so as well. To this end, we perform a continuation of the

diwhirl solutions in the parameter cr, starting from the FENE-P solutions (cr = 0). If the

solutions persist at cr = 1, then we will have obtained solitary solutions for the FENE-

CR model in circular Couette flow. Figure 63 shows the results of these computations.

Both values of L that we chose exhibit turning points at small values of cr, indicating that

these solutions do not exist for the FENE-CR model, at least for the parameter values

153

0.0 0.2 0.4 0.6 0.8 1.0cr

0.000

0.002

0.004

0.006

0.008

0.010

||vr||

L=3.57L=4.74

Figure 63: Diwhirl solution amplitudes as a function of the parameter cr for two differentwavelengths. Weθ = 25.15 and the other parameters are as in figure 58. The existenceof turning points demonstrates that these solutions cannot be extended to the FENE-CRmodel.

that we have chosen. Note, though, that in the limit of b → ∞, both the FENE-P and

FENE-CR models are equivalent to the Oldroyd-B model. Hence, at sufficiently large

values of b, the FENE-CR model should yield these localized solutions as well. What

these computations show is that the parameter values at which the solutions come into

existence depends very strongly on the details of the model being used.

The work of Al-Mubaiyedh et al. (2000) has shown that non-isothermal effects intro-

duce a new mode of instability in circular Couette flow. While this mode of instability

does not appear to be relevant for the diwhirls, it is still of interest to quantify the effect

of viscous dissipation at the core of the diwhirl, where we would expect dissipation to be

largest. In dimensionless form the viscous dissipation, scaled by the product of the shear

154

modulus G and the characteristic shear rate γs, is given by the relation

Φ =(τ +WeθS(∇v +∇v

t)):∇v. (152)

Here Φ represents the work done by viscous dissipation per unit time, per unit volume

of fluid. In figure 64, we show a density plot of Φ for a point on the upper branch at

L = 4.74. For the sake of clarity, we have only shown a twentieth of the wavelength,

centered around the symmetry plane. As expected, Φ has its largest amplitude in the core.

2.25 2.35 2.450

0.2

0.4

0.6

0.8

1

z

r

Figure 64: Intensity plot of the dimensionless viscous dissipation for Weθ = 23.73 on theupper branch at L = 4.74. Light areas represent areas of large viscous dissipation anddark areas represent regions of low viscous dissipation. The horizontal axis is stretchedby a factor of two relative to the vertical axis for clarity.

To get an estimate of the magnitude of the viscous dissipation, we can compute the

work done on a fluid element as it traverses the core of the diwhirl, and use that to calcu-

late its temperature rise in the absence of heat loss to the neighboring fluid. This can be

155

calculated using

Ev =

∫ t1

t0

Φdt. (153)

Since the diwhirl is axisymmetric and vz = 0 at the symmetry plane, we can replace dt

by dr/vr and write

Ev =

∫ r1

r0

Φ

vrdr. (154)

Once Ev is known, we can calculate the temperature rise using

∆T =Gγs(R2 −R1)Ev

ρCp

, (155)

where ρ is the mass density of the polymer solution, Cp its specific heat capacity at

constant pressure on a mass basis, and the product Gγs(R2 − R1) is required to convert

Ev to dimensional form. We performed this integral using Simpson's rule for the data

in figure 155 choosing (arbitrarily) r0 = 0.98 and r1 = 0.2. From the data in Groisman

and Steinberg (1998), we use a solvent viscosity of 0.1 Pa s, and a relaxation time of

1 s. As a first (and rough) approximation, we assume the heat capacity and density of the

sugar syrup to be the same as that of water (≈ 4.186× 103J kg−1C−1 and 1000 kg m−3

respectively). This yields a temperature rise of about 10−4C in one cycle. For a PIB/PB

Boger fluid such as the one used in the experiments by Baumert and Muller (1999),

however, the temperature rise would be much higher. For instance, using the data for

their medium viscosity Boger fluid (ηs = 6.5 Pa s, λ = 0.23 s, ρ = 880 kg m3, and Cp ≈

2.0 × 103J kg−1C−1) yields a temperature rise of 0.27C if the geometrical parameters

are taken to be the same as those used by Groisman and Steinberg (1998). Given the

low magnitude of the temperature rise in the PAA Boger fluids, it would appear that

156

the experiments of Baumert and Muller (1999) are more likely to be subject to non-

isothermal effects than those of Groisman and Steinberg (1998). However, since similar

structures are seen in both fluids, this would seem to indicate that non-isothermal effects

do not significantly influence either the diwhirls or the flame patterns.

5.4.4 Self-sustaining mechanism

Since there is no stationary bifurcation in circular Couette flow, the diwhirl solutions that

we have computed are part of an isolated branch that does not connect directly to the base

state. Therefore, the sustaining mechanism for these patterns must be inherently nonlin-

ear. Following their experiments, Groisman and Steinberg (1998) proposed one such

mechanism. They argued that the difference in symmetry between inflow and outflow

results in the elastic forces performing net positive work on the fluid. While this argu-

ment shows that finite amplitude stationary structures that exhibit significant asymmetry

between inflow and outflow are physically plausible, it does not explain the mechanism

by such structures sustain themselves. Having the detailed velocity and stress fields avail-

able to us from our computations, we propose a more complete mechanism. Figure 65

shows a vector plot of v at the axial centerline of the vortex. The azimuthal velocity field

has a parabolic structure near the outer cylinder, similar to the velocity field in the outer

half of the channel in Dean flow. This velocity field results in an unstable stratification of

azimuthal normal stress (Joo and Shaqfeh, 1992b). We therefore propose the following

fully nonlinear mechanism for the diwhirls: a finite amplitude perturbation near the outer

cylinder results in a locally parabolic velocity profile. This in turn creates an unstable

stratification of hoop stress (visible in figure 65(b) for r <∼ 0.99), as in Dean flow, which

drives inward radial motion. As the fluid moves inward, it accelerates azimuthally due to

157

0.97

1.0 1.0

r r

0.97θ θ

(a) (b)

Figure 65: (a) Vector plot of v near the outer cylinder at the center of the diwhirl structure(oblique arrows) and the base state (straight arrows). The length of the arrows is propor-tional to the magnitude of the velocity. The axial velocity is identically zero in the basestate, and is zero by symmetry at the center of the diwhirl. (b) Principal stress directionsat the same location as for (a). The Couette flow stress is not shown because it is verysmall in comparison. This figure shows how fluid elements at larger radii are pulled downand forward sustaining the increase in vθ.

the base state velocity gradient. The azimuthal tension in polymer chains drags the fluid

at larger r forward and down (figure 65(b)). This maintains the increase in vθ and results

in a self-sustaining mechanism (figure 66). Thus we see that, in common with other elas-

tic instabilities, the mechanism is based on the inward radial force associated with tensile

stresses along curved streamlines. This mechanism shows that we have come a full cir-

cle. We began our search for stationary, finite amplitude solutions in circular Couette

flow by starting from Dean flow. We now find that these solutions sustain themselves by

a mechanism similar to the one resulting in the primary instability in Dean flow.

158

local increase in vθ near outer cylinder

unstable stratificationof τθθ, generates inward radial motion

fluid moves azimuthally due to base state shear

tension in the streamlines pulls fluid near the outerouter cylinder forward and downFigure 66: Nonlinear self-sustaining mechanism for the diwhirl patterns.

5.4.5 Stability

We now address the stability of the diwhirl patterns with respect to axisymmetric and

non-axisymmetric perturbations. We determine this by finding the eigenvalues ω in equa-

tion 142 for axisymmetric perturbations or equation 145 for non-axisymmetric perturba-

tions. These computations are expensive, so we only perform them for a few points along

the upper branch for L = 4.74 in figure 58. Here, we report the results from one such

computation, performed at Weθ = 23.87. To resolve this point, we needed 16 radial

elements and 14 axial elements using fifth order polynomials in both directions in each

element, which is a slightly coarser mesh than the one shown in figure 50, and results in

a system with 53687 unknowns for axisymmetric disturbances, and double that number

for non-axisymmetric perturbations. The computation requires the storage of two sparse

matrices, the preconditioner for one linear system, and the Krylov basis. This was well

above the storage capacity available to us on a single computer, so we stored the Krylov

basis and one of the matrices on one machine and the second matrix and preconditioner

159

on another, and used Message Passing Interface (MPI) subroutines to communicate be-

tween the two machines. We used a 600 vector basis (ARPACK would not converge if

significantly fewer vectors were used) and asked for the most unstable eigenvalues.

For axisymmetric disturbances, we found two pairs of complex conjugate eigenvalues

that had positive real parts. The eigenvectors corresponding to one pair had an irregular

grid scale structure, suggesting that they are part of the continuous spectrum of eigen-

values (Graham, 1998; Wilson et al., 1999; Renardy, 2000). These modes are expected

to be stable, but since the eigenvectors are nonintegrable (Graham, 1998), they will not

converge exponentially in a spectral element scheme and can display spurious instability.

The structure of the other two eigenvectors is shown in figure 67. They show that although

the branch is unstable, the destabilizing disturbance has significant amplitude only near

the ends of the domain, where the flow is essentially circular Couette flow. The base

circular Couette flow has a minimum critical Weθ of 20.37 with respect to axisymmet-

ric disturbances, and so is linearly unstable at Weθ = 23.87. Hence, it is not surprising

that the portion of flow pattern where the flow is essentially circular Couette would be

susceptible to destabilizing disturbances. What is interesting, however, is that the core of

the pattern, where the diwhirl lies, is entirely unaffected. This shows that the diwhirl pat-

tern is dynamically distinct from the oscillatory finite wavelength axisymmetric pattern

arising from the linear instability of circular Couette flow.

For non-axisymmetric disturbances, the stability picture is slightly more complicated.

We performed computations with n = 1 and n = 2 and found three pairs of unstable

complex conjugate eigenvalues. Their growth rates are summarized in table 7, while

their structures are shown in figure 68 and 69. For n = 1, we see that there are two

modes that have their largest amplitude close to the core of the diwhirl (figures 68 (a) and

160

(b)), while the third (figure 68 (c)) has a large amplitude away from the core. The third

mode is directly related to the linear instability of circular Couette flow with respect to

non-axisymmetric disturbances with n = 1. For n = 2, the picture is slightly different.

There is still one mode (figure 68(a)) that is largely concentrated outside the core, and

which therefore seems related to the linear instability of circular Couette flow. The mode

in figure 68 (b) is largely concentrated at the core, and shows similarities to figures 68 (a)

and (b). In addition, there is non-localized mode (figure 69(c)) that is absent for n = 1.

n = 1 n = 21. 3.878× 10−2 ± 9.276× 10−3 i 3.674× 10−2 ± 1.742× 10−3 i2. 3.590× 10−2 ± 1.843× 10−2 i 2.968× 10−2 ± 8.189× 10−2 i3. 3.279× 10−2 ± 4.878× 10−2 i 2.788× 10−2 ± 5.722× 10−5 i

Table 7: Growth rates for the unstable nonaxisymmetric modes at Weθ = 23.87, L =4.74, S = 1.2, and b = 1830.

5.4.6 Dean flow revisited

In the previous sections, we have demonstrated that stationary, localized solutions of

very large wavelength exist in circular Couette flow. Here, we return to Dean flow to

investigate whether such solitary solutions are possible there. We do this by tracking

long wavelength steady states that bifurcate from the base state in Dean flow. To make

comparison with the circular Couette flow results easier, we choose b = 1830, S = 1.2,

and ε = 0.2, which are the same parameter values that we used for circular Couette

flow. In figure 70, we show the bifurcation diagram for L = 1.963. This diagram is

similar to figure 54, in that there is a large subcritical region, and the nontrivial branch

exhibits a turning point, i.e., there is a critical value of the Weissenberg number below

which the nontrivial solution does not exist. In figure 71, we show an density plot of

161

z

r

z

r

z

r

(a)

(b)

(c)

Figure 67: Density plot of vr showing the (a) real and (b) imaginary parts of the desta-bilizing axisymmetric disturbance, and (c) the streamlines of the base diwhirl. Note thatthe core of the diwhirl is entirely unaffected by the disturbance. The parameters areWeθ = 23.87, S = 1.2, ε = 0.20, b = 1830, and L = 4.74.

162

t=0 t=T/4

t=0 t=T/4

t=0 t=T/4

(a)

(b)

(c)

r r

r r

r r

z z

z z

z z

Figure 68: Density plot of the perturbation radial velocity for the three non-axisymmetricunstable eigenmodes with n = 1. The parameters are identical to those in figure 67.

t=0 t=T/4

t=0 t=T/4

t=0 t=T/4

(a)

(b)

(c)

r r

r r

r r

z z

z z

z z

Figure 69: Density plot of the perturbation radial velocity for the three non-axisymmetricunstable eigenmodes with n = 2. The parameters are identical to those in figure 67.

163

the radial velocity at Weθ = 15.486 (marked by an open circle in figure 70) and at

Weθ = 13.930 (marked by a filled circle in figure 70). The figure shows that rather than

becoming localized, the nonlinear patterns have a tendency to split into two vortices, each

one having half the wavelength of the original vortex. In fact, for the particular case of

L = 1.963, we were not able to advance the continuation beyond Weθ = 13.94 on the

upper branch. Examination of figure 71(b) reveals why: it is clear from the symmetry of

the solution that the L = 1.963 path crosses a branch of solutions with L = 1.963/2 in

a pitchfork bifurcation. This intersection is the reason the continuation fails beyond this

point. We wish to stress here that the absence of localization in Dean flow is not limited

to the particular case of L = 1.963: we have seen this for larger values of L as well.

One possible reason for this could be that localized structures would have areas in the far

field that would be very similar to base state Dean flow, much like the case with solitary

steady states in circular Couette flow where the far field is essentially base state circular

Couette flow (see figure 57). However, since Dean flow is linearly unstable to stationary

disturbances of smaller wavelengths, there would be a tendency for vortex structures of

smaller wavelengths to form in the far field and prevent true localization.

As with circular Couette flow in figure 59 , we can plot an existence boundary for the

non-trivial Dean flow solutions. This curve is shown in figure 72. Note that there is a

discontinuity at L = 1.795. At this wavelength, we get a collision with the L = 1.795/2

branch in a pitchfork bifurcation as was the case with L = 1.963, but this time, the col-

lision occurs before the turning point is reached. Therefore, no turning point exists for

this value of L, resulting in the discontinuity. For reference, we also show two linear

stability curves of Dean flow. The curve on the left (the dashed line) is the usual linear

stability curve of Weθ,c versus L. The curve on the right (the dotted line) is the linear

164

13.0 14.0 15.0 16.0 17.0 18.0 19.0Weθ

0.0000

0.0020

0.0040

0.0060

||vr||

Figure 70: Bifurcation diagram for Dean flow at b = 1830, S = 1.2, ε = 0.2, andL = 1.963. We could not continue the branch beyond Weθ = 13.94, for reasons discussedin section 5.4.6. Radial velocity profiles corresponding to the points marked by the filledand open circles are shown in figure 71.

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

z

r

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

z

r

(a)

(b)

Figure 71: Density plots of the radial velocity at the points circled in figure 70. (a)Weθ = 15.486 (b) Weθ = 13.930.

165

stability curve with respect to disturbances which have two vortices, each of wavelength

L/2. This curve is trivially obtained from the regular linear stability diagram by multi-

plying the horizontal coordinate by a factor of two. At the intersection of the two curves,

denoted by Lt, is a codimension 2 point, where disturbances of wavelength Lt and Lt/2

bifurcate simultaneously. We would expect interactions between modes to play a signif-

icant role in the nonlinear behavior close to this wavelength. Note that the discontinuity

in the nonlinear existence curve occurs close to L = Lt.

0.5 1.5 2.5L

12.0

13.0

14.0

15.0

16.0

17.0

18.0

19.0

20.0

21.0

22.0

We θ,

c

Linear Stability (single vortex) Linear Stability (two vortices) Nonlinear Existence boundary

L t

Figure 72: Linear stability curves and existence boundaries for nonlinear solutions inDean flow at S = 1.2, ε = 0.2, and b = 1830. The filled triangle shows the data atL = 1.795 where the solution terminates via collision with the L/2 branch in a pitchforkbifurcation before the turning point.

5.5 Conclusions

In this chapter, we have computed stationary, nontrivial solutions in viscoelastic circu-

lar Couette flow by numerical continuation from stationary bifurcations in Couette-Dean

166

flow by using a FENE-P model. These stationary solutions show very strong spatial lo-

calization with a marked asymmetry between inflow and outflow, exist only for large

values of the polymer extensibility parameter and large wavelengths, show a hysteretic

character in the Weissenberg number, and are isolated from the base Couette flow branch

of solutions, all of which are characteristics shared by similar patterns (“diwhirls” or

“fl ame patterns” ) observed experimentally by Groisman and Steinberg (1997, 1998) and

Baumert and Muller (1999). Direct comparison shows that the streamlines and magni-

tudes of the radial velocity for the patterns we have computed are very similar to the

corresponding experimentally measured quantities. The only major difference between

the experimentally observed patterns and our solutions is in the degree of hysteresis they

exhibit: the experimentally observed diwhirls exist at Weissenberg numbers as low as 10,

whereas the computed diwhirls do not exist at Weissenberg numbers lower than about

23.3. We attribute this discrepancy to the approximate nature of the FENE-P model.

We have also performed computations with the FENE-CR model. These show that the

FENE-P and FENE-CR model produce different results in both the linear and nonlinear

regimes. In particular, the diwhirls are not predicted with FENE-CR model, at least for

the parameter values that we have examined. Note that the “exact” FENE molecular

model is better approximated by the FENE-P model than the FENE-CR model (Herrchen

and Ottinger, 1997). Our result demonstrates that the choice of constitutive equation is

critical in modeling nonlinear pattern formation in viscoelastic flows.

We have used the velocity and stress fields generated from our computations to pro-

pose a self sustaining mechanism for the diwhirls. The mechanism is related to the mech-

anism of instability in viscoelastic Dean flow and arises from a finite amplitude pertur-

bation giving rise to a locally parabolic profile of the azimuthal velocity near the outer

167

cylinder at the symmetry axis of the vortices. Strictly speaking, we find that the computed

diwhirls are unstable with respect to axisymmetric and non-axisymmetric perturbations.

The axisymmetric perturbations only have significant amplitude away from the core of

the diwhirl, whereas the non-axisymmetric perturbations can introduce modulations in

the core itself. Although unstable, the qualitative and quantitative similarities between

the computed structures and the experimentally observed diwhirls and flame patterns in-

dicates a direct link between the two.

Finally, we investigated whether structures similar to diwhirls or flame patterns could

form in Dean flow by tracking steady states bifurcating from the base flow in Weissenberg

number. Our computations demonstrate that long wavelength solutions in Dean flow tend

to show transitions to shorter wavelength structures rather than becoming localized.

Along with the solutions arising from the linear instability of the circular Couette flow

base state, we propose that the solitary solutions we have computed form building blocks

for spatiotemporal dynamics in the flow of viscoelastic liquids. We believe that com-

putations presented here are a first step towards understanding these complex nonlinear

dynamics.

168

Chapter 6

Conclusions and future work

In this thesis, we have performed analyses of some of the instabilities observed in the

flows of viscoelastic fluids. In chapter 2, we showed that multiplicity in the slip law does

not necessarily imply multiplicity in the flow curve. We performed simulations with the

UCM, PTT, Newtonian, and shear thinning models, all of which yielded similar results,

thus showing that the result did not depend on the particular slip model being used.

In chapter 3, we performed an analysis of the buckling instability in elongational flow

by modeling the polymer filament as a thin elastic membrane enclosing a passive fluid

that only serves to enforce the incompressibility constraint. We demonstrated that such

a set up will show instabilities that are similar to those seen in the elongational flow of

polymeric liquids. This simple model makes several assumptions that need to be checked

by direct numerical simulation of elongational flow. In particular, the numerical results

would need to show stress localization at the free surface, and regions of compressive

hoop stress near areas where the filament has buckled. It is only recently that such nu-

merical simulations have been attempted (Rasmussen and Hassager, 1999) for a UCM

169

liquid with no solvent viscosity. A direct comparison between our results and these sim-

ulations is not possible for reasons which we outline below.

Firstly, the computations in Rasmussen and Hassager (1999) were performed for a

non-axisymmetric base state. In particular, they added a small (0.5% of the original

radius) non-axisymmetric perturbation to the unstretched filament and then performed

the stretching. Thus, their work cannot be classified as a true stability analysis of the

stretching of an axisymmetric filament. Secondly, the stress contours for their simulations

at high Deborah numbers exhibit boundary layers in the component along the direction

of elongation, not the hoop stress. However, the high Deborah number results are only

shown at low values of the Hencky strain (2.5 for the case when the Deborah number

equals 1, and smaller strains at higher Deborah numbers), since numerical difficulties

prevented exploration of the regimes of larger Hencky strain.

Given the problems outlined in the preceding paragraph, we propose that future sim-

ulations take the following suggestions into account. In performing the stability analysis

of filament stretching, the primary difficulty is that the base state is time dependent. One

way of handling this problem is to perform what is called a “momentary stability” anal-

ysis, first described by Shen (1961) in the context of time dependent parallel shear flows,

and recently used for stability analysis of the spreading of surfactant coated films (Matar

and Troian, 1999). In this scheme, the governing equations are linearized about a base

state, just as in traditional linear stability analysis. A small disturbance is added to the

base flow at some reference time t0, and the linearized equations are then integrated si-

multaneously with the base state equations. The energy in the disturbance relative to its

initial energy at time t0 is then compared with the energy in the base state, relative to its

value at t0. If the disturbance grows faster than the base state, the flow is considered to

170

be unstable. In the context of filament stretching, this technique will ensure that the base

state being considered is purely axisymmetric.

The second problem that needs to be addressed is the numerical difficulty in achieving

large values of the Hencky strain at high Deborah numbers. One improvement that can

be made with relative ease is to use the FENE-P or FENE-CR model for the simulations.

Both are more realistic models for polymer solutions than the UCM equation, and are also

easier to simulate. Yet another improvement would be to take solvent viscosity into ac-

count. Besides improving the stability of the calculations, solvent viscosity would make

the simulations more representative of the experiments, since Spiegelberg and McKin-

ley (1998) used polymer solutions where the solvent and polymer contributions to the

viscosity were similar in magnitude.

In chapter 4, we demonstrated how an axial flow of small magnitude can signifi-

cantly stabilize the purely elastic instability in Dean flow. The next logical step would

be to perform experiments to verify that the stabilization mechanism works in practice.

As demonstrated by Joo and Shaqfeh (1994), a single concentric cylinder geometry can

be used to generate both Dean and circular Couette flow, simply by inserting a small

impenetrable block that spans the gap between the cylinders and extends across the en-

tire length. Moving both cylinders at equal angular velocities approximates Dean flow,

whereas moving them in opposite directions at equal angular speeds approximates circu-

lar Couette flow. By adjusting the relative angular velocities of the cylinders, it is possible

to get the entire range of flows between Dean and circular Couette. Axial flow can be

added either by imposing a pressure drop or by oscillating the inner cylinder. Thus, using

this apparatus, it would be possible to investigate the effect of axial flow on both Dean

and circular Couette flows. In a recent paper, Grillet et al. (1999) have described how

171

forward and reverse roll coating flows can be approximated in an eccentric cylinder ge-

ometry. It follows that if we make this modification in the set up described above, we can

determine the effect of axial forcing on free surface coating flows as well.

From the purely computational point of view, there are still some issues that remain

to be addressed. Firstly, as shown by Al-Mubaiyedh et al. (2000), non-isothermal effects

create a new mechanism of instability in viscoelastic circular Couette flow. The instabil-

ity associated with this mechanism sets in at a Weissenberg number that is an order of

magnitude lower than the one at which the purely elastic instability does. Their results

also showed that incorporation of additional relaxation times can have a large effect on

the Weissenberg number at which the instability sets in, although the mechanism of in-

stability is not affected. Given this fact, it seems logical to suppose that the mechanism of

suppression would work even when more than one relaxation time is taken into account.

This supposition must be checked with computations, as also what effect the axial flow

has on the non-isothermal mode of instability.

Finally, in chapter 5, we used numerical continuation from Couette-Dean flows to ac-

cess isolated, long wavelength, stationary, axisymmetric steady states in circular Couette

flow. We showed that these steady states are remarkably similar to the diwhirls (Grois-

man and Steinberg, 1997, 1998) and flame patterns (Baumert and Muller, 1999) observed

in experiments. Our computations demonstrated that such structures are absent in Dean

flow. There are several future directions that are worth exploring, within the context of the

FENE-P constitutive equation. Firstly, these solitary structures are unstable with respect

to axisymmetric and non-axisymmetric time dependent disturbances. While simulation

of fully three dimensional viscoelastic flow is far too computationally intensive, it would

be interesting to perform two dimensional axisymmetric time integrations of the unstable

172

steady states in order to gain insight into the interactions between the diwhirls and the

axisymmetric instability in circular Couette flow. Secondly, we have eased the computa-

tional burden by assuming that the structures that we are trying to compute have certain

symmetry properties and only modeled half the computational domain. This excludes

solutions that do not satisfy these properties. For instance, Baumert and Muller (1999)

have observed that the vortices in the flame patterns have a tendency to merge. Removing

the symmetry constraint may help us detect this phenomena in our simulations.

The main discrepancy between our calculations and the experimental results lies in

the fact that the solutions we compute lose existence at Weθ = 23.3, whereas diwhirls

and flame patterns have been observed at Weissenberg numbers as low as 10. There could

be several reasons for this lack of agreement. Firstly, we have not been able to access the

regime of large b due to limitations in our numerical scheme, and as discussed in chap-

ter 5, the point at which the diwhirls lose existence moves to lower values of Weθ as b

increases. Secondly, the FENE-P model that we have used for our simulations only incor-

porates a single relaxation time. Using the multimode Oldroyd-B, Giesekus, and Phan-

Thein and Tanner models, Al-Mubaiyedh et al. (2000) demonstrated that incorporating

multiple relaxation times can significantly change the point at which the linear instability

sets in. It is quite possible that incorporation of multiple relaxation times could have an

effect on the characteristics of the nonlinear solutions as well. Performing computations

with multimode constitutive equations such as the ones mentioned above can help us bet-

ter understand the effect of additional relaxation times. Another approach to including

multiple relaxation times is to use a FENE or Hookean chain, and perform microstructural

simulations using a technique like Operator Splitting Coefficient Matching (OSCM) (Jen-

drejack et al., 2000) or the Adaptive Lagrangian Particle Method (ALPM) (Gallez et al.,

173

2000) to get the stress. These techniques, however, are still in their infancy. ALPM

has been successfully used for a strong flow (a 1:4:1 contraction) for a FENE chain at

low Weissenberg numbers (Gallez et al., 2000), but requires enormous computational

power. OSCM appears to be less computationally intensive, but is still in the test phase,

with simulations being restricted to obtaining the stress field for a given velocity profile.

Non-isothermal effects could be another possible source of the quantitative discrepancy

between the experiments and our results. In chapter 5, we demonstrated that there is a

large difference between polyacrylamide based Boger fluid and polyisobutylene based

ones in terms of thermal effects. Since localized patterns are observed in both fluids, it is

unlikely that thermal effects play a role in the mechanism. However, they may still play

a qualitative role, for instance by changing the location of the turning point. Computa-

tions using non-isothermal models, such as the model developed by Crochet and Naghdi

(1969), or the more recent models based on phase space kinetic theory for dilute solutions

of dumbbells under non-isothermal conditions (Ottinger and Petrillo, 1996; Curtiss and

Bird, 1996), can elucidate the role of thermal effects.

In the computations described in chapter 5, the chief bottleneck was the development

of an effective preconditioning technique. The preconditioner that we developed took us

close the limits of what is achievable using a single computer. Further progress, such as

adding additional relaxation times, will require the use of more advanced numerical tech-

niques. A promising area in this regard is parallel computing. In performing the Newton

iterations, the computation of the Jacobian is an easily parallelizable step, since we can

simply split the domains between processors. However, the ILUT∗ preconditioner that

we have developed is not easy to parallelize, either in its computation or its application.

Therefore, it is critical to develop parallelizable preconditioners that are effective. In this

174

context, approximate inverse preconditioners (Benzi et al., 2000; Duff and van der Vorst,

1999; Benzi and Tuma, 1998; Chow and Saad, 1998) are worth investigating. The ILU

class of preconditioners are implicit preconditioners, in the sense that their application

requires the solution of a linear system of equations. On the other hand, approximate

inverse preconditioners are explicit, i.e., they involve the construction of a matrix that

approximates the inverse of the original matrix. Both the construction and application

involve the action of a matrix on a vector, and these are steps that are easily paralleliz-

able. Recently, Leriche and Phillips (2000) have discussed preconditioners for spectral

element solutions to viscoelastic problems that are based on linear finite element precon-

ditioning. The linear finite element systems are solved using Schur complement methods

or overlapped Schwartz methods, both of which are parallelizable. This last precondi-

tioner, however, is used in the context of time integration, where operator splitting is

used to update the convection terms in the constitutive and momentum equations explic-

itly. How effective it will be for a steady state continuation scheme where the momentum,

continuity, and constitutive equations are coupled remains to be determined.

175

Appendix A

Introduction to finite elasticity

This section of the appendix gives an overview of a few of the basic equations of finite

elasticity. This material is condensed from Corneliussen and Shield (1961).

A.1 The basic equations of finite elasticity

Consider a homogeneous elastic body which is isotropic in its undeformed state. The

coordinates of a typical point in the undeformed state in a fixed Cartesian coordinate

system x are given by (x1, x2, x3). Let (v1, v2, v3) denote the coordinates of this point in

an arbitrary curvilinear coordinate system. We assume that a transformation rule of the

form

xi = xi(v1, v2, v3) (156)

exists. Let r denote the position vector from the origin of the system x to a point of the

body. The covariant base vectors of the system v of the undeformed body are given by

g(k) =∂r

∂vk= r,k, (157)

176

where in this notation, a comma followed by k denotes differentiation with respect to vk.

In future, we will drop the parentheses on the subscripts of base vectors for convenience.

An element of length ds is given by

ds2 = dxkdxk = gijdvidvj, (158)

where gij are the covariant components of the metric tensor of the system v. The con-

travariant components of the metric tensor are denoted by gij and are the components of

the inverse of the matrix gij. As before, g denotes the determinant of the matrix gij .

At some future time t,the body has been deformed, and the coordinates of a typical

point in the system x are (X1, X2, X3). The curvilinear system moves with the body (i.e.

we can think of the vi as particle labels), and we have

Xk = Xk(v1, v2, v3, t). (159)

Thus, at time t, the covariant base vectors of the system v are

Gk = R,k. (160)

The element of length dS is given by

dS2 = dXkdXk = Gijdvidvj, (161)

where the Gij are the covariant components of the metric tensor of of the system v at time

t. Gij denote the contravariant components of the metric tensor and G the determinant of

Gij .

For some materials, the stress work acts to change a state function called the strain

energy. If stress work is done, the strain energy changes by the same amount, and this

strain energy is recoverable as stress work. The strain energy function W gives the change

177

in strain energy from its value at the reference state, per unit volume of the reference state.

It is a function of the three strain invariants I1, I2 and I3,

W = W (I1, I2, I3),

with the invariants being given by

I1 = grsGrs

I2 = grsGrsI3

I3 = G/g. (162)

The contravariant components of the stress tensor are now given by

τ ij = Φ gij +ΨDij + P Gij, (163)

where

Φ = 2√

I3∂W

∂I1,

Ψ = 2√

I3∂W

∂I2,

P = 2√

I3∂W

∂I3,

Dij = I1 gij − gir gjsGrs. (164)

For an incompressible material I3 = 1, and

W = W (I1, I2).

For certain rubber-like materials, the strain energy function takes the Mooney form

W = C1(I1 − 2) + C2(I2 − 3), (165)

178

where C1 and C2 are positive constants. A special case of the Mooney material is the

neo-Hookean material, whose strain energy function takes the form

W = C1(I1 − 3). (166)

In the absence of body forces, the equations of motion are given by

ρ f j = ∇iτij , (167)

where the f j are the components of the acceleration vector in the system v.

A.1.1 The equations of membrane elasticity

In this section, we look at the simplifications that result when the equations of finite

elasticity are applied to a thin membrane. Let (v1, v2) be the coordinates of a general

curvilinear coordinate system defined on a surface in space. Suppose that a is the position

vector from the origin of a Cartesian coordinate system to a point on the surface, the base

vectors of this system are

aα =∂a

∂vα= a,α . (168)

We now define a unit vector a3(v1, v2) perpendicular to the surface at each point, in the

direction a1 × a2. The distance from the point (v1, v2) on the surface to a point on the

normal to the surface at that point is denoted by v3, and is considered positive in the

direction of a3. If we now consider a point on the middle surface of a shell of thickness

h with v3 = 0, the position vector can be written as

r = a(v1, v2) + v3a3(v1, v2),

179

where

−1

2h(v1, v2) ≤ v3 ≤

1

2h(v1, v2). (169)

Thus, in the undeformed (or reference) state, we get

gαβ = aαβ, gα3 = 0, g33 = 1,

g = a,

gαβ = aαβ, gα3 = 0 , g33 = 1,

where aαβ are the covariant components of the surface metric tensor

aαβ = aα · aβ, (170)

a = |aαβ|, and aαβ is the inverse of the matrix whose elements are aαβ .

Now consider the surface at the later time t, when it has been deformed. Let the

position vectors of the deformed surface be given by A(v1, v2). The base vectors of the

deformed surface are given by

Aα = A,α . (171)

As before, we define at each point on the surface, a unit vector A3 perpendicular to

the surface at that point and in the direction A1 × A2, with the same convention for

positivity as before. The unit normal vector N is thus given by (A1 ×A2)/|A1 ×A2|.

The covariant components bαβ of the curvature tensor B are then given by

bαβ =N ·Aα,β =N ·A,αβ. (172)

180

For a shell composed of an isotropic material, whose principal radii of curvature of

the deformed middle surface are large compared to the thickness, and whose thickness

varies slowly with v1 and v2, it can be shown that the position vector to a point on the

deformed membrane can be written as

R = A(v1, v2) + λ3(v1, v2) v3A3(v

1, v2), (173)

where λ3 is positive and is the extension ratio in the direction of the normal to the middle

surface. The components of the metric tensor are now given as

Gαβ = Aαβ, Gα3 = 0, G33 = λ23,

G = λ23A,

Gαβ = Aαβ, Gα3 = 0, G33 =1

λ23,

where Aαβ and Aαβ are respectively the covariant and contravariant components of the

surface metric tensor and A = |Aαβ|. The strain invariants are given as

I1 = aαβAαβ + λ23,

I3 = λ23A/a, (174)

I2 = I3

(aαβA

αβ +1

λ23

). (175)

For an incompressible material, I3 = 1, which implies that W =W (I1, I2) and that

λ3 =√(a/A). (176)

This in turn implies that the remaining two invariants are given by

I1 = aαβAαβ + a/A,

I2 = aαβAαβ +A/a. (177)

181

The contravariant components of the stress tensor in the coordinate system v is then given

by

nαβ =√(a/A3)h

((Aaαβ − aAαβ

)Φ +

(ADαβ − aAαβ D33

)Ψ), (178)

where h is the thickness of the membrane, and

Dαβ =a

Aaαβ +

(aαβ aµν − aαµ aβν

)Aµν

D33 = aµν Aµν .

Let P = pjaj be the external force acting on the membrane. Then the equations of

motion for the membrane are given by

∇α nαβ + pβ = h√

a/Aρ0 fβ,

bαβ nαβ + p3 = h√

a/Aρ0f3, (179)

where ρ0 is the density of the undeformed membrane.

182

Appendix B

Base state solutions and matrix

components in Dean flow

We present the expressions for the base state velocities and stresses, as well as the nonzero

entries of the matrices L and A for the axisymmetric case.

B.1 Base state solutions

The following are the steady state velocities and polymer stresses of an Oldroyd-B fluid.

DAP flow

vr = 0 (180)

vθ = r(1− r) (181)

vz =Wez

Wpr(1− r)ε1/2 (182)

τrr = 0 (183)

183

τrθ =Wp (1− 2r)

ε1/2(184)

τrz = Wez(1− 2r) (185)

τθθ =2(1− 2r)2Wp2

ε(186)

τθz =2(1− 2r)2WezWp

ε1/2(187)

τzz = 2(1− 2r)2We2z (188)

DMAC flow: The equations correpond to the regime where ω = O(ε1/2) and t =

O(ε−1/2)with all other variables scaled as in section 4.3. We define ω1 = ε−1/2ω = O(1)

and t1 = ε1/2t = O(1). The steady axial flow (DAC) equations are obtained by setting

ω1 = 0.

vr = 0 (189)

vθ = (r − r2) (190)

vz =Wez(1− r)ε1/2

Wpcos(ω1t) (191)

τrr = 0 (192)

τrθ =Wp (1− 2 r)

ε1/2(193)

τrz = −Wezcos(ω1t1) +Wpω1 sin(ω1t1)

(1 + ω21Wp2)(194)

τθθ =2Wp2(1− 2r)2

ε(195)

τθz =WezWp(−1 + 2r)(2 cos(ω1t1) +Wpω1(3 +Wp2ω21) sin(ω1t1))

ε1/2(1 +Wp2ω12)2(196)

τzz =−We2z

(1 + 4Wp2ω21)(1 +Wp2ω21)(−1− 4ω21Wp2 − cos(2ω1t1) (197)

+2ω21Wp2 cos(2ω1t1)− 3ω1Wp sin(2ω1t1))

184

B.2 Disturbance equations

The non-zero entries of WpA(t) in equation (105) for the DMAC cases are given below.

The operator D represents the derivative w.r.t. the gap coordinate r. The matrixE(t) has

−1 on the diagonal corresponding to the constitutive equations and zeros elsewhere. The

EVSS formulation (not shown here) is derived by substituting

τrr = Σrr + 2WpD vr,

τrθ = Σrθ +WpD vθ,

τrz = Σrz + i αWp vr +WpD vz,

τθθ = Σθθ,

τθz = Σθz +Wp vθ,

τzz = Σzz + 2 i αWp vz,

where the components of Σ represent the elastic part of the extra stress tensor τ . The

equations for DAC flow are recovered by setting ω1 = 0.

WpA1,1 = 1 + iWez α (1− r) cos(ω1t1)

WpA1,7 =2 iWez Wpα (cos(ω1t1) +Wpω1 sin(ω1t1))

(1 +Wp2 ω21)− 2WpD

185

WpA2,1 = Wp (−1 + 2r)

WpA2,2 = 1 + i αWez (1− r) cos(ω1t1)

WpA2,7 = −Wp22 (1 +Wp2 ω21)

2 + 2 iWez α (−1 + 2 r) cos(ω1t1)

(1 +Wp2 ω21)2

−Wp2iWez Wpα(−1 + 2 r)ω1 (3 +Wp2 ω21) sin(ω1t1)

(1 +Wp2 ω21)2

+

Wp2 (−1 + 2r)D

WpA2,8 = iαWez Wpcos(ω1t1) +Wpω1 sin(ω1t1)

(1 +Wp2 ω21)−WpD

WpA3,1 = Wez cos(ω1t1)

WpA3,3 = 1 + iWez α (1− r) cos(ω1t1)

WpA3,7 =−i αWp

(1 + 4Wp2 ω21)(1 +Wp2 ω21)

((1 +We2z +Wp2 ω21)(1 + 4Wp2 ω21)

−We2z (−1 + 2Wp2 ω21) cos(2ω1t1))

−i αWp3We2z Wpω1 sin(2ω1t1)

(1 + 4Wp2 ω21)(1 + ω21Wp2)

+Wez Wp (cos(ω1t1) +Wpω1 sin(ω1t1))D

(1 +Wp2 ω21)

WpA3,9 = −WpD

WpA4,2 = 2Wp (−1 + 2 r)

WpA4,4 = 1 + i αWez (1− r) cos(ω1 t1)

WpA4,7 = 8Wp3 (−1 + 2 r)

WpA4,8 =−2 iWez Wp2 α

(1 +Wp2ω21)((−1 + 2 r)(2 cos(ω1t1)

+Wpω1 (3 +Wp2 ω21) sin(ω1t1)))+ 2Wp2 (−1 + 2 r)D

186

WpA5,2 = Wez cos(ω1t1)

WpA5,3 = Wp (−1 + 2 r)

WpA5,5 = 1 + i αWez (1− r) cos(ω1t1)

WpA5,7 = 2Wez Wp22 cos(ω1t1) +Wpω1(3 +Wp2 ω21) sin(ω1t1)

(1 +Wp2 ω21)

WpA5,8 =i αWp

(1 + 4Wp2 ω21)(1 +Wp2 ω21)((1 +We2z +Wp2 ω21)(1 + 4Wp2 ω21)

−We2z (−1 + 2Wp2 ω21) cos(2ω1t1) + 3We2z Wpω1 sin(2ω1 t1)) +

Wez Wp (cos(ω1t1) +Wpω1 sin(ω1t1))

(1 +Wp2 ω21)D

WpA5,9 = iWez Wp2 α (−1 + 2 r)2 cos(ω1t1) +Wpω1 (3 +Wp2 ω21) sin(t1 ω1)

(1 +Wp2 ω21)2

+Wp2 (−1 + 2 r)D

WpA6,3 = 2Wez cos(ω1t1)

WpA6,6 = 1 + iWez α(1− r) cos(ω1t1)

WpA6,9 =−2 iWpα

(1 + 4Wp2 ω21)(1 + 4Wp2 ω21)

((1 +We2z +Wp2 ω21)(1 + 4Wp2 ω21)

−We2z (−1 + 2Wp2 ω21) cos(2 ω1t1))

−2 iWpα3We2z ω1Wp sin(2ω1t1)

(1 + 4Wp2 ω21)(1 + 4Wp2 ω21)

+2Wez Wpcos(ω1t1) +Wpω1 sin(2ω1t1)

(1 +Wp2 ω21)D

WpA7,1 = D WpA7,3 = i α WpA7,4 = −1 WpA7,7 = S Wp(−α2 +D2

)WpA7,10 = −D

WpA8,2 = D WpA8,5 = i α WpA8,8 = S Wp(−α2 +D2

)

187

WpA9,3 = D WpA9,6 = i α WpA9,9 = S Wp(−α2 +D2

)WpA9,10 = −α

WpA10,7 = D WpA10,9 = i α

DAP flow: Below are given the nonzero components of the linear operator WpL for DAP

flow.The components of the momentum and continuity equations are the same as for the

DMAC case, as are the terms in the matrix E.

WpL1,1 = 1 +Wez i α r (1− r)

WpL1,7 = 2 i, Wez Wpα(−1 + 2 r)− 2WpD

WpL2,1 = Wp (−1 + 2 r)

WpL2,2 = WpL1,1

WpL2,7 = −2Wp2 − 2 iWez Wp2 α (−1 + 2 r)2 +Wp2 (−1 + 2 r)D

WpL2,8 = iWez Wpα (−1 + 2 r)− WpD

WpL3,1 = Wez (−1 + 2 r)

WpL3,3 = WpL1,1

WpL3,7 = −2WezWp+ 8 i rWe2z Wpα(1− r)− i αWp (1 + 2We2z)

+Wez Wp (−1 + 2 r)D

WpL3,9 = iWez α (−1 + 2 r)D

WpL4,2 = 2Wp (−1 + 2 r)

WpL4,4 = WpL1,1

188

WpL4,7 = 8Wp3(−1 + 2 r)

WpL4,8 = −4 iWez Wp2 α(1− 2 r)2 + 2Wp2(−1 + 2 r)D

WpL5,2 = Wez (−1 + 2 r)

WpL5,3 = Wp (−1 + 2 r)

WpL5,5 = WpL1,1

WpL5,7 = 8Wez Wp2(−1 + 2 r)

WpL5,8 = −i αWp (1 + 2We2z) + 8 i rWe2z Wp (1− r) +Wez(−1 + 2 r)D

WpL5,9 = −2 iWez Wp2 α(1− 2 r)2 +Wp2(−1 + 2 r)D

WpL6,3 = 2Wez (−1 + 2 r)

WpL6,6 = WpL1,1

WpL6,7 = 8We2z Wp (−1 + 2 r)

WpL6,9 = −2 i αWp (1 + 2We2z (1− 2 r)2) + 2Wez Wp (−1 + 2 r)D

189

Appendix C

Velocity and Stress scalings for

Couette-Dean flow

We found base state solutions to the FENE-P equation starting from the analytical so-

lutions for the Oldroyd-B model in Couette-Dean flow. We also used these solutions to

obtain scalings for the velocity and time. For the Oldroyd-B model, the azimuthal veloc-

ity in dimensional form can be written as the sum of a Couette contribution, v∗θ,c, and a

Dean contribution v∗θ,d, where

v∗θ,c = ΩR2

((1− ε)

ε(1− ε+ rε)(2− ε)−(1− ε)(1− ε+ rε)

ε(2− ε)

), (198)

and

v∗θ,d =−Kθε

2R22ηt

((1− ε)2 log(1− ε)

ε3(1− ε+ r ε) (2− ε)−(1− ε+ rε)

ε3 (2− ε)

((1− ε)2 log(1− ε)

+ε(2− ε) log(1− ε+ rε))) , (199)

190

with r being the radial coordinate scaled by the gap width, so that r = 0 is the inner

cylinder and r = 0 is the outer cylinder. Adding these two equations gives

v∗θ =−Kθε

2R22ηt

fd +ΩR2(1− ε)fc, (200)

where

fc =(1− ε)

ε(1− ε+ rε)(2− ε)−(1− ε)(1− ε+ rε)

ε(2− ε), (201)

and

fd =(1− ε)2 log(1− ε)

ε3(1− ε+ r ε) (2− ε)−(1− ε+ rε)

ε3 (2− ε)

((1− ε)2 log(1− ε)

+ ε(2− ε) log(1− ε+ rε)) . (202)

If we now choose

Vd∗ =−Kθε

2R22ηt

(203)

to be a characteristic velocity for Dean flow and

Vc∗ = ΩR2 (1− ε), (204)

which is the velocity at the inner cylinder, to be a characteristic velocity for circular

Couette flow, can define parameter δ, which measures the relative contribution of the

azimuthal pressure gradient to the total flow as

δ =Vd∗

Vd∗ + Vc

∗ . (205)

We will only consider the case where Vd∗ and Vc

∗ have the same sign, i.e., where the

pressure gradient and inner cylinder rotation act to drive the flow in the same direction.

191

This means that δ = 0 is circular Couette flow and δ = 1 is Dean flow. With this definition

of δ, we can write the azimuthal velocity as

v∗θ = (V∗c + V ∗d )(δ fd + (1− δ) fc). (206)

We choose the velocity scale to be the magnitude of the shear rate at the outer cylinder

times the length scale. Since Vd∗ and Vc

∗ are both taken to be positive, the shear rate at

the outer cylinder is negative, so that the magnitude of the shear rate is −γ2, where γ2 is

the shear rate at the outer cylinder. This gives the velocity scale as

v∗ = (V ∗c + V ∗d )v, (207)

where

v =

(δ2 (1− ε)2 log(1− ε) + ε (2− ε)

ε2(2− ε)+ (1− δ)

2 (1− ε)

2− ε

). (208)

Therefore, the azimuthal velocity can be written in dimensionless form as

vθ =δ fd + (1− δ) fc

v. (209)

The Weissenberg number is the product of the characteristic shear rate, εR2 v∗, and

the relaxation time. The non-zero components of the polymer stress tensor and the radial

pressure gradient can be computed from the relations

τrθ = Weθ

(∂vθ∂r−

vθr − 1 + 1/ε

), (210)

τθθ = −2Weθτrθ

(vθ

r − 1 + 1/ε−

∂vθ

∂r

), (211)

∂p

∂r= −

τθθr − 1 + 1/ε

. (212)

Having calculated the stresses, the components of the 〈QQ〉 for the Oldroyd-B model

can be obtained using the Kramers equation as

〈QQ〉 = τ + I. (213)

192

We use these components as the initial approximations for the FENE-P model.

193

Appendix D

Time Integration of viscoelastic

flows

In chapter 5, we have described steady state solution tracking for viscoelastic flows. Here,

we will describe time integration of viscoelastic flows where inertia is important. This has

applications in the simulation of the effect of polymers on turbulence, and also in studying

mixing flows. Our goal here is to gain expertise in the use of spectral methods in time

integration of viscoelastic flows, and for that reason, we will work with a model problem.

We consider here the classical problem of Rayleigh-Benard convection in which a layer of

fluid of infinite extent in both horizontal directions is heated from below. For simplicity,

we assume free slip conditions at both vertical boundaries. There are several convection

patterns possible with these boundary conditions, but we will restrict ourselves to the

two dimensional rolls whose axes are aligned with one of the horizontal directions. We

choose this direction to be the y axis, the horizontal direction perpendicular to it to be the

x axis, and the vertical direction to be the z axis. Thus, the flow is in the x−z plane. The

194

streamfunction for this flow is given by Chandrasekhar (1961) as

ψ0(x, z) =A

ksin(πz) sin(kx), (214)

where A is the maximum velocity of the flow, k = 2πL

, where L is the wavelength of

the cell. In equation 214, length has been nondimensionalized by the depth of the layer

so that the upper boundary is at z = 1 and the lower boundary is at z = 0. Figure 73

shows a contour plot of equation 214. The roll cells are themselves susceptible to a two

dimensional, time dependent instability as the temperature drop is increased. Solomon

and Gollub (1988) modeled the stream function of the flow that develops after the onset

of the instability as

ψ(x, z, t) =A

ksin(πz) sin(kx) + εA cos(ωt) cos(kx) sin(πz), (215)

where ω is the frequency of the instability, and ε = (Ra − Rac)1/2, where Ra is the

Rayleigh number, and Rac is the value of Ra at the onset of the instability. This time-

periodic flow pattern creates regions within the flow cell where passive scalars have

chaotic trajectories (see figure 74).

0 1 20

1

x

z

Figure 73: Streamlines for equation 214 with A = 1 and k = π.

195

0.0 0.5 1.0 1.5 2.0x

0.0

0.2

0.4

0.6

0.8

1.0

z

Figure 74: Poincare map of the trajectory of a point in the flow field given by equa-tion 215. The parameters chosen were k = π, ω = 2π, and ε = 0.25.

D.1 Governing equations

In the Rayleigh-Benard problem, the convection patterns are driven by heating the fluid

from below. Our interest lies only in obtaining the velocity profiles corresponding to

equation 215, and we do this by the computationally simpler approach of adding forcing

terms to the momentum equations. For Newtonian fluids, the governing equations are the

Navier-Stokes equations and the continuity equation. In dimensionless form, they are

Re

(∂u

∂t+ u

∂u

∂x+ w

∂u

∂z

)= −

∂p

∂x+

∂2u

∂x2+

∂2u

∂z2+ fx, (216)

Re

(∂w

∂t+ u

∂w

∂x+ w

∂w

∂z

)= −

∂p

∂z+

∂2w

∂x2+

∂2w

∂z2+ fz, (217)

∂u

∂x+

∂w

∂z= 0. (218)

Here, u and w are respectively the x and z components of the velocity, and p is the

pressure. The velocity has been scaled by A, distance by the height of the layer, and

196

time by the ratio of the distance scale to the velocity scale. The forcings fx and fz can

be obtained by requiring that the governing equations be satisfied by the desired velocity

field and zero pressure. We calculate them to be

fx =1

4 k(−(π (−4 ε k π Re cos(ω t) cos2(k x) +

4 ε k cos(k x) cos(π z) ((k2 + π2) cos(ω t)

−ω Re sin(ω t))+sin(k x) (2 (−2+ ε2 k2) πRe cos(k x)+ ε2 k2 π Re cos(2ω t−k x)+

ε2 k2 π Re cos(2ω t+ k x) + 4 k2 cos(π z) + 4 π2 cos(π z)−

2 ε k π Re sin(ω t− k x) + 2 ε k π Re sin(ω t+ k x))))) (219)

fz = sin(π z) ((k2 + π2) cos(k x) + πRe cos2(k x) cos(π z) (1 + k2 ε2 cos2(t ω))

+ sin(k x) (−(k (k2 + π2) ε cos(t ω)) + k2 πRe ε2 cos(π z) cos2(t ω) sin(k x) +

Re (π cos(π z) sin(k x) + k ε ω sin(t ω)))) (220)

When polymer is added to the system, the momentum conservation equations are

modified to incorporate the contribution of the polymer stresses:

Re

(∂u

∂t+ u

∂u

∂x+ w

∂u

∂z

)= −

∂p

∂x+

∂2u

∂x2+

∂2u

∂z2+ fx (221)

+1

WeS

(∂τp,xx∂x

+∂τp,xz∂z

),

Re

(∂w

∂t+ u

∂w

∂x+ w

∂w

∂z

)= −

∂p

∂z+

∂2w

∂x2+

∂2w

∂z2+ fz (222)

+1

WeS

(∂τp,xz

∂z+

∂τp,zz

∂z

),

where τp,xx, τp,xz, and τp,zz are the three components of the polymer stress τp, We =

λA/d is the Weissenberg number, λ is the relaxation time of the polymer, and S is the

ratio of the solvent to polymer viscosity. We choose the FENE-P constitutive equation

197

for the polymer with the same scalings for the conformation tensor 〈QQ〉 as in chapter

5. Once again, the stress tensor is calculated from the Kramers relation. We use periodic

boundary conditions in the x direction. The z direction is periodic as well, except that

here, the period length is 2. This means that the solutions exhibit symmetries in this

direction. In particular, u, p, 〈QQ〉xx, and 〈QQ〉yz can be represented in terms of even

functions, while w and 〈QQ〉xz can be represented in terms of odd functions.

D.2 Numerical Method

Periodic boundary conditions naturally suggest the use of a Fourier series representation.

Here, we use Fourier series in both the x and z directions. We take advantage of the

symmetry in the z direction by representing the even functions as a Fourier cosine series

in the vertical direction, and the odd functions as a Fourier sine series. The primary

representation for all variables is in Fourier space. We compute Fourier expansions of

nonlinear terms by transforming the variables involved into physical space, computing

the nonlinear terms, and transforming back into Fourier space. This can be done very

efficiently using a the Fast Fourier Transform (FFT). A one dimensional FFT on an N

point grid requires O(N log2(N)) operations, where N is the number of grid points.

Transforming data on a two dimensional grid with Nx points in the x direction and Nz

points in the z direction requires Nz transformations of one dimensional data that is Nx

points long, followed by Nx transformations involving data that is Nz points long. Thus,

the work required for transforming two dimensional data is a semi-linear function of

the number of grid points (NxNz). We use the highly efficient public domain package,

FFTW (Frigo and Johnson, 1999), to perform these transformations. Equation 215 gives

198

a velocity field that has stagnation points at x = L/2, z = 0 and x = L/2, z = 1 (i.e., the

midpoints of the upper and lower walls). Polymer molecules are highly stretched in the

neighborhood of these points, which gives rise to sharp stress gradients and requires the

use of high resolution.

We use a time splitting scheme to perform the integration. For purposes of numerical

stability, we found it necessary to add an isotropic artificial diffusion term to the con-

stitutive equations. Similar observations have been made in the context of simulating

high Reynolds number Poiseuille flow of a polymer solution in a channel (Sureshkumar

and Beris, 1995) and turbulent flow in a channel (Sureshkumar et al., 1997). Thus, the

modified constitutive equation is written

〈QQ〉(1) =1

We

(〈QQ〉

1− tr(〈QQ〉)/b− I

)+D∇2〈QQ〉, (223)

where D is the diffusion coefficient. The diffusion coefficient is chosen to have the form

D = a(∆h)2

t∗, (224)

where ∆h is the mesh size, t∗ is a time scale, and a is a constant. Thus, in the limit

∆h → 0 (i.e., an infinitesimally small mesh size) with a fixed, D = 0, so that the

magnitude of the diffusion becomes smaller as the mesh is refined. Since the flow is

periodic in both x and z, we do not need to specify additional boundary conditions for

equation 223.

We use a splitting scheme for the time integration. The first stage involves updating

〈QQ〉 in two steps. The first of these involves an explicit update of the convection and

nonlinear terms using an Adams-Bashforth scheme. Denoting the vector (u, w) by v, this

199

can be written as

〈QQ〉∗ = 〈QQ〉(n) +∆ t

1∑j=0

βj[−v · ∇〈QQ〉+ 〈QQ〉 · ∇v + (∇v)t · 〈QQ〉

+1

We

(〈QQ〉

1− tr(〈QQ〉)/b− I

)](n−j), (225)

where β0 = 3/2 and β1 = −1/2, and the superscripts denote values at the corresponding

timestep. The second step involves a semi-implicit update to include the diffusion term.

We do this using a Crank-Nicholson method:

〈QQ〉(n+1) − 〈QQ〉∗

∆t=

D

2(∇2〈QQ〉(n+1) +∇2〈QQ〉(n)). (226)

Note that although equation 226 is an implicit equation for 〈QQ〉(n+1), it is trivially

solvable in Fourier space, since the individual Fourier modes decouple.

The next stage involves the calculation of v(n+1). This is done in three steps. The first

step involves the update of the convection and stress terms. We use an Adams-Bashforth

scheme for the convection terms and Crank-Nicholson for the stress gradients:

v − v(n)

∆t= −

1∑j=0

βj(v · ∇v)(n−j) +

1

2WeSRe

(∇ · τ (n+1)p +∇ · τ (n)p

). (227)

The second step is a pressure correction, given by the equation

ˆv − v

∆t= −

1

Re∇p(n+1). (228)

This equation can be used to obtain the pressure by requiring that ˆv be divergence free,

which yields the Poisson equation

∇2p =∆t

Re∇ · v. (229)

In flows which have non-periodic directions, the boundary conditions for the pressure

equation must be chosen with care in order to preserve the overall temporal accuracy of

200

the integration scheme, and Karniadakis et al. (1991) present a discussion on high-order

pressure boundary conditions for such flows. In our case, however, the only indetermi-

nacy is in choosing the coefficient for the pressure mode with zero wavenumbers in both

directions. What we choose this coefficient to be is completely irrelevant, since this has

no effect on the pressure gradient, which is the quantity that appears in the momentum

equation.

The final step in the integration scheme is to account for the viscous terms. We use a

Crank-Nicholson scheme for these terms. The update is given by

v(n+1) − ˆv

∆t=

1

2Re(∇2v(n+1) +∇2v(n)). (230)

As in the case of the diffusion correction in equation 226, equations 229 and 230 are

trivially solvable in Fourier space.

The scheme as outlined above is second order in time. The time step is chosen based

on the requirements of accuracy and stability. The stability of the scheme is governed

by the Courant-Freidrichs-Levy (CFL) condition for the two explicit steps (equation 225

and 227). We choose the value of t∗ in equation 224 to be the largest timestep permitted

by the CFL condition. This essentially makes the diffusivity independent of the actual

choice of timestep for a given spatial resolution.

We can get a rough estimate of the total work required per time step by keeping track

of the number of FFTs that need to be performed. Given the initial vectors v and 〈QQ〉

in Fourier space, we need to perform a total of 15 two dimensional transformations to

get their values and those of their derivatives in two directions in physical space, in order

to compute the nonlinear terms. Once the nonlinear terms are computed, a total of 5

transformations need to be performed to get their values in Fourier space, yielding a total

of 20 transforms per timestep. In the absence of artificial diffusion in the constitutive

201

equation, this number is reduced to 17, since in this case, the update in equation 225

gives the physical space components directly.

D.3 Results and Discussion

We restrict our discussion here to the necessity of using artificial diffusion in integrating

the constitutive equation. For our test case, we used a resolution of 144 points in the

horizontal direction and 72 points in the vertical direction and started our integration with

initial conditions being the Newtonian velocity field and zero values for the components

of the conformation tensor. We set ∆t to 10−4, which is well below the upper limit set

by the CFL condition. The Weissenberg and Reynolds numbers were both chosen to be

1, and the results are shown in figure 75, for various values of the diffusivity multiplier,

a. The quantity plotted is the value of 〈QQ〉xx at x = L/4 and z = 0.5. We see from the

figure that the integration blows up if artificial diffusion is not used, or if the diffusion

constant a is much smaller than 10−3. At this value of a, the diffusion coefficient, D

was calculated to 1.640 × 10−5. The integration for a = 10−3 was stable indefinitely.

Figure 75 also shows that the quantitative effect of artificial diffusion is quite small: the

curves for the integrations with and without artificial diffusion virtually coincide before

the numerical instability sets in.

To determine the effect of increasing mesh resolution, we refined the mesh to a grid

of 256 points in the horizontal direction and 128 points in the vertical direction. We then

integrated the governing equations for the same parameter values and time step as the

coarser mesh, with a = 10−4. At this value of a, the diffusion coefficient D has a value

of 9.223× 10−7 for the fine mesh, and 1.640× 10−6 for the coarse mesh. The results are

202

shown in figure 76. Although both integrations blow up due to numerical instability, it is

clear that the fine mesh with the smaller diffusion coefficient is stable for a longer period

of time than the coarse mesh. This shows that refining the mesh improves numerical

stability even if the diffusion coefficient is lowered.

0.0 5.0 10.0 15.0 20.0t

−1.0

0.0

1.0

2.0

3.0

4.0

5.0

<Q

Q>

xx

*

a=10−3

a=10−4

a=10−5

a=10−6

a=0

Figure 75: Illustration of the effect of adding a diffusion term to the constitutive equationon a mesh of 144×72. The parameter values used were We = 1, Re = 1, ε = 0.2, k = π,ω = 2π, b = 10, and S = 6.67.

A possible question to ask is if the numerical instability would still appear if the con-

stitutive and momentum equations are decoupled. To answer this question, we integrated

the constitutive equation using the steps outlined in equations 225 and 226, with the ve-

locity field supplied by equation 215, with the same parameters as for the coupled case.

The results, shown in figure 77 for a = 0 and a = 10−3, show that the instability is

indeed associated with the integration of the constitutive equation, since the case with

a = 0 blows up. In fact, comparing figures 75 and 77 shows that the instability is delayed

203

0.0 5.0 10.0 15.0t

−1.0

0.0

1.0

2.0

3.0

4.0

5.0

<Q

Q>

xx

*

144 x 72 256 x 128

Figure 76: Effect of mesh refinement on numerical stability. Runs with both meshes wereperformed with a = 10−4 and the same parameter values as in figure 75.

0.0 5.0 10.0 15.0 20.0t

0.0

1.0

2.0

3.0

<Q

Q>

xx

*

a=10−3

a=0

Figure 77: Effect of artificial diffusion when equation 223 in integrated in a known ve-locity field. The parameters and mesh are identical to those in figure 75. Note the earlierblow up when compared to the coupled case.

204

in the coupled case. This may possibly be due to the stabilizing influence of the viscous

terms in the momentum equations.

205

Appendix E

Branch tracing in three dimensional

plane Couette flow

E.1 Introduction and Formulation

Analysis of turbulence in plane Couette flow is complicated by the absence of a linear

instability, i.e., plane Couette flow is stable to infinitesimal perturbations at all Reynolds

numbers. This means that the transition to turbulence must occur as a finite amplitude

effect. Homotopy (see the description in section 5.1) offers one approach to access non-

trivial steady states in plane Couette flow that may have relevance to coherent structures

seen in turbulence. This approach was taken by Nagata (Nagata, 1988, 1990, 1997) who

tracked finite amplitude solutions from Taylor-Couette flow to plane Couette flow. A

slightly different approach was taken by Waleffe (Waleffe, 1995, 1997), who started by

proposing a self sustaining mechanism to maintain turbulence, which we describe below

in some detail.

206

Consider the schematic diagram of plane Couette flow shown in figure 2. We shall

call the direction of flow (the x direction) as the streamwise direction, the direction of ve-

locity change (the y direction) as the wall-normal direction, and the neutral direction (the

z direction) as the spanwise direction. The Waleffe mechanism consists of three steps. In

the first step, rolls with their axes in the streamwise direction redistribute the streamwise

momentum to create spanwise fluctuations in the streamwise velocity. The presence of in-

flections due to this spanwise fluctuation causes a wake-like instability (Drazin and Reid,

1981), which takes the form of vortices that have their axes oriented in the wall-normal

direction. The mean shear then advects these vortices into the streamwise direction, rein-

forcing the original streamwise vortices and completing the self-sustaining mechanism.

In a later work (Waleffe, 1998), Waleffe extended this mechanism into a continuation

strategy for obtaining non-trivial solutions in plane Couette flow by adding an explicit

forcing term, f , in the Navier-Stokes equations of the form

f = κFr

Re2[0, γ cos(βy) cos(γz), β sin(βy) sin(γz)], (231)

where Re is the Reynolds number, β = π/2, κ = (β2 + γ2)/γ, and Fr is a continuation

parameter.

Waleffe considered plane Couette flow, with free-slip boundary conditions. In dimen-

sionless form, the governing equations are the Navier-Stokes equations

Dv

Dt= −∇p+

1

Re∇2v + f , (232)

∇ · v = 0, (233)

where p is the pressure and v = (u, v, w) is the velocity vector. The boundary conditions

207

are the so-called free-slip boundary conditions

∂u

∂y= 1, v =

∂w

∂y= 0 (234)

at the lower and upper boundaries in the wall-normal direction, taken to be at y = −1

and y = 1 respectively. Waleffe looked for solutions that satisfied certain symmetry

constraints (listed in Waleffe (1997)), and was able to show the existence of a subcritical

bifurcation at Fr = 5, Re = 150, α = 0.49, and γ = 1.5. By tracking this branch as Fr

was reduced, he found that the solution existed at Fr = 0. In the absence of forcing, the

self-sustaining mechanism discussed above drives the rolls.

Our ultimate interest here is in investigating what effect polymers have on the Waleffe

solutions. Since these structures are three-dimensional, we reduce the computational

burden by splitting the problem into two parts. First, we rewrite equation 233 to take into

account the polymer stresses as

Dv

Dt= −∇p+

1

Re∇2v + f +

1

ReWeS∇ · τ , (235)

∇ · v = 0, (236)

where τ is the polymer component of the stress tensor, We is the Weissenberg number,

and S is the ratio of the solvent to polymer viscosity. We would like to examine the effect

of both free slip and no slip boundary conditions, so we write the boundary conditions as

Λ(u− 1)− (1− Λ)

(∂u

∂y− 1

)= 0 at y = 1,

Λ(u+ 1) + (1− Λ)

(∂u

∂y− 1

)= 0 at y = −1,

v = 0 at y = ±1,

Λw + (1− Λ)∂w

∂y= 0 at y = ±1, (237)

208

so that Λ = 1 is no slip and Λ = 0 is free slip. The difference in the boundary conditions

on u at y = 1 and y = −1 is to ensure that they are identical when u is replaced by −u

and y by −y. As before, we have scaled the polymer component of the stress tensor by

the shear modulus G. Our strategy will be to treat the polymer contribution as a forcing

term in the momentum equations. Thus, for a given value of the polymer stress tensor, we

plan to use equation 236 to solve for v and p. We will then use v in an as yet unspecified

polymer constitutive equation to update τ , and iterate back and forth till the velocity and

stress fields are consistent. Here we describe the first step in this process, which is the

solution of equations 237 with no polymer component, i.e., with τ = 0.

E.2 Discretization

We use a three-dimensional spectral method, with Chebyshev collocation in the y di-

rection, and Fourier collocation in the x and z directions. It is common to use Fourier

collocation based on an even number of grid points. However, the first derivative matrix

for this discretization has two zero eigenvalues, one corresponding to the constant mode,

and one corresponding to the cos(Nx/2) where N is the number of collocation points.

The second eigenmode gives rise to spurious solutions in a collocation scheme. For this

reason, we use Fourier collocation based on an odd number of collocation points. This

scheme includes the cos(Nx/2) mode, and thus only has one zero eigenvalue, which

corresponds to the constant mode. As in chapter 4, we use a primitive variable formula-

tion, and define pressure on a staggered grid in the non-periodic (y) direction. Thus, our

unknowns are the three components of the velocity and the pressure.

If we denote the collocation order in the x direction by Nx, in the y direction by

209

Ny, and in the z direction by Nz, equations 236 yield a system of 3(Nx + 1)(Ny +

1)(Nz + 1) + (Nx + 1)(Nz + 1)Ny nonlinear equations for the three components of the

velocity and the pressure. As in chapter 5, we construct solutions to these equations by

using pseudo-arclength continuation. Since the Navier-Stokes equations do not contain

mixed derivatives, the Jacobian matrix, J , is sparse: for instance, the x derivative of a

variable at a collocation point only depends on the value of the variable at the collocation

points which have the same values of y and z as the point where the derivative is being

computed. Thus, we can store the matrix in sparse format, i.e., we only store the non-zero

entries and information about the row and column numbers of these entries. We solve

the linear systems arising from the Newton iteration using GMRES (Saad and Schultz,

1986). Once again, it is necessary to precondition the system in order to make GMRES

converge, and we use the dual threshold incomplete LU decomposition preconditioner,

or ILUT (Saad, 1996) described in section 5.3. We use a drop tolerance of 10−3 and in

each row of the preconditioner, keep twice the number of entries as in the corresponding

row of J .

The continuation scheme needs a known solution as a starting point, and the Couette

flow solution, u(x, y, z) = y, v = 0, w = 0, valid for Fr = 0, provides a convenient

one. Although this solution is valid for all values of Λ, the continuation method fails for

Λ = 0. This is because, for free slip boundary conditions, u(y, z) + U is a solution to

the governing equations for any constant U . Thus, the absolute value of u is not fixed,

which results in failure to converge. Adding a small component of no-slip, i.e., setting Λ

to some non-zero number, removes this degeneracy, and all the results presented here will

be for Λ different from zero. Since we do not impose any symmetry constraints, we do

not know beforehand whether the loss of stability is through a stationary bifurcation, or

210

through a Hopf bifurcation. For this reason, we determine the unstable eigenvalues using

ARPACK (Lehoucq et al., 1997), using the technique described earlier in section 5.3, for

determining the stability of the diwhirls.

E.3 Preliminary Results

We describe the results from our simulations using Re = 150, β = π/2, γ = 1.5, and

α = 0.49. These are identical to the values used by Waleffe (1997) in his quest for co-

herent structures in Couette flow. The base flow is two-dimensional, homogeneous in

the x direction, and has a y − z dependence. We use 15 Chebyshev collocation points

in the wall normal direction, and 15 Fourier collocation points in the z direction. Us-

ing fewer points gives rise to spurious unstable eigenvalues, and using more collocation

points, particularly in the wall-normal direction, results in ill-conditioned matrices. In

the streamwise direction, we use three Fourier collocation points, since the base flow has

no variation in this direction, and we are, for now, only interested in capturing the struc-

ture of the destabilizing disturbance close to the bifurcation point. Keeping three Fourier

collocation points is equivalent to keeping the cos(αx), sin(αx), and the constant modes

in the x direction.

We first show results for the run with Λ = 0.002, which is very close to the free slip

case. Figure 78 shows the streamwise component of the velocity in the y − z plane. The

redistribution of the streamwise momentum due to the forcing is evident. Consistent with

the findings of Waleffe (1997), we find that the base flow becomes unstable to a three-

dimensional disturbance at Fr = 5. However, since we do not assume any symmetries,

we find that two modes bifurcate close to Fr = 5. The modes have different growth rates,

211

and are thus not degenerate. The structure of these two modes is shown in figure 79,

where we have plotted the streamwise components of the two unstable eigenvectors in

the x−z plane at y = 0. Both modes are consistent with the mechanism of the instability,

i.e., the generation of vortices that have their axes in the y direction due to the existence

of inflection points in the spanwise direction. The difference in the symmetries between

the two modes becomes evident if we plot the streamwise velocity component of the two

unstable eigenvectors as a function of x at y = 0 and z = π/γ. This plot is shown in

figure 80. Clearly, this component in the first eigenvalue (which has the higher growth

rate of the two) has a non-zero mean, while in the case of the second eigenvalue, the

mean is zero. The fact that two eigenvalues bifurcate very close to each other has the

unfortunate effect of making the test function method of detecting stationary bifurcations

(discussed in section 5.3) very unreliable. Although it is possible to construct a test

function that changes sign close to the bifurcation point, the sign change occurs abruptly,

and the test function quickly reverts back to the original sign, so that it is easy to miss the

bifurcation unless the step size is very small.

As we increase Λ, the bifurcation changes from stationary to Hopf, suggesting that

a Takens-Bogdanov point exists close to Λ = 0, where the two real eigenvalues collide

and are transformed to a complex conjugate pair. This phenomenon was discussed in

section 4.5.2 in the context of Dean flow. In figure 81, we show the real and imaginary

parts of the streamwise velocity component of the unstable mode in the x − z plane at

y = 0. Note the similarity between this and figure 79. The similarity in the structure of

the unstable modes between Λ = 0.002 and Λ = 0.1 lends support to the existence of a

Takens-Bogdanov point.

212

z

y

0 1 2 3-1

-0.5

0

0.5

1

Figure 78: Contours and density plot of the streamwise velocity u in the y − z plane inthe base state. The value of Re = 150, Λ = 0.002, and Fr = 5.08.

E.4 Proposals for future work

In the preceding paragraphs, we have shown how base solutions can be tracked, and

bifurcations detected in plane Couette flow with a two-dimensional forcing. The next

step is to actually track the fully three-dimensional (non-trivial) branches. Since the test

function approach fails in this case, it will be necessary to approximate a point on the

non-trivial branch by adding a small component of the unstable eigenvector to the base

solution. We can then get an exact solution on the branch by using this approximation

as a first guess and performing a Newton iteration keeping the coefficient of the cos(αx)

or sin(α x) mode at a single point fixed to prevent convergence to the trivial branch.

Once a solution on the non-trivial branch is found, the solution, the branch can then be

tracked as desired. Since the bifurcation at larger values of Λ is of the Hopf type, non-

trivial solutions for these values of Λ will have to be accessed by continuation from lower

213

x

z

0 5 100

1

2

3

x

z

0 5 100

1

2

3

(a)

(b)

Figure 79: Contours and density plot of the u component of the two unstable eigenvectorsfor the flow shown in figure 78 in the x − z plane at y = 0. The mode shown in (a) hasthe higher growth rate of the two unstable modes.

214

0.0 5.0 10.0x

−0.06

−0.04

−0.02

0.00

0.02

u

Eigenvalue − 1Eigenvalue − 2

Figure 80: Plot of the profile of the u component of the streamwise velocity of the twounstable eigenvectors for the flow shown in figure 78 at y = 0, and z = π/γ. Theeigenvalue corresponding to eigenvector 1 has the higher growth rate. Eigenvector 2 issimply a vertically shifted and scaled version of eigenvector 1.

values of Λ, just as the non-trivial stationary solutions in viscoelastic circular Couette

flow were accessed by continuation of non-trivial solutions in viscoelastic Couette-Dean

flow in chapter 5.

One problem we had, even in the limited analysis that we performed, was the poor

condition number of the Jacobian matrix when the number of Chebyshev modes was

increased. A possible solution to this is to use a spectral element discretization in the

wall-normal direction, while preserving the Fourier discretization in the two periodic

directions. As the number of degrees of freedom increases, so will computational re-

quirements, and the parallelizable preconditioners discussed at the end of chapter 6 may

provide improvements in performance by permitting efficient parallelization of the solu-

tion phase.

215

x

z

0 5 100

1

2

3

x

z

0 5 100

1

2

3

(a)

(b)

Figure 81: Contours and density plot in the x − z plane of the u component of the realand imaginary part of the unstable eigenvector at Re = 150, Λ = 0.1, and Fr = 5.01.

216

Nomenclature

The following three tables summarize the notation used in the thesis. Since chapters 2

and 3 deal with subjects that follow a different notation from the material in the other

chapters, we have provided the notation for these chapters separately from the rest of the

thesis. The variables are listed in case insensitive alphabetical order, so the descriptions

of some of these variables may refer to later entries in the table. Sometimes, for the sake

of convenience, the same symbols are used in the text for dimensionless and dimensional

quantities. Such variables are specified as being either dimensional or dimensionless,

depending on where they are used in the text.

Notation used in Chapter 2

Variable Description

a1 parameter for the slip model (dimensional)

A2, A3, β scaled parameters for the slip model (dimensionless)

C capacitance, equals Vb/πR2u∗t∗ (dimensionless)

D diameter of die (dimensional)

De Deborah number (dimensionless)

Continued on next page

217

Continued from previous page


H dimensionless number, equals 8ηu∗/GR for Newtonian

and (3 + 1/n)K1/n/a1G1/n−1R for power law fluid

G shear modulus (dimensional)

P pressure (dimensional or dimensionless)

Pb barrel pressure (dimensionless)

Q volumetric flow rate (dimensional)

Q0 volumetric flow rate of polymer at die exit

(dimensional or dimensionless)

Qp volumetric rate of displacement of the piston


R radius of the die (dimensional)

t∗ residence time, Qp/πR2L (dimensional)

u∗ scaling factor for the velocity, equals G/a1 (dimensional)

us slip velocity (dimensional or dimensionless)

Vb barrel volume (dimensional)

γA apparent shear rate (dimensional)

η shear viscosity of the melt (dimensional)

κ equals Gχ (dimensionless)

λ relaxation time (dimensional)

λ1 t∗u∗/L (dimensionless)

λ2 t∗G/η (dimensionless)


218



Λ L/D (dimensionless)

ρ density (dimensional)

ρ0 density at reference pressure (dimensional)

τ stress tensor (dimensional or dimensionless)

τc2, τc3 lower and upper critical shear stresses on the flow curve


τw wall shear stress (dimensional or dimensionless)

χ compressibility (dimensional)

Notation used in Chapter 3


a position vector in the undeformed state

(dimensionless)

a∗ position vector in the undeformed state

(dimensional)

a determinant of matrix with components aαβ

(dimensionless)

aαβ , aαβ covariant and contravariant components

of the metric tensor in the

undeformed state (dimensionless)


219



A position vector in the deformed state

(dimensionless)

A∗ position vector in the deformed state

(dimensional)

A determinant of matrix with components Aαβ

(dimensionless)

Aαβ , Aαβ covariant and contravariant components

of the metric tensor in the

deformed state (dimensionless)

De Deborah number (dimensionless)

e1,e1, e3 Cartesian unit vectors (dimensionless)

E dimensionless strain energy

F strain energy functional

I1 strain invariant (dimensionless)

l amount by which cone is stretched

(dimensionless)

L length of the truncated cone (dimensional)

n azimuthal wavenumber (dimensionless)

r1, r2 radius of cone at x3 = 0 and x3 = L

(dimensional)

s slope of the cone, (r1 − r2)/L


220



(dimensionless)

v velocity vector (dimensionless)

v velocity vector (dimensional)

vi component i of v

v1, v2 surface coordinates (dimensionless)

v1∗, v2∗ surface coordinates (dimensional)

V0 volume of cone in undeformed state

(dimensionless)

Vl volume of cone in deformed state

(dimensionless)

We Weissenberg number (dimensionless)

x1, x2, x3 Cartesian coordinate directions

z equals v2 cos(φ) (dimensionless)

γ rate of deformation tensor, equals∇v + (∇v)t(dimensionless)

ˆγ rate of deformation tensor (dimensional)

ε rate of elongation

η0 viscosity of UCM fluid

θ azimuthal angle (dimensionless)

λ relaxation time for UCM fluid

(dimensional)

λ1, λ2 stretch factors for the cone


221



in the base state (dimensionless)

λ1, λ2, λ3 perturbation stretch factors

for the cone (dimensionless)

Λ Lagrange multiplier (dimensionless)

τ stress tensor


τ stress tensor (dimensional)

τij component ij of the stress tensor

φ cone half angle, equals tan−1(s)

(dimensionless)

Notation used in remainder of main text


b maximum polymer extension squared (dimensionless)

c complex growth rate, equals cr + i ci (dimensionless)

cr continuation parameter linking

FENE-P and FENE-CR models (dimensionless)

Cp specific heat capacity at constant

pressure on a mass basis (dimensional)

d gap width, equals R2 −R1 (dimensional)


222



Ev viscous dissipation per unit volume (dimensional)

F c spring force (dimensional)

G shear modulus (dimensional)

H spring constant (dimensional)

J Jacobian matrix (dimensionless)

Kθ azimuthal pressure gradient (dimensional)

L axial wavelength (dimensionless)

n azimuthal wavenumber (dimensionless)

n equals ε1/2n (dimensionless)

N number density of dumbbells (dimensional)

p pressure (dimensional or dimensionless)

Pz axial pressure drop (dimensional)

Q dumbbell end to end vector (dimensional)

Q0 maximum spring extension (dimensional)

Q2 tr(〈QQ〉) (dimensional or dimensionless)

〈QQ〉 ensemble average of conformation tensor


〈QQ〉ij component ij of 〈QQ〉 (dimensional or dimensionless)

r shifted and scaled radial coordinate (r∗ −R1)/(R2 −R1)

r∗ radial coordinate (dimensional)

R1 inner cylinder radius (dimensional)


223



R2 outer cylinder radius (dimensional)

S ratio of solvent to polymer viscosity

t time (dimensionless)

T temperature (dimensional)

T time period (dimensionless)

v velocity vector (dimensional or dimensionless)

vi component i of vector v (dimensional or dimensionless)

V axial velocity amplitude (dimensional)

Wez axial Weissenberg number (dimensionless)

Weθ azimuthal Weissenberg number (dimensionless)

Wp equals ε1/2Weθ

Wpc critical value of Wp for onset of instability (dimensionless)

Wpc,min minimum critical value of Wp for onset of instability

(dimensionless)

z axial coordinate

α axial wavenumber (dimensionless)

β Floquet multiplier (dimensionless)

γ rate of deformation tensor, equals∇v + (∇v)t


γij component ij of γ (dimensional or dimensionless)

γ magnitude of shear rate (dimensional or dimensionless)


224



γs characteristic shear rate (dimensional)

δ relative contribution of azimuthal pressure gradient

to total forcing (dimensionless)

ε gap width (dimensionless)

ε elongation rate (dimensional)

ρ density (dimensional)

σ Floquet exponent (dimensionless)

τ stress tensor (dimensional or dimensionless)

τij component ij of stress tensor τ


ζ friction coefficient due to hydroynamic drag (dimensional)

η general symbol for shear viscosity (dimensional)

ηp polymer viscosity (dimensional)

ηs solvent viscosity (dimensional)

ηt total viscosity, equals ηs + ηp (dimensional)

κ equals (∇v)t (dimensional)

λ relaxation time (dimensional)

ψ distribution function (dimensionless)

Ψ1 First normal stress coefficient (dimensional)

Ψ2 Second normal stress coefficient (dimensional)


225



ω frequency of axial forcing (dimensionless)

Ω angular velocity of inner cylinder rotation (dimensional)

226

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