Planar Flow of Dilute Polymer Solutions
Transcript of Planar Flow of Dilute Polymer Solutions
Shear & Extensional Effects in Internal Flows of
Dilute Polymer Solutions
by
Shamsur Rahman
A thesis submitted in conformity with the requirements
for the degree of Master of Applied Science
Department of Mechanical & Industrial Engineering
University of Toronto
© Shamsur Rahman 2011
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Abstract
Shear & Extensional Effects in Internal Flows of Dilute Polymer Solutions
Shamsur Rahman
Master of Applied Science, 2011
Department of Mechanical & Industrial Engineering
University of Toronto
Shear and extensional flows of dilute polymer solutions were studied experimentally in an
attempt to understand the mechanism of polymer-induced drag reduction. A flowcell capable of
simulating the dynamics of a turbulent boundary layer, involving the motion of counter-rotating
vortices, was designed and fabricated. The pressure drop across the flowcell was measured for
different flow arrangements, first with a Newtonian fluid and then with drag reducing, dilute
polymer solutions. The pressure drop in excess of the Newtonian baseline, after accounting for
viscous effects, was used as a measure of elastic effects.
With the dilute polymer solutions, elastic effects were observed both in shear, extensional, as
well as presheared extensional flows. These effects can be attributed to additional normal
stresses generated by shearing. For extensional flows, the observed effects were independent of
elongation rates, indicating that a conclusion regarding the mechanism of drag reduction cannot
be made from the flowfield investigated.
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Table of Contents
Chapter 1: Introduction ................................................................................................................1
1.1 Previous Work with Drag-Reducing Fluids...............................................................................5
1.2 Research Objectives...................................................................................................................8
Chapter 2: Experimental Methodology .....................................................................................10
2.1 Conceptual Design ...................................................................................................................10
2.2 Design Considerations .............................................................................................................15
2.2.1 Main Channel Geometry ................................................................................................16
2.2.2 Side Channel Geometry .................................................................................................19
2.2.3 Channel Lengths.............................................................................................................20
2.2.4 Exit Channel Length.......................................................................................................21
2.2.5 Pressure Tap Locations ..................................................................................................21
2.2.6 Pressure Dropo & Choice of Test Fluids .......................................................................22
2.3 Flowfield with Final Design ....................................................................................................23
2.4 Fabrication of Flowcell ............................................................................................................27
Chapter 3: Test Fluids .................................................................................................................28
3.1 Non-Newtonian Fluids.............................................................................................................28
3.1.1 Boger Fluids ...................................................................................................................29
3.1.2 Rheometry ......................................................................................................................29
3.2 Shear Viscosity of Test Fluids .................................................................................................31
3.3 Critical Concentration..............................................................................................................33
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3.4 Relaxation Time & First Normal Stress Difference ................................................................35
3.4.1 First Normal Stress Difference.......................................................................................35
3.4.2 Oldroyd-B Model ...........................................................................................................36
3.5 Elastic Modulus .......................................................................................................................39
3.6 Summary of Fluid Prpoerties ...................................................................................................42
Chapter 4: Experimental Results and Discussions ...................................................................43
4.1 Main Channel and Side Channel Combined Flow...................................................................43
4.1.1 Newtonian Results..........................................................................................................45
4.1.2 Results with PEO Solutions ...........................................................................................47
4.2 Main Channel Flow without Side Flow...................................................................................51
4.3 Analyses of Results..................................................................................................................53
4.3.1 Flow Instability in Shear ................................................................................................53
4.3.2 Hole Pressure Error ........................................................................................................55
4.4 Side Channel Flow without Main Channel Flow.....................................................................58
4.4.1 Elastic Effects in Extension............................................................................................62
4.4.2 Numerical Analysis ........................................................................................................64
4.4.3 N1 Effect .........................................................................................................................68
4.5 Comparison with Prior Work...................................................................................................71
Chapter 5: Concluding Remarks................................................................................................74
5.1 Summary ..................................................................................................................................74
5.2 Conclusions..............................................................................................................................76
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5.3 Future Work .............................................................................................................................76
Chapter 6: References .................................................................................................................78
Appendix A: Numerical Simulation ...........................................................................................81
Appendix B: Fluid Mechanics ....................................................................................................84
Appendix C: Oldroyd-B Model ..................................................................................................91
Appendix D: Pressure Transducer.............................................................................................95
Appendix E: Engineering Drawings of Flowcell .......................................................................99
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List of Figures
Figure 1.1 Setup of turbulent burst experiment (Reproduced from Kim et al, 1971).....................2 Figure 1.2 Photographic plate showing H2 bubble lines (Reproduced from Kim et al, 1971) .......3 Figure 1.3 (a) Formation of a hairpin vortex. (b) A group of hairpin vortices being lifted up from the surface. (c) A pair of counter-rotating vortices exerting upward force on the fluid. (Reproduced from Davidson, 2004) ................................................................................................5 Figure 1.4 Setup of presheared extensional flow experiment. (Reproduced from James et al, 1987) ................................................................................................................................................7 Figure 2.1 Fully developed flow through a rectangular channel. Parabolic velocity profile........11 Figure 2.2 Planar extensional flowfield. The streamlines are typically hyperbolic......................12 Figure 2.3 Cross-slot geometry in three-dimensions (Reproduced from Winter et al., 1979)......12 Figure 2.4 Setup of experimental flowcell....................................................................................14 Figure 2.5 Setup of experimental apparatus..................................................................................14 Figure 2.6 Comparison between velocity profiles for flow in a rectangular channel with different aspect ratios, and flow between two parallel plates. Channel width = 2a, height = 2d .................15 Figure 2.7 Streamwise component of velocity along vertical centreplane ...................................25 Figure 2.8 Enlarged view of streamwise velocity at the main channel entrance ..........................25 Figure 2.9 Transverse component of velocity along vertical centreplane. Color legend shows velocity in metres per second (m/s). ..............................................................................................26 Figure 3.1 Working principle of a cone-and-plate rheometer.......................................................30 Figure 3.2 Steady shear viscosity measurements for test fluids ...................................................32 Figure 3.3 Determination of intrinsic viscosity ............................................................................34 Figure 3.4 a) Step input in strain and the corresponding stress relaxation of b) a Newtonian fluid and c) a viscoelastic fluid and solid (Reproduced from Macosko, 1994)......................................35 Figure 3.5 First normal stress difference measurements in response to steady shearing for the 1200 ppm PEO solution in PEG solvent........................................................................................38
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Figure 3.6 Elastic modulus measurements in response to small-amplitude oscillations ..............40 Figure 4.1 Three-dimensional view of streamlines for combined flow from the main and the side channels showing how flow from the side channels is superposed on main channel flow ...........44 Figure 4.2 Measured pressure drop and prediction by COMSOL, at two different shear rates, corresponding to main channel flow rates of 4 ml/s (Re=7) and 5.5 ml/s (Re=8).........................46 Figure 4.3 Normalized pressure drop measurements for the viscoelastic test fluids and the Newtonian solvent at a wall shear rate of 1100 s-1, corresponding to a main channel flow rate of 4 ml/s and a Reynolds number of 7. ..............................................................................................48 Figure 4.4 Normalized pressure drop measurements for the viscoelastic test fluids and the Newtonian solvent at a wall shear rate of 1500 s-1, corresponding to a main channel flow rate of 5.5 ml/s and a Reynolds number of 8. ...........................................................................................48 Figure 4.5 Normalized pressure drop measurements for the viscoelastic test fluids and the Newtonian solvent at a wall shear rate of 2000 s-1, corresponding to a main channel flow rate of 7.5 ml/s and a Reynolds number of 11. .........................................................................................49 Figure 4.6 Normalized pressure drop measurements for the viscoelastic test fluids and the Newtonian solvent for variation in main channel flow with no flow from the side channels. The Reynolds numbers corresponding to the lowest and highest flow rates are 2 and 11 respectively ........................................................................................................................................................52 Figure 4.7 Pressure measurement in (a) a Newtonian fluid and (b) a viscoelastic fluid (Reproduced from Bird et al., 1987)..............................................................................................55 Figure 4.8 Three-dimensional view of streamlines for side channel flow only............................58 Figure 4.9 Pressure measuring arrangement for the side flow experiment...................................59 Figure 4.10 Normalized pressure drop measurements for two viscoelastic test fluids and the Newtonian solvent for variation in extensional rates with side channel flow with no flow from the main channels. For the 1200 ppm fluid, the Deborah number corresponding to the extension rates are also shown. ......................................................................................................................60 Figure 4.11 Uniaxial Trouton ratio for a Boger fluid (a semidilute solution of 0.31 wt% polyisobutylene in polybutene) stretched over a range of extensional rates and plotted as a function of Hencky strain. (Reproduced from McKinley and Sridhar, 2002) ...............................63 Figure 4.12 Side view of section from flowcell showing intersection of the main, side and the slanted exit channels ......................................................................................................................65 Figure 4.13 Numerical results for velocity in the flow direction along channel centerline, obtained from COMSOL. ..............................................................................................................65
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Figure 4.14 Calculated values of Hencky strain in the flowfield plotted as a function of Deborah number ........................................................................................................................67 Figure 4.15 Measurements of First Normal Stress Difference for the 1200 ppm PEO solution in PEG Solvent. Solid line shows 1N from the Oldryod-B model fitted to the first six data points ........................................................................................................................................................70 Figure 4.16 Comparison of elastic pressure drop for the 1200 ppm PEO solution in PEG with extrapolated values of N1 corresponding to the wall shear rate downstream of the side flows. ....70 Figure A1 Tetrahedral mesh in channel geometry........................................................................81 Figure A2 Enlarged section showing mesh at the entrance to the main channel..........................82 Figure A3 Comparison between analytical results for flow in a rectangular channel with aspect ratio a/d=6.7 and channel height 2d, flow between two parallel plates, and numerical results from COMSOL...................................................................................................................83 Figure B1 Turbulent flow over a flat plate ...................................................................................87 Figure C1 Characterization of the original Boger fluid prepared by Boger, 1977. The blue symbols represent the shear stress and the red symbols represent the first normal stress difference. (Reproduced from James, 2009)..................................................................................93 Figure D1 Low pressure calibration curve using a column of water ............................................97 Figure D2 High pressure calibration curve using a column of water ...........................................97 Figure E1 Exterior view of flowcell .............................................................................................99 Figure E2 Interior view showing the channels in the flowcell .....................................................99 Figure E3 Side view of flowcell .................................................................................................100 Figure E4 Front view of flowcell................................................................................................100
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List of Tables Table 2.1 Summary of channel dimensions used in the final design of the flowcell ....................25 Table 3.1 Fluid properties of test fluids ........................................................................................43
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Nomenclature
a Channel half-width [m]
b Parameter determining type of extension
*c Crtitical concentration [ppm]
or xy Strain
O Initial strain
shear rate [s-1]
C Critical shear rate [s-1]
D Downstream channel wall shear rate [s-1]
U Upstream channel wall shear rate [s-1]
xy shear rate in the x-y plane [s-1]
d Channel half-height [m]
D Deformation rate tensor [s-1]
D Upper convected derivative of the deformation rate tensor [Pa.s-1]
De Deborah number
P Pressure drop [Pa]
elasticP Elastic pressure drop [Pa]
NewtonianP Newtonian pressure drop [Pa]
t Change in time [s]
v Velocity gradient tensor [s-1]
Tv Transpose of the vcelocity gradient tensor [s-1]
xi
xV Change in velocity in the flow direction [m/s]
y Displacement in the transverse direction [m]
Hencky strain
Extensional rate [s-1]
max Maximum extensional rate [s-1]
F Force [N]
F Inertial force [N]
F Viscous force [N]
G Dynamic storage modulus (slastic modulus) [Pa]
G Dynamic loss modulus (shear modulus) [Pa]
h Height of main and side channels [mm]
Uh Upstream channel height [mm]
Dh Downstream channel height [mm]
Shear viscosity [Pa.s]
E Extensional viscosity [Pa.s]
P Polymer viscosity [Pa.s]
PEG Viscosity of PEG Solvent [Pa.s]
S Solvent viscosity [Pa.s]
SP Specific viscosity [Pa.s]
Intrinsic viscosity [Pa.s]
Cone angle [rad]
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l Stretched length of fluid filament [m]
El Inlet length [mm]
ol Unstretched length of fluid filament [m]
L Main channel length [mm]
xL Length-scale in flow direction [m]
yL Length-scale in transverse direction [m]
SL Side channel length [mm]
eL Slanted exit channel length [mm]
or 1 Relaxation time [s]
2 Retardation time [s]
P Polymer contribution to the relaxation time [s]
M Applied torque [N.m]
1N First normal stress difference [Pa]
2N Second normal stress difference [Pa]
Kinematic viscosity [m2/s]
P Pressure [Pa]
OP Stagnation pressure [Pa]
1P Upstream pressure [Pa]
2P Downstream pressure [Pa]
*P Hole pressure error [Pa]
MQ Flow rate [ml/s]
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SQ Side channel flow rate [ml/s]
r Cone radius [m]
Re Reynolds number
ReI Reynolds number based on fluid acceleration
Density [kg/m3]
t Time [s]
T Observation time [s]
Stress tensor [Pa]
0 Initial stress [Pa]
Upper convected derivative of the stress tensor [Pa.s-1]
S Solvent contribution to the stress tensor [Pa]
P Polymer contribution to the stress tensor [Pa]
P Upper convected derivative of the solvent contribution to the stress tensor [Pa.s-1]
W Wall shear stress [Pa]
xy Shear stress in the x-y plane [Pa]
xz Shear stress in the x-z plane Normal stress in the x-direction [Pa]
yz Shear stress in the y-z plane [Pa]
xx Normal stress in the x-direction [Pa]
yy Normal stress in the y-direction [Pa]
zz Normal stress in the z-direction [Pa]
*u Friction velocity [m/s]
xiv
U Bulk velocity [m/s]
U Free-stream velocity [m/s]
v Velocity [m/s]
Savgv , Side channel average velocity [m/s]
xv x-component of velocity [m/s]
yv y-component of velocity [m/s]
zv z-component of velocity [m/s]
Vol Volume [m3/s]
w Main channel width [mm]
Sw Side channel width [m]
Wi Weissenberg number
CWi Critical Weissenberg number
Frequency of oscillation [s-1]
Angular velocity [rad/s]
*y Spatial parameter for turbulent boundary layer
1
Chapter 1: Introduction
Drag reduction, also known as Toms’ effect, has been a topic of interest in fluid mechanics over
the last 60 years. In 1948, Toms discovered that the addition of a small amount of high-
molecular-weight polymer to a Newtonian turbulent flow reduced the wall shear stress by up to
70%. Considering the extent by which an inertia-driven phenomenon such as turbulence can be
interrupted by a polymer concentration as low as 5 parts per million, this observation was quite
remarkable, more so because the addition of such low quantities of polymer hardly causes any
change in the fluid’s viscosity, implying that the phenomenon is not a viscous effect. Since its
discovery, drag reduction has been extensively used in industrial pipe flows such as those in long
transcontinental oil pipelines, as an effective method to reduce power consumption. In addition,
this phenomenon has applications in many other areas where it is desirable to increase the flow
rate without increasing the pumping costs, such as in the hoses of fire fighting equipment (Sellin
and Ollis, 1980; Khalili et al., 2002).
Despite the widespread application of the technique, the mechanism of drag reduction has not
been fully understood (White and Mungal, 2008). However, it is believed that this mechanism
pertains to the fluid elasticity caused by the long polymer chains. The induced elasticity,
although weak, is believed to be sufficient to provide resistance to fluid stretching during vortex
formation in the turbulent boundary layer, where vortices are generated though a process known
as the turbulent burst.
Turbulent bursts were characterized in an experiment conducted by Kim et al. (1971). Using
hydrogen bubble time-streak markers in a turbulent flow of water over a flat surface,
Chapter 1: Introduction 2
they observed the formation of streamwise vortices in the viscous sublayer, the region closest to
the wall in the turbulent boundary layer. With a Newtonian fluid, the vortex is lifted up from the
wall and grows in the buffer layer where inertial forces compete with viscous forces.
Subsequently, the vortex enters the log region, where inertial stresses completely outweigh
viscous stresses and the flow is overwhelmingly turbulent, and breaks up with the onset of even
more chaotic fluctuations and the cycle starts again. This process, starting with the formation of a
vortex in the viscous sublayer and ending with its break-up in the log region, is termed the
turbulent burst. Figure 1.1 shows the bubble markers and Figure 1.2 is an image from a
photographic plate showing the streamwise vortex just after formation.
Figure 1.1 Setup of turbulent burst experiment (Reproduced from Kim et al, 1971)
Chapter 1: Introduction 3
Figure 1.2 Photographic plate showing H2 bubble lines (Reproduced from Kim et al, 1971)
Donohue et al (1972) conducted a set of experiments to determine if the degree of drag reduction
can be correlated with the rate of turbulent burst. As turbulent bursts can be characterized as
streaky structures traveling at low speeds, they decided to use the spacing between streaks as a
measure of the amount of bursting. Using a dye-injection method for flow visualization, they
compared the spacing between low speed streaks in a turbulent channel flow between water and
a drag-reducing fluid: a dilute solution of polyethylene oxide in water. They observed an
increase in streak spacing with the polymer solution compared to that with the solvent at the
same flow rate, indicating fewer bursts in the polymer solution. A separate experiment involving
pressure drop measurements in a turbulent pipe flow with water and the same polymer solutions
was conducted in order to determine the percentage of drag reduction. A correlation of the results
from the two experiments showed that an increase in the amount of drag reduction corresponds
to a decrease in the rate of turbulent bursting.
Chapter 1: Introduction 4
Although results from Donohue et al (1972) confirmed that drag reduction is related to the
turbulent burst, the exact mechanism by which polymer molecules interact and suppress the
bursting process was still not understood. In order to relate the interaction of polymer molecules
with the turbulent boundary layer, it is necessary to closely examine the underlying fluid motion
during vortex formation in a burst. Particular emphasis needs to be given to the first stage of the
bursting process which involves the lifting of fluid away from the wall, requiring a force to be
exerted on the fluid in the perpendicular direction. Prevention of lifting should therefore suppress
turbulent burst. Hence, it is necessary to understand the details of this mechanism.
Fluid lifting in a turbulent burst takes place via formation of counter-rotating vortices known as
Hairpin vortices. When a flow at high Reynolds number is perturbed, the vorticity field
associated with the flow rearranges in such a way that a pair of counter-rotating vortices exerts
an upward force on the fluid causing the fluid to be lifted up from the wall in the shape of
hairpins, and so creates “hairpin vortices”. The formation of hairpin vortices is an extensional
motion, as shown in Figure 1.3, and it is this mechanism that causes lifting of fluid away from
the wall.
Chapter 1: Introduction 5
Figure 1.3 (a) Formation of a hairpin vortex. (b) A group of hairpin vortices being lifted up from
the surface. (c) A pair of counter-rotating vortices exerting upward force on the fluid. (Reproduced from Davidson, 2004)
1.1 Previous Work with Drag Reducing Fluids
Drag reducing fluids are dilute polymer solutions, i.e, a solution in which the macromolecules
are so few that there is no interaction between them. The only interaction in these fluids is
between the polymer chains and the solvent. Several researchers (Metzner and Metzner 1970,
Chauveteau 1981, James and Saringer 1980) reported that, in laminar flow of dilute polymer
solutions, the onset of elastic effects usually take place when the extensional rate, , exceeds the
inverse of the longest relaxation time, , of the fluid, i.e,
1 ,
Chapter 1: Introduction 6
corresponding to a Deborah number )( De greater than unity. For a drag-reducing
concentration of an aqueous polymer solution, the longest relaxation time is of the order of 1
millisecond, and thus the critical extensional rate required for elastic effects to take place is
~ 1000 s-1. However, measurements by Muller & Gyr (1986) showed that the extensional rate
in a turbulent burst is only about 50 s-1, a value almost two orders of magnitude lower than the
critical extensional rate required in laminar flow. This discrepancy can be explained by realizing
that in a laminar extensional flow, a macromolecule generally enters the extensional flowfield
from a “strain-free” region where it undergoes zero or very little deformation and hence the
polymer chain remains close to its equilibrium configuration before being extended by the
extensional field. However, in the turbulent boundary layer, the macromolecule is subjected to
considerable shearing near the wall and is already partially extended before reaching the
extensional flowfield. James et al. (1987) conducted an experiment to investigate whether
preshearing has an effect on the laminar extensional flow of dilute polymer solutions.
The apparatus for their experiment, as shown in Figure 1.4, was a rectangular channel with an
axisymmetric orifice channel placed in the bottom wall at a downstream location in the channel.
The fluid was presheared in the rectangular channel before entering the orifice where it was
extended. The pressure drop was measured between the location of the fluid entering the orifice
and that of the fluid leaving the orifice. The shear rate was varied by varying the flow rate in the
rectangular channel while the extensional rate was varied by varying the flow rate through the
orifice by controlling the flow restriction in the orifice.
Chapter 1: Introduction 7
Figure 1.4 Setup of presheared extensional flow experiment.
(Reproduced from James et al, 1987)
The results from this experiment showed that, even with preshearing, the extensional rates
required for obtaining elastic effects were much higher than the value measured in the turbulent
boundary layer. For an upstream shear rate of 800 s-1, extensional rates greater than 600 s-1 were
required to produce elastic effects with a polymer concentration of 20 ppm. Moreover, this
experiment did not completely resemble the turbulent boundary layer because the motion of
counter-rotating vortices was not simulated in this experiment. Also, in this experiment, in order
to undergo extension, the fluid had to flow into the wall while in the turbulent boundary layer the
fluid undergoes extension while being lifted away from the wall.
Planar vs Uniaxial Extension
Since it is generally accepted that drag reduction is an elastic effect in extension, it is worthwhile
to investigate whether this phenomenon can be correlated to elasticity in uniaxial extension, the
simplest and the most common form of extensional flow. James and Yogachandran (2006)
demonstrated that the breaking length of fluid filaments under uniaxial extension can be a
Chapter 1: Introduction 8
measure of the fluid’s elasticity. They then attempted to correlate drag reduction to this measure
of elasticity in uniaxial extension. However, no correlation could be established. The reason was
attributed to the fact that, in uniaxial extension, the polymer chains are extended a considerable
amount of their original length; however, in drag reduction, the chains are not extended as much
because shear is the dominant mode of deformation in the wall region where the polymer is
operative. By performing numerical simulation of polymerstretching in the turbulent boundary
layer, Terrapon et al. (2004) showed that the polymer chains are stretched a large fraction of their
full extension but not stretched to their fullest extent. This is important because extensional
stresses are roughly proportional to the cube of the effective length of the polymer chain. Their
simulation results also indicated that the polymer chains are highly extended in regions of planar
extension which is preceded by shearing.
1.2 Research Objectives
Results from the two experimental works described above indicate that neither pure uniaxial
extension nor axisymmetric extension coupled with preshearing was able to accurately model the
mechanism of drag reduction. No attempt has yet been made, however, to simulate this
phenomenon by creating a planar extensional flowfield coupled with preshearing, or to model the
extensional motion involving counter-rotating vortices. As planar extensional motion created by
counter-rotating vortices is a crucial step in the formation of vortices in the turbulent boundary
layer, an accurate simulation of these motions should lead to a representative model of the
underlying dynamics of the turbulent boundary layer. The objective of the present study is
therefore to understand the mechanism of drag reduction by examining the effect of fluid
elasticity in a laminar flowfield created to simulate the first stage of a turbulent burst. With a
Chapter 1: Introduction 9
laminar flow at low Reynolds number, it should be possible to identify elastic responses in the
flowfield. These responses are not easy to identify in a turbulent flow because in turbulent flows
inertial effects are dominant and hence elastic effects are difficult to separate experimentally
from inertial effects . The goals of the research are then:
To design and fabricate a flowcell capable of simulating a turbulent burst using a laminar
flow.
To minimize the Reynolds number in this flow in order to eliminate inertial effects.
To measure the pressure drop and flow rate in this flowcell first with a Newtonian fluid
and then with dilute polymer solutions, and explain the difference in results.
10
Chapter 2: Experimental Methodology
In order to run a controlled experiment with the desired flowfield, an approach involving the
following steps was adopted:
Design and fabrication of a flowcell capable of simulating with a laminar flow the fluid
mechanics of preshearing and planar extension in a turbulent boundary layer. Previous
work and numerical simulation will be used to determine the optimum geometry of the
flowcell.
A mechanism to vary shear and extensional rates.
Testing Newtonian and non-Newtonian fluids, specifically, dilute polymer solutions.
Comparison of measurements between the fluids to determine the elastic effects.
2.1 Conceptual Design
The first objective of the design problem is to establish a planar extensional flowfield combined
with preshearing. In order that the dynamics correspond to a turbulent burst, this flowfield should
be able to generate a minimum shear rate of 1000 s-1, a minimum extensional rate of 50 s-1, and
resemble the motion of counter-rotating vortices responsible for causing extensional motion in
the turbulent boundary layer. In addition, the Reynolds number needs to be low enough to ensure
that inertial effects can be neglected.
Preshearing can be achieved by flow through a wide rectangular channel built into the flowcell.
If this flow is fully developed, the velocity profile will be parabolic, with the highest shear rate at
the walls. As the wall shear rate is proportional to the flow rate in the channel, the required
Chapter 2: Experimental Methodology 11
shear rates can be achieved by adjusting the flow rates. Figure 2.1 below shows the schematic of
a fully-developed shear flow through a rectangular channel.
Figure 2.1 Fully developed flow through a rectangular channel. Parabolic velocity profile
A planar extensional flowfield, on the otherhand, can be set up in several ways. One common
method is to use a stagnation point flow. This type of flow is particularly preferred because the
shear stresses are identically zero in this flowfield, except at confining walls, causing the flow to
be purely extensional away from the walls. A planar extensional flowfield is shown in Figure
2.2. A particular technique for generating a stagnation point flow involves flow in a cross-slot
device, as shown in Figure 2.3. With opposing inlet and outlet flows, a stagnation point is created
at the centre of the device.
Figure 2.2 Planar extensional flowfield. The streamlines are typically hyperbolic
Chapter 2: Experimental Methodology 12
Figure 2.3 Cross-slot geometry in three-dimensions (Reproduced from Winter et al., 1979)
A modified flowfield consisting of a horizontal bottom plate and the top half of a stagnation
point flowfield can be used to effectively simulate the planar extensional flowfield created in the
first stage of the turbulent burst cycle. In this way, the effects of the counter-rotating vortices of a
turbulent burst can be achieved by the hyperbolic streamlines of stagnation point flow exerting
an upward force and corresponding extensional stresses. This flowfield retains the prime features
of a purely extensional flow as well as the linear dependence of the normal stresses on the spatial
dimensions. Also, extensional rates can be varied by varying the flowrate through the flowfield.
This concept of a presheared planar extensional flowfield led to the design of the flowcell shown
in Figure 2.4.
Chapter 2: Experimental Methodology 13
This flowcell consists of three inlets and one outlet. The flow in the main channel is a gravity-
driven flow from an overhead reservoir and its purpose is to provide preshearing, to generate
shear rates greater than 1000 s-1. The flow from the side channels is generated by a pressurized
container and is meant to establish a planar extensional flowfield with extensional rates higher
than 50 s-1. This flow, superimposed on the shear flow from the main channel, is meant to create
the motion of counter-rotating vortices in the turbulent boundary layer. The exit channel has
been slanted at an angle of 13° to the horizontal in order to resemble the lifting motion of a
streamwise vortex as illustrated in Figure 1.2 from Kim et al (1971). All the channels have
rectangular cross-sections.
The flow rate in the main channel and the combined flow rates in side channels are the quantities
to be controlled in each experiment. The flow rate in the main channel is controlled using a valve
before the entrance of the main channel, as shown in Figure 2.5. The flow rate in the side
channels is controlled by the pressure in the container which generates this flow. The fluid
exiting the flowcell is collected and weighed over time using a digital balance and a timer in
order to determine the total flow rate.
This proposed flowcell should be a good model because it incorporates all of the essential
features present in the first stage of a turbulent burst. The side flows simulate extensional motion
caused by counter-rotating vortices while the main flow ensures that the flow is presheared, high
enough to produce elastic effects. Thus, the flowcell should be a good first attempt to model
turbulent burst using a laminar flow.
Chapter 2: Experimental Methodology 14
Figure 2.4 Setup of experimental flowcell
Figure 2.5 Setup of experimental apparatus
Chapter 2: Experimental Methodology 15
2.2 Design Considerations
Although rectangular channel flow is present in all components of the flowcell, the calculations
to design the flowcell can be greatly simplified if the flow can be modelled as pressure-driven
flow between two parallel plates. For this assumption to be valid, the channel aspect ratio of
width to height has to be high enough for the influence of the side walls to be negligible. In order
to test the validity of this assumption, the analytical solution for flow in a rectangular channel,
with various aspect ratios, was compared with the solution for flow between two parallel plates.
The details of this comparison can be found in Appendix B1. The results are shown in Figure
2.6.
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2
y/d
Ve
loci
ty (
m/s
)
Rectangular channel with a/d = 6.7
Rectangular channel with a/d = 4.0
Rectangular channel with a/d = 3.0
Rectangular channel with a/d = 2.5
Flow between parallel plates
Figure 2.6 Comparison between vertical velocity profiles midway between the side walls for
flow in a rectangular channel with different aspect ratios, and flow between two parallel plates. Channel width = 2a, height = 2d
Chapter 2: Experimental Methodology 16
As the plot shows, at an aspect ratio greater than 3, the velocity profile calculated assuming 2D
flow between two parallel plates is in agreement with the analytical solution for the velocity
profile in a rectangular channel. Hence, an aspect ratio of 6.7 was chosen as the design aspect
ratio to neglect the influence of the side walls. Moreover, negligible influence of the side walls
means that the assumption of pressure-driven flow between parallel plates can be used to
perform design calculations for the flowcell. This assumption was therefore used in the
subsequent design calculations.
Computational fluid dynamics (CFD) is another method for analyzing the flow in the flowcell
geometry. The advantage of this method is that no simplifying assumptions are required and flow
behaviour, predicted by numerically solving the Navier-Stokes equations, can be visualized in all
sections of the flowcell, including in regions close to the side walls. Hence, simulation results
were used to validate the design of the flowcell and verify that many of the desired flow
parameters are achievable. For this purpose, results from numerical simulations performed using
COMSOL Multiphysics 4.0, are provided in the relevant sections to validate a number of design
decisions. Details of the software package and numerical computations can be found in
Appendix A2.
2.2.1 Main Channel Geometry
In order to eliminate inertial effects, it is desirable to have as low a Reynolds number as possible.
Hence, one of the objectives during the design of the flowcell was to minimize the Reynolds
number. In the present work, the Reynolds number, Re, based on the of the main channel is
given by:
Chapter 2: Experimental Methodology 17
W
QVh M
Re , (2.1)
where is the fluid density, is the fluid viscosity, h is the height, and V is the average
velocity given by:
WhQV M / , (2.2)
where MQ is the flow rate.
Because the Reynolds number is proportional to the flow rate and inversely proportional to the
viscosity, lowering the flow rate and increasing the viscosity of the fluid are two methods to
reduce the Reynolds number. As the primary objective is to model the first stage of a turbulent
burst, one estimate of the experimental Reynolds number is the local Reynolds number in the
viscous sublayer. This Reynold’s number can be estimated from the definition of *y , given in
Appendix B as:
yu
y ** , (2.3)
where *u is the friction velocity, and is the kinematic viscosity of the fluid. Thus the quantity
*y is analogous to the definition of Reynolds number and can be used as an estimate of the
design Reynolds number. In the viscous sublayer, *y ranges from 0 to 5. Therefore, a Reynolds
number of 5 in the flowcell should correspond to the flow in the viscous sublayer.
As discussed in section 1.1, for dilute aqueous polymer solutions, the onset of drag reduction
takes place at a shear rate of 1000 s-1. This critical value is based on the reciprocal of the
Chapter 2: Experimental Methodology 18
relaxation time of aqueous polymer solutions with relaxation times in the order of 1 ms. As most
viscoelastic fluids have relaxation times greater than 1 ms, the value of 1000 s-1 should be an
upper bound for the critical shear rate and should be high enough for the onset of drag reduction
with most viscoelastic fluids. Therefore, the flowcell should be able to generate a shear rate
considerably higher than this value in order to successfully simulate drag reduction effects.
The wall shear rate, W , can be derived from the approximation of flow between two parallel
plates as:
2
6
Wh
QMW . (2.4)
That is, the wall shear rate is inversely proportional to the square of the channel height. Hence,
the wall shear rate can be increased by reducing the channel height or increasing the flow rate.
The above equation should provide a reasonably accurate estimate of the centreline wall shear
rate in the flowcell, half-way from both the sidewalls.
To allow sufficient tolerance for machining of the flowcell, a minimum height of h = 1.5 mm
was chosen as the height of both the main and the side channels. A main channel flow rate range
of 4 ml/s to 7.5 ml/s was chosen because this range falls within the range of flow rates that can
be generated by the overhead reservoir. These values of h and MQ combined with the design
aspect ratio of W/h = 6.7, generates, according to equation 2.4, a wall shear rate range of W
from 1067 s-1 to 2000 s-1, which meets the design objective of having a wall shear rate greater
than 1000 s-1. From equation 2.1, the Reynolds numbers corresponding to this geometry and flow
Chapter 2: Experimental Methodology 19
rates range from 2 to 11. This range is comparable with the Reynolds numbers in the viscous
sublayer.
2.2.2 Side Channel Geometry
As discussed in section 1.1, the extensional rate in turbulent burst is approximately 50 s-1. In the
flowcell, extensional motion is created when the flow from the side channels is superimposed on
the flow from the main channels. This motion takes place in the region where the side channels
meet the main channel, as shown in Figure 2.4. The extensional rate is given by (Coventry &
Mackley, 2008):
,2hW
Q
S
S (2.5)
where SQ is the combined flow rate in both side channels, and h and SW are the side channel
height and width respectively. Thus, the extensional rate is proportional to the side channel flow
rate and inversely proportional to the side channel height and the square of the side channel
width. The obvious way of increasing the extensional rate is to increase the side flow rate, but
the maximum flow rate is limited by the pressure limit of the tank used to drive the side flow.
Therefore, a maximum side flow rate of 1 ml/s, corresponding to the tank pressure limit, was
used in the calculations. To maintain geometric conformity and to allow sufficient machining
tolerance, the side channel height was kept the same as the main channel height, i.e, h = 1.5 mm.
The side channel width therefore needs to be as small as possible in order to maximize the
extensional rate. To allow for sufficient machining tolerance, the side channel width, SW , was
chosen to be 2 mm. Substituting these values of SQ , h and SW into equation 2.5 yields a
Chapter 2: Experimental Methodology 20
maximum extensional rate of = 333 s-1, which is well above the required extensional rate in a
turbulent burst. From equation 2.1, the maximum Reynolds number corresponding to this
geometry and flow rate is 11. This value is comparable to the value of 5 in the viscous sublayer,
and thus ensures that inertial effects can be neglected.
2.2.3 Channel Lengths
To ensure that a parabolic velocity profile is established in the vertical centreplanes and the shear
rate at the wall is maximum, it is necessary for the flow in the main channel to become fully
developed before reaching the flow from the side channels. The minimum length, El , of the main
channel required for the flow to become fully developed can be calculated using the equation for
fully developed laminar flow in a rectangular channel assuming uniform flow at the entrance
(Schilchting, 1960, p. 171):
416.0 2Vh
lE , (2.6)
where is the kinematic viscosity of the fluid.
Using the design channel height, the maximum main channel flow rate, and a kinematic viscosity
of 8.5 x 10-5 m2/s, the maximum inlet length was calculated as El = 18 mm. This value of
viscosity is the kinematic viscosity of a 33.3% solution of polyethyelene glycol in water. The
justification for using this viscosity will be provided in the section describing the choice of
fluids. To ensure that the length is long enough for the flow to become fully-developed, a main
channel length of L = 28 mm was chosen.
Chapter 2: Experimental Methodology 21
Similarly, a side channel length of LS = 10 mm was chosen to ensure that the side channel flow is
also fully-developed.
2.2.4 Exit Channel Length
The exit channel length was chosen as 13 mm. This distance allows enough space to install a
pressure tap in the exit channel wall, leaving a workable machining distance on either side of the
tap.
2.2.5 Pressure Tap Locations
The pressure drop along the channel will be measured at each flow rate. For a Newtonian
laminar flow, this pressure drop arises due to the fluid’s viscosity. For a polymeric liquid,
however, the pressure drop can be caused by both viscosity as well as elasticity. For such a fluid,
the extra pressure drop obtained above values for a Newtonian fluid with the same viscosity is a
measure of the fluid’s elasticity. Therefore, if this experimental design is indeed a model for
turbulent burst, the elastic pressure drop obtained should be a function of the amount of drag
reduction.
As shown in Figures 2.4 and 2.5, the pressure difference between a location upstream in the main
channel flow and a location downstream of the flow from the side channels is measured. These
locations would ensure that the measured pressure drop is between a region where the fluid has
not been extended and a region where it has been subjected to planar extension. The criterion for
choosing the locations was obtaining pressure drop readings across a portion of the channel
where the flow has been subjected to both preshearing and planar extension. To do this, the high
Chapter 2: Experimental Methodology 22
pressure tap was placed at a distance of 11 mm from the main channel entrance while the low
pressure tap was placed 6.5 mm from the slanted channel exit.
2.2.6 Pressure Drop & Choice of Test Fluids
The pressure transducer used in the experiment is a Honeywell differential pressure sensor which
generates an electrical signal based on the deflection of its silicone membrane. This was the
transducer that was available in the laboratory and therefore was used for this exploratory work.
This device was calibrated using a column of water. Detailed specifications of the pressure
transducer as well as the results of the calibration can be found in Appendix D. This pressure
sensor can provide a minimum pressure reading of 0.1 psi in its linear range. Hence, it is
necessary to ensure that the pressure drop in the channel is high enough to be measured by the
transducer. High values would also ensure that the readings are not affected by noise associated
with weak signal. The pressure drop, P , along the flowcell can be estimated using the
following equation based on flow between parallel plates:
3
12
wh
LQP M (2.7)
Although water is usually the fluid of choice in experiments involving channel flows. However,
using water in the flowcell gives pressure drops from 0.005 psi to 0.01 psi, values much below
the minimum 0.1 psi of the pressure transducer. To increase the pressure drop to a measurable
level, it is necessary to increase the viscosity of the fluid to about 20 times that of water. Since
drag reduction is observed with solutions of very low polymer concentration, the addition of the
very small quantities of a drag-reducing polymer would not have much effect in altering the fluid
viscosity. Hence it was decided that a viscous solvent would be used instead of water.
Chapter 2: Experimental Methodology 23
Experiments conducted by Dontula et al (1998) showed that dissolving moderate quantities of
polyethylene glycol (PEG) in water can increase the viscosity of the solution by up to 200 times
that of water while the fluid remains Newtonian. PEG is a low molecular weight polymer and
hence non-Newtonian effects are not observed as long as the concentration is below
approximately 43% by weight. The advantage of using an aqueous PEG solution over other
viscous liquids like glycerol or silicone oils is that a PEG solution is capable of dissolving water-
soluble polymers, a property necessary for the present work.
For the present experiments, polyethylene glycol, with a molecular weight of 8000, was
dissolved in water to prepare a fluid with a concentration of 33.3% by weight. Dontula et al
(1998) reported this fluid has a viscosity 86 times that of water. Hence, this fluid was used as the
solvent for dissolving drag-reducing polymers. The density of PEG is almost identical to that of
water, i.e, = 1000 kg/m3. This fluid is subsequently referred to as the PEG solvent.
For a flow rate of 7.5 ml/s and viscosity of 0.086 Pa.s, numerical simulation using COMSOL
gives a pressure drop of 0.7 psi along the channel between the pressure taps. This value is within
the measurable range of the pressure transducer.
2.3 Flowfield with Final Design
Since the final design of the flowcell is now complete, this design geometry, with dimensions
summarized in Table 2.1, can be used in COMSOL to obtain numerical velocity profiles in
different parts of the flowcell. These velocity profiles are important in understanding the
Chapter 2: Experimental Methodology 24
flowfield and to observe if the flow behaviour is as expected. The PEG solvent’s fluid properties,
namely the values of the viscosity and density, were used to obtain the simulation results. For
running these simulations, a uniform flow velocity of 0.5 m/s, corresponding to a flow rate of 7.5
ml/s, was used as the main channel entrance flow velocity while a uniform flow velocity of 0.2
m/s, corresponding to a flow rate of 0.6 ml/s, was used as the side channel entrance flow velocity
in each side channel. These velocities were chosen to correspond to the upper limits of the design
flow rates in order to achieve maximum shear and extensional rates in the flowcell.
Geometric Parameter Value Unit
Main channel width, W 10 mm
Main channel height, h 1.5 mm
Main channel length, L 28 mm
Side channel width, WS 2 mm
Side channel height, h 1.5 mm
Side channel length, LS 10 mm
Exit channel height, h 1.9 mm
Exit channel length, Le 13 mm
Slant angle 13 degrees
Table 2.1 Summary of channel dimensions used in the final design of the flowcell
Figure 2.7 is a snapshot from COMSOL showing the magnitude of the flow velocity along the
vertical centreplane of the main channel. An enlarged view of the streamwise (x-component)
velocity at the entrance section is shown in Figure 2.8. The velocity magnitudes are shown in the
Chapter 2: Experimental Methodology 25
accompanying color legends, which indicate that a uniform flow velocity of 0.5 m/s at the
entrance becomes fully developed with a maximum of about 0.75 m/s.
Figure 2.7 Streamwise component of velocity along vertical centreplane
Figure 2.8 Enlarged view of streamwise velocity at the main channel entrance
Chapter 2: Experimental Methodology 26
Figure 2.9 shows the streamwise velocity component (z-direction) in the side channel vertical
centreplane. This plot confirms that the flows in the side channels are fully developed and are
equal and in opposite directions.
Figure 2.9 Streamwise component of velocity in the vertical centreplane of the side channels. .
Color legend shows velocity in metres per second (m/s).
The above velocity plots provide some insight into the expected flow behavior in the flowcell.
The flow in both the main and the side channels become fully developed, confirming that the
channel lengths are sufficiently long, even for the highest flow rates to be used in the
experiments. Further numerical simulation using COMSOL will be used in subsequent chapters
to analyze the experimental results, in particular to calculate strain and strain rates, which are
useful in explaining the cause of elastic effects.
Chapter 2: Experimental Methodology 27
2.4 Fabrication of Flowcell
Detailed engineering drawings of the parts and assemblies required to fabricate the flowcell were
prepared using SolidWorks. The detailed drawings for each part can be found in Appendix E.
The material was polycarbonate, chosen for its visual transparency, water-resistance and ability
to form chemical bonds with adhesives. The flowcell was constructed at the University of
Toronto’s MIE machine shop.
28
Chapter 3: Test fluids
As described in section 2.2.6, a 33.3% by weight of polyethylene glycol solution dissolved in
water was chosen as the inelastic fluid for establishing the Newtonian baseline, against which
pressure drop measurements of dilute polymer solutions will be compared. Several viscoelastic
fluids were prepared by dissolving different concentrations of polyethylene oxide in the
polyethylene glycol solvent. This chapter describes the fluid characterization tests conducted on
these fluids and the results. These tests were carried out to measure the relevant viscous and
elastic properties of these viscoelastic fluids. The rheological concepts behind these fluid
properties, as well as the laboratory techniques used to conduct the tests are presented first.
3.1 Non-Newtonian Fluids
The most important fluid property that differentiates a non-Newtonian fluid from a Newtonian
fluid is viscosity. For simple shearing, the viscosity, , is defined as:
xy
xy
, (3.1)
where xy is the shear stress in the x - y plane and xy is the shear rate defined as
y
vxxy
, (3.2)
where xv is the velocity component in the flow (x) direction. Unlike Newtonian fluids, the
viscosity of most non-Newtonian fluids decreases with shear rate, i.e, the fluids are shear-
thinning. There also exist fluids, such as a concentrated suspension of corn starch in water,
whose viscosity increases with shear rate, and as such are termed as shear-thickening fluids.
Chapter 3: Test Fluids 29
Another important property of non-Newtonian fluids is elasticity. Many polymeric liquids
exhibit behaviours such as stringiness which indicate that these fluids possess elasticity in
addition to viscosity.
3.1.1 Boger Fluids
Boger fluids are dilute polymer solutions whose viscosity remains almost constant with respect
to shear rate. This property makes these fluids special because it enables elastic effects to be
clearly separated from viscous effects. Although most polymer solutions and melts are inherently
shear thinning, the polymer concentrations in Boger fluids are low enough that the variation in
viscosity can be ignored. These fluids were first introduced by Boger in 1977 and, since then,
they have been an effective means of studying elastic effects of polymer solutions (Boger, 1977).
In an experiment conducted with two fluids: a Boger fluid and a Newtonian fluid with the same
viscosity, the difference in outcomes at the same flow rate can be attributed to elasticity alone. In
experiments where the viscosities between the fluids are different, the results can still be
compared by making use of appropriate dimensionless groups. Thus, Boger fluids have made it
possible to determine whether an observed non-Newtonian effect is caused by shear thinning, or
elasticity, or both (James, 2009). Therefore, although drag-reducing fluids are usually not Boger
fluids, because drag reduction is caused by elasticity, Boger fluids can be used to understand the
role of elasticity in causing drag reduction.
3.1.2 Rheometry
Shear rheometers are the most common instruments used to characterize non-Newtonian fluids.
A wide range of rheological shear properties including viscosity, first normal stress difference,
Chapter 3: Test Fluids 30
and viscous and elastic moduli can be measured by these instruments. In the present study, a
cone-and-plate rheometer was used to characterize the test fluids.
A cone-and-plate rheometer consists of a fixture, as shown in Figure 3.1, mounted over a flat
plate leaving a small gap for the fluid sample to be inserted in the space between the fixture and
the plate. The cone angle is typically between 0.5 to 2 degrees, while the diameter is usually
between 2 cm to 6 cm.
Figure 3.1 Working principle of a cone-and-plate rheometer
As the cone rotates with a constant angular velocity, , it generates a uniform shear rate, ,
throughout the fluid:
, (3.3)
where is the cone angle. Thus a wide range of shear rates can be obtained depending on the
range of angular velocity of the machine’s motor.
The shear stress, , in the fluid can be expressed in terms of the torque, M , according to the
following relationship (Macosko, 1994):
Chapter 3: Test Fluids 31
32
3
R
M
, (3.4)
where R is the cone radius. Using equation 3.1, equation 3.4 can be rewritten to obtain an
expression for the fluid viscosity, :
32
3
R
M
(3.5)
Using the dimensions of the cone geometry, the rheometer measures the torque in order to
determine the viscosity for each angular velocity and calculates the shear rate corresponding to
this angular velocity and thus produces viscosity measurements at different shear rates.
3.2 Shear Viscosity of Test Fluids
The drag reducing polymer used in this work was polyethylene oxide (PEO) with a molecular
weight of 4 million. PEO is well-known to cause drag reduction and has been used in many prior
studies (for example James et al. 1987, and Scrivener 1974). Solutions with concentrations of
100 ppm, 500 ppm, 750 ppm, 1000 ppm, and 1200 ppm PEO dissolved in the PEG solvent were
prepared. Shear viscosity measurements of these fluids were made with an AR2000 rheometer
using a 60 mm 0.5 degree cone-and-plate fixture at a temperature of 25° C. The temperature was
chosen to coincide with the laboratory temperature when the flowcell experiments were
conducted. Figure 3.2 shows the viscosity data as a function of shear rate. The test fluids were
sheared up to a shear rate of 2000 s-1, corresponding to the maximum shear rate expected in the
flowcell.
The plot shows that the viscosity of PEG solvent alone is independent of the shear rate,
confirming that the solvent exhibits Newtonian behavior in this range of shear rates. The
Chapter 3: Test Fluids 32
constant viscosity of PEG solvent was found to be 85 mPa.s, which is within 1% of the value of
86 mPa.s reported by Dontula et al (1998) for a solution with the same concentration of PEG.
With this value of viscosity, the pressure drop in the main channel should range from 2800 Pa to
5500 Pa, corresponding to a range of 0.4 psi to 0.8 psi. This pressure drop is within the 0.1 psi –
1 psi measurable range of the pressure sensor.
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
10 100 1000 10000Shear Rate (s-1)
Vis
cosi
ty (
Pa.
s)
1200 ppm PEO in PEG Solvent
500 ppm PEO in PEG Solvent
100 ppm PEO in PEG Solvent
PEG Solvent
Figure 3.2 Steady shear viscosity measurements for test fluids
Also shown in Figure 3.2 are the viscosity measurements for the 100 ppm, 500 ppm, and the
1200 ppm PEO solutions. These results indicate that the addition of the 100 ppm PEO increases
the viscosity of the PEG solvent by 4%, 500 ppm by 20%, and 1200 ppm by 50%. The solutions’
viscosities were virtually constant indicating that the fluids are Boger fluids.
Chapter 3: Test Fluids 33
3.3 Critical concentration
As described earlier, drag reducing fluids are dilute polymer solutions, i.e, a solution in which
the macromolecules are so few that there is no interaction between them. The only interaction in
these fluids is between the polymer chains and the solvent. An experimental measure of the
boundary separating dilute and semi-dilute solutions is defined by the critical concentration, *c .
A widely accepted definition of *c , provided by Graessley (1980), is:
][
77.0*
c , (3.6)
where ][ is the intrinsic viscosity defined as
csp
c
0lim][ (3.7)
where c is the concentration, and sp is the specific viscosity defined as:
s
ssp
, (3.8)
where is the overall viscosity and s is the solvent viscosity. The quantity csp
is known as
the reduced viscosity.
The critical concentration of the PEO/PEG solution was determined by plotting the reduced
viscosity, cSP
, versus the concentration for four different concentrations, as shown in Figure
3.3.
Chapter 3: Test Fluids 34
520
522
524
526
528
530
0 200 400 600 800 1000 1200 1400
Concentration (ppm)
Red
uce
d V
isco
sity
(η
sp/c
)
ηsp=(η-ηs))/ηs
[η]
Figure 3.3 Determination of intrinsic viscosity
The intrinsic viscosity was evaluated by extrapolating the plotted data in Figure 3.3 to zero
concentration to obtain the intercept of 521 ppm-1.
The critical concentration, *c , was then obtained, using equation 3.6, as
ppmc 148077.0*
Thus, the test fluids were all within the dilute regime for this polymer/solvent combination, even
though their concentrations are much higher than the O(10) ppm concentration of well-known
aqueous drag-reducing fluids. Because the solutions are dilute, the test fluids can be considered
Boger fluids.
Chapter 3: Test Fluids 35
3.4 Relaxation Time & First Normal Stress Difference
When a Newtonian fluid is subjected to a step increase in strain, as shown in Figure 3.4 a), the
stress, , relaxes instantly to zero, as shown in Figure 3.4 b). However, when a viscoelastic fluid
is subjected to the same deformation, the stress decays as shown in Figure 3.4 c) (Macosko,
1994, p.110).
Figure 3.4 a) Step input in strain and the corresponding stress relaxation of b) a Newtonian fluid
and c) a viscoelastic fluid and solid (Reproduced from Macosko 1994, p.110).
A viscoelastic fluid’s relaxation time indicates how quickly the fluid relaxes and is another
measure of the fluid’s elasticity. In this work, relaxation time was a key property used to
characterize fluids and to explain experimental results.
3.4.1 First Normal Stress Difference
The first normal stress difference, N1, of a viscoelastic fluid is defined as the difference between
the normal stress components in the flow direction and in the direction perpendicular to the flow
direction, i.e:
yyxxN 1 , (3.9)
Chapter 3: Test Fluids 36
where xx and yy are respectively the normal stress components in the streamwise and
transverse directions. The first normal stress difference is a measure of fluid elasticity in shear
and its value is identically zero for Newtonian fluids.
By measuring the axial force, F , a cone-and-plate rheometer can produce values for 1N
according to the following expression (Macosko, 1994):
21
2
R
FN
, (3.10)
where R is the cone radius. However, this axial force generated by weakly elastic fluids in shear
is usually very small and often lower than the measurable range of the force transducer in the
rheometer. Therefore, if the test fluids are able to generate large enough normal stresses in shear,
it should be possible to obtain 1N measurements for these fluids.
3.4.2 Oldroyd-B model
The Oldroyd-B constitutive equation is a mathematical model commonly used to predict the
behaviour of Boger fluids. It has been derived from the dynamics of a dilute suspension of bead-
spring dumbbells in a viscous fluid, resembling the dynamics of polymer chains dissolved in a
viscous solvent (Prilutski et al, 1983). This model is particularly appropriate for Boger fluids
because the separate contributions of the solvent and the polymer viscosities are included in the
constitutive equation, according to:
SP , (3.11)
where is the fluid viscosity, P is the contribution to the viscosity by the polymer and S is
the solvent viscosity.
Chapter 3: Test Fluids 37
For steady shear flow, the Oldroyd-B equation predicts 1N to have a quadratic dependence on
shear rate, i.e:
21 2 PN (3.12)
where is the shear rate, is the relaxation time. Constitutive equations of the Oldroyd – B
model can be found in Appendix C.
Equation 3.12 can be used to determine a fluid`s relaxation time from measurements of 1N .
Therefore, measurements of first normal stress difference, 1N , were made under steady shearing
using an AR2000 cone-and-plate rheometer and attempt was made to obtain a relaxation time.
Reliable 1N measurements could be obtained for only one of the fluids, the 1200 ppm solution,
the data for which are shown in Figure 3.5. The figure is a plot of 21 / N vs shear rate, . If the
fluid is an Oldroyd-B fluid the quantity 21 / N should be independent of the shear rate.
Chapter 3: Test Fluids 38
0.001
0.01
100 1000Shear Rate (s-1)
N1/γ
2
Figure 3.5 First normal stress difference measurements in response to steady shearing for the
1200 ppm PEO solution in PEG solvent
As is evident from the data, this fluid follows the Oldroyd-B prediction in the range of shear
rates below 150 s-1, making the quantity 21 / N independent of shear rate in this range. Using
this constant value of 21 / N , together with the polymer viscosity, P , equation 3.12 can be
used to determine a relaxation time, , for this fluid as:
382 2
1
P
Nms
Chapter 3: Test Fluids 39
Thus, the relaxation time for the 1200 ppm solution, as determined from 1N measurements is
approximately 38 milliseconds.
3.5 Elastic modulus
One other measure of a fluid`s elasticity is a quantity known as the elastic modulus. This
quantity, defined in oscillatory shear flow, was also used to characterize the test fluids.
When a fluid is sheared sinusoidally with a small amplitude 0 at a frequency , the stress
response is also sinusoidal.
For a strain input: )sin()( 0 ttxy ,
the stress-response, consisting of an in-phase and an out-of-phase component, is given by:
),cos(")sin(')(
0
tGtGtxy
(3.13)
where xy is the output shear-stress, 'G is known as the dynamic storage modulus or the elastic
modulus, and "G is known as the dynamic loss modulus or the shear modulus. For a viscoelastic
fluid, 'G is a measure of the fluid’s elasticity while "G is a measure of its viscosity. For a
Newtonian fluid, xy is always 90° out of phase with xy and therefore, by equation 3.13, 'G is
identically zero for a Newtonian fluid.
Measurements of 'G and "G can be obtained from a cone-and-plate rheometer, which determines
these quantities by applying an oscillatory strain and measuring the amplitude of the torque
response and its phase shift with the applied strain (Collyer and Clegg, 1998).
Chapter 3: Test Fluids 40
Figure 3.6 shows measurements of the elastic modulus, 'G , in response to small amplitude
oscillatory shear, performed using the ARES rheometer with a 5 cm 0.5 degree fixture at 25° C.
For all concentrations shown, 'G increases with the frequency of oscillation, , indicating that
the solutions possess elasticity. Moreover, at each frequency, 'G values for the 1200 ppm is
almost 70% higher than that of the 500 ppm indicating that the 1200 ppm solution is
considerably more elastic than the 500 ppm solution. As shown in the plot, 'G values are below 1
Pa for all the fluids in the range of frequencies used. These values agree with measurements
made by Dontula et al. (1998) for PEO/PEG solutions with similar concentrations. Reproducible
'G measurements could not be obtained for the 100 ppm solution indicating that the elasticity of
this solution is below the measurable range of the rheometer. Similarly no reproducible
measurements could be obtained for the Newtonian solvent, 'G readings for which should be
identically zero.
0.01
0.1
1
1 10 100
ω (1/s)
G' (
Pa)
1200 ppm
1000 ppm
750 ppm
500 ppm
Figure 3.6 Elastic modulus measurements in response to small-amplitude oscillations
Chapter 3: Test Fluids 41
For small frequencies, i.e, as 0 , the Oldroyd-B model predicts 'G to have a quadratic
dependence on the frequency, ie:
2' PG , as 0 , (3.14)
where and P once again are respectively the fluid’s relaxation time and the polymer
contribution to the viscosity. The derivation of the above equation is given in Appendix C.
According to this equation, a logarithmic plot of 'G versus , as that in Figure 3.6, should have
a slope of 2. But a slope of 2 is not observed in this plot, even though several other properties
confirmed that the fluids are Boger fluids. This discrepancy can be explained by noting that the
above relationship holds only for very small values of . Thus, it is possible that a slope of 2
can be obtained at lower frequencies than were used in these measurements; however, at lower
frequencies, reliable measurements could not be obtained for any of the fluids, as lower values of
'G most likely fall outside the measurable range of the rheometer.
Chapter 3: Test Fluids 42
3.6 Summary of Fluid Properties
Table 5.1 is a summary of the relevant fluid properties determined from the fluid characterization
measurements described above for the test fluids.
Fluid Fluid
Viscosity, (Pa.s)
Polymer Viscosity, P
(Pa.s)
Solvent Viscosity, S
(Pa.s)
RelaxationTime,
(s)
Density, (kg/m3)
PEG Solvent 0.085 - - - 1000
100 ppm PEO in PEG 0.088 0.003 0.085 - 1000
500 ppm PEO in PEG 0.103 0.018 0.085 - 1000
1200 ppm PEO in PEG 0.128 0.043 0.085 0.038 1000
Critical Concentration: *c = 1480 ppm
Table 3.1 Fluid properties of test fluids
43
Chapter 4: Experimental Results & Discussions
Results of pressure drop measurements from the flowcell for the Newtonian and viscoealstic
fluids are presented and discussed in this chapter. Results from numerical analyses, performed in
order to explain some of the results, are also discussed where applicable.
4.1 Main Channel & Side Channel Combined Flow
A presheared planar extensional flowfield was established by combining flows in the main
channel and the side channels. Figure 4.1 shows the streamlines in such a flowfield as predicted
by COMSOL for a Newtonian fluid. As shown by this plot, the streamlines from the side
channels bend towards the exit channel as they meet the streamlines from the main channel, and
push the main channel streamlines towards the centre of the channel. Although the streamlines
do not resemble the motion of counter-rotating vortices in the turbulent boundary layer, they
demonstrate that extensional motion is present in the flowfield in the region where the side flow
is superimposed on the main channel flow.
In order to ensure that the flowfield was symmetric, the flow rate from each side channel was
measured individually. Keeping one channel blocked, the flow rate from the other channel was
measured, and vice versa. Equal flow rates in each channel confirmed that the flowfield was
symmetric.
Chapter 4: Experimental Results & Discussions 44
Figure 4.1 Three-dimensional view of streamlines for combined flow from the main and the side
channels showing how flow from the side channels is superposed on main channel flow
The extensional rate, , in this flowfield was estimated as:
,2hW
Q
S
S
where SQ is the combined flow rate in both side channels, SW is the side channel width, and h
is the side channel height.
Drag reduction is thought to be a phenomenon caused by elasticity in extension. Therefore if the
designed flowfield is a model of the beginning of turbulent burst, an elastic effect that depends
on the rate of extension in this flowfield should have a correlation with the amount of drag
reduction. It is therefore necessary to study the effect of extensional rates on the pressure drop, at
different values of preshearing. The wall shear rate, W , that indicates the amount of presheraing,
was calculated using equation 2.4 as:
2
6
Wh
QMW ,
Chapter 4: Experimental Results & Discussions 45
where MQ is the main channel flow rate, W is the main channel width and h is the main channel
height.
For each test, the main channel flow rate was set to a value corresponding to a certain wall shear
rate, while the side channel extensional rate was varied to vary the extensional rates. By
repeating the tests at different main channel flow rates, the effect of varying the shear rate was
studied.
4.1.1 Newtonian Results
The first experiments were conducted with PEG solvent in order to establish a Newtonian
baseline. Figure 4.2 shows pressure drop measurements for the PEG solvent at two main channel
flow rates, at shear rates above the critical value of 1000 s-1 required for the onset of drag
reduction. The pressure drop is plotted versus extensional rate, , and the corresponding side
channel flow rate, SQ . The pressure drop predicted by COMSOL, for a Newtonian fluid with the
same density and viscosity as the PEG solvent, is also shown in the graph.
Chapter 4: Experimental Results & Discussions 46
2500
3000
3500
4000
100 150 200 250 300 350 400 450
Pre
ssu
re D
rop
(P
a)
Wall Shear Rate = 1500/s
Wall Shear Rate = 1100/s
COMSOL
Qs (ml/s) = 0.6 0.9 1.2 1.5 1.8 2.1 2.4
έ (1/s) =
Re = 7
Re = 8
Fluid: 33.3% solution ofpolyethylene glycol in water (PEG solvent)
Figure 4.2 Measured pressure drop and prediction by COMSOL, at two shear rates,
corresponding to main channel flow rates of 4 ml/s (Re=7) and 5.5 ml/s (Re=8), for various side channel flow rates.
The plot shows that the pressure drop increases linearly with side channel flow rate and
extensional rate at both shear rates, although the variation is slight. The pressure drop at a shear
rate of 1500 s-1 is approximately 36% higher than that at 1100 s-1. As expected, this difference
remains almost constant in the range of extensional rates studied, because for a Newtonian fluid,
the pressure drop, P , is proportional to the main channel flow rate, i.e:
3
12
Wh
LQP M ,
where is the fluid viscosity, and L is the length between the pressure tap locations. Because
the main channel flow rate was increased to increase the wall shear rate, the pressure drop
increased proportionately.
Chapter 4: Experimental Results & Discussions 47
The increase in pressure drop with extensional rate is caused by the increase in the side channel
flow rates required to increase the extensional rate. By continuity, this increases the flow rate in
the exit channel and thus increases the pressure drop. Since the side channel flow rate is only a
fraction of the main channel flow rate, the increase in the total flow rate is small and hence the
increase in the pressure drop is also slight. Since PEG solvent is an inelastic fluid, the variation
of pressure drop with shear and extensional rates is caused entirely by the fluid’s viscosity and
hence is linear.
The pressure drop predicted by COMSOL is in agreement with the experimental data, indicating
that the software is capable numerically of solving this flowfield for a Newtonian fluid.
4.1.2 Results with PEO Solutions
The experiments described above were then repeated with two viscoelastic fluids: 100 ppm and
500 ppm PEO dissolved in the PEG solvent. The 100 ppm concentration was chosen because this
is the minimum concentration where the solution viscosity can be discerned from the solvent
viscosity. The 500 ppm solution was chosen because 'G measurements show that this fluid is
considerably elastic. Since these fluids had higher viscosities than the solvent, P was
normalized by multiplying it by /PEG , the ratio of viscosities, to eliminate viscous effects and
thereby reveal any elastic effects. The normalized pressure drop,
PEGPx , therefore represents
the pressure drop of a fluid with the same viscosity as the PEG solvent. Figures 4.3 to 4.5 show
plots of this quantity versus extensional rate for different shear rates, corresponding to three main
Chapter 4: Experimental Results & Discussions 48
channel flow rates. All of these shear rates are higher than the critical value of 1000 s-1, required
for the onset of drag reduction.
2000
2500
3000
3500
4000
0 50 100 150 200 250 300 350 400 450
Extension Rate (s-1)
∆P
x (η
PE
G /
η)
(Pa)
500 ppm
100 ppm
PEG Solvent
Preshearing:
Wall shear rate = 1100 s-1
Figure 4.3 Normalized pressure drop measurements for the viscoelastic test fluids and the
Newtonian solvent at a wall shear rate of 1100 s-1, corresponding to a main channel flow rate of 4 ml/s and a Reynolds number of 7.
Chapter 4: Experimental Results & Discussions 49
3000
3500
4000
4500
5000
0 50 100 150 200 250 300 350 400 450
Extension Rate (s-1)
∆P
x (η
PE
G /
η)
(Pa)
500 ppm
100 ppm
PEG Solvent
Preshearing:
Wall shear rate = 1500 s-1
Figure 4.4 Normalized pressure drop measurements for the viscoelastic test fluids and the
Newtonian solvent at a wall shear rate of 1500 s-1, corresponding to a main channel flow rate of 5.5 ml/s and a Reynolds number of 8.
5000
5500
6000
6500
7000
0 50 100 150 200 250 300 350 400 450
Extension Rate (s-1)
∆P
x (η
PE
G /
η)
(Pa)
500 ppm
100 ppm
PEG Solvent
Preshearing:
Wall shear rate = 2000 s-1
Figure 4.5 Normalized pressure drop measurements for the viscoelastic test fluids and the
Newtonian solvent at a wall shear rate of 2000 s-1, corresponding to a main channel flow rate of 7.5 ml/s and a Reynolds number of 11.
Chapter 4: Experimental Results & Discussions 50
In the above plots, since the pressure drop readings have been normalized to eliminate viscous
effects, any data above the PEG solvent’s baseline signify a non-viscous effect, i.e, an elastic
effect. Thus, elastic effects are observed at all shear rates with the 500 ppm solution and at shear
rates greater than 1500 s-1 with the 100 ppm solution. It should be noted from these results that
the magnitudes of the observed effects are dependent on the shear rates and not on the
extensional rates. If the elastic effect was an extensional one, it would vary non-linearly with
extensional rate, a behavior characteristic of extensional effects but absent in the present results.
The observed elastic effect, however, grows with increasing shear rates. At a shear rate of 1100
s-1,
PEGPx is about 8% higher than the PEG solvent for the 500 ppm solution, whereas the
difference is about 15% at a shear rate of 1500s-1, and almost 20% at a shear rate of 2000s-1. This
observation led to the need for investigating the effect of shearing, independently, on the
measured pressure drop.
For the three shear rates tested, the pressure drop varies linearly with extensional rate in the
range of extensional rates tested. This linear increase in pressure drop with extensional rate,
observed both with the Newtonian PEG solvent as well as the polymer solutions, is a viscous
effect caused by the increase in the side channel flow rate.
In this experiment, the Reynolds number, defined as Vh
Re , has a maximum value of 11. For
channel flows, the transition from laminar to turbulent flows takes place at a Reynolds number of
around 1500 and hence Re = 11 present in the experiment should be low enough to assume that
inertial effects can be neglected. However, since the velocity in the flow direction is changing,
Chapter 4: Experimental Results & Discussions 51
inertial forces may still play a role in this flowfield. To examine the role of inertia in this case, a
more appropriate form of Reynolds number is:
2Re
VhI
, (4.1)
where V is the change in velocity in the flow direction. A derivation of this Reynolds number
can be found in Appendix B6. For the experiment with combined flow in the main and side
channels, the change in velocity can be obtained using the flow rates and the dimensions of the
flowcell. The maximum Reynolds number obtained with V is ReI = 0.7. Thus inertial inertial
effects are neglected in the first instance.
4.2 Main Channel Flow without Side Flow
As the elastic effects observed in the side flow experiment varied with the wall shear rate, a
different experiment involving flow in the main channel only, with no flow in the side channels,
was conducted to study the effect of shearing alone. The results from this experiment are shown
in Figure 4.6. In order to eliminate viscous effects, the quantity
PEGPx is plotted once again;
this time, however, the independent variable is the shear rate in the main channel.
Chapter 4: Experimental Results & Discussions 52
0
1000
2000
3000
4000
5000
6000
7000
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Shear Rate (s-1)
∆P
x (η
PE
G /
η)
(Pa)
500 ppm
100 ppm
PEG Solvent
Re = 2
Re = 11
Qside = 0
Figure 4.6 Normalized pressure drop measurements for the viscoelastic test fluids and the
Newtonian solvent for variation in main channel flow with no flow from the side channels. The Reynolds numbers corresponding to the lowest and highest flow rates are 2 and 11 respectively.
The results show that an elastic effect is observed with the 500 ppm PEO solution in PEG as the
pressure drop, after compensating for the difference in viscosities between the fluids, increases
above the Newtonian baseline beyond a certain shear rate. The above data also shows an onset of
elastic effects, as reported in the literature for dilute polymer solutions. This onset takes place at
a shear rate of approximately 1000 s-1, beyond which the effect appears to grow with increasing
shear rate. No effect is observed with the 100 ppm solution, the data for which falls on the
Newtonian baseline.
Chapter 4: Experimental Results & Discussions 53
Since there is no flow from the side channels, there is no identifiable extensional motion taking
place in this flowfield. The elastic effect observed with the 500 ppm solution therefore has to be
attributed to shearing. Since the highest shearing takes place at the walls, it is necessary to
examine the wall shear rates and the elastic stresses generated by shear.
4.3 Analyses of Results
The results of the main channel experiment show an elastic effect that grows with the wall shear
rate. The next step is therefore to understand and explain the cause of this effect. However, first,
it is necessary to ensure that the observed phenomenon is not caused by a flow instability.
4.3.1 Flow Instability in Shear
The shear flow of a Boger fluid can become unstable when the shear rate exceeds a critical value.
Hence, in order to differentiate between an elastic effect and a flow instability, it is necessary to
know this critical value. The value is given in terms of the Weissenberg number, ,Wi defined as
the product of the fluid’s characteristic time and the rate of shear, i.e:
, Wi (4.2)
Phan-Thien (1985) derived an expression for the critical shear rate of an Oldroyd-B fluid using
the geometry of a cone-and plate rheometer. For a Boger fluid with viscosity and polymer
viscosity P , the critical Weissenberg number, cWi , for the onset of shear instability to take
place is given by:
5
2 PcWi (4.3)
Chapter 4: Experimental Results & Discussions 54
For flow in a cone-and-plate rheometer, since the shear rate is proportional to the angular
velocity, , in this particular context, the Weissenberg number, Wi , can be written as:
, Wi (4.4)
The critical shear rate, c , can then be obtained as:
5
2 Pc , (4.5)
where is the fluid relaxation time and is the cone-angle of a cone-and-plate fixture.
As indicated by equation 4.5, the critical shear rate depends on the fluid’s relaxation time, which,
unfortunately, could not be determined for the 500 ppm PEO solution in PEG solvent. However,
the relaxation time for the 1200 ppm solution can be used to obtain a lower bound for the critical
shear rate because the critical shear rate varies with the inverse of the relaxation time. As the 500
ppm solution is certainly less elastic than the 1200 ppm, it is reasonable to assume that the
relaxation time for the 500 ppm solution is shorter than that of the 1200 ppm solution. Hence, for
a cone angle of 0.5 degrees, and for the fluid properties in Table 4.1, the critical shear rate can be
calculated as:
2505103.05
018.02
5.0038.0
180
x
x
xc 1s
Thus, the critical shear rate of the 500 ppm solution is at least 2500 1s . This values is higher
than the maximum shear rate in the experiment, and hence the possibility of a shear instability
affecting the results can be ruled out.
Chapter 4: Experimental Results & Discussions 55
4.3.2 Hole Pressure Error
The observed elastic effects may have been caused by a phenomenon known as hole pressure
error. To understand this concept, it is necessary to identify the differences between measuring
the pressure of a Newtonian fluid and that of non-Newtonian fluid using a pressure tap.
When a Newtonian fluid flows between two parallel plates, the pressure, 1P , measured by a
pressure transducer flush mounted on the wall is the same as the pressure, 2P , measured by a
recessed transducer placed at the end of a pressure tap at the same downstream location as the
flush-mounted transducer, i.e, 21 PP , as shown in Figure 4.7 (a).
Figure 4.7 Pressure measurement in (a) a Newtonian fluid and (b) a viscoelastic fluid
(Reproduced from Bird et al., 1987)
However, because normal stresses can develop in a viscoelastic fluid, the reading given by a
pressure transducer is the pressure P plus yy , the component of normal stress acting on the
surface of the transducer. At the transducer surface, yy is generally lower than it is at the surface
of the flush-mounted transducer, making the readings of the two transducers different, i.e,
0)()( 21 yyyy PP , as indicated in Figure 4.7 (b). This difference in pressure readings is
Chapter 4: Experimental Results & Discussions 56
known as the hole pressure error, *P . For pressure taps with circular cross-section, this quantity
is given by (Bird et al., 1987, p. 68):
W
xyxy
dNN
P
0
21*
3
1, (4.6)
where 1N & 2N are respectively the first and the second normal stress differences. Since 1N is
the dominant quantity and 2N is usually taken as one-tenth of 1N , ie, 21 1.0 NN (Bird et al.,
1987) equation 4.6 can be written as
)( 1* NfP (4.7)
where )( 1Nf is the function representing the right-hand side of equation 4.6. )( 1Nf will be
used in the subsequent sections to refer to this function.
An attempt will now be made to apply the principles of hole pressure error to flow in the main
channel. In this setup, the two pressure taps are located in different channel sections, as shown
earlier in Figure 2.4. However, since the channel heights at the two locations are different, the
wall shear rates are also different. More specifically, by equation 2.4, the wall shear rate, ,W
varies with the inverse square of the channel height, h , i.e:
2
1
hW .
Using the subscripts U and D to denote the upstream and the downstream sections respectively,
the ratio of heights from the channel geometry is:
79.09.1
5.1
mm
mm
h
h
D
U .
Hence, in the experiment involving only the main channel flow, the ratio of wall shear rates is
Chapter 4: Experimental Results & Discussions 57
6.179.0
122
U
D
WD
U
h
h
,
Because the upstream wall shear rate is 60% higher than that in the downstream section, the first
normal stress difference, 1N , is higher upstream.
As mentioned, for a viscoelastic fluid, presence of normal stresses affects the measured pressure
at the location of each pressure tap because the normal stress component, yy , contributes to the
pressure reading. As the first normal stress difference 1N is different in the locations of the two
pressure taps, so is the quantity yy . For low shear rates and correspondingly small values of 1N ,
the difference in yy between the upstream and the downstream sections is small and hence does
not appear as an effect in the measured pressure drop. As 1N increases with the wall shear rate,
this difference in yy becomes larger and measurable, causing the onset of elastic effect in Figure
4.6. Also, according to the Oldroyd-B equation, 1N increases as the square of the shear rate and
thus the observed elastic effect also grows non-linearly with shear rate.
It is of interest to see if the observed increase in the pressure drop can be related to the
magnitude of 1N . This, however, could not be done because reliable measurements for 1N could
not be obtained for this particular test fluid, the 500 ppm PEO solution with the available
rheometers.
Chapter 4: Experimental Results & Discussions 58
4.4 Side Channel Flow without Main Channel Flow
Since extensional flow, in combination with shear flow, did not produce any elastic effect, the
next step in the investigation was to examine if extensional flow alone was able to produce an
effect. To do this, an experiment involving side channel flow only was carried out. This
experiment was conducted by blocking the entrance to the main channel and having a pressure-
driven flow in the side channels. Figure 4.8 show the streamlines obtained from COMSOL for
this flowfield. The streamlines pattern is similar to that for stagnation point flow, indicating that
an extensional flowfield has been created. Although this setup was considerably different from
the previous two experiments, the arrangement for measuring the pressure difference was the
same as before, as shown in Figure 4.9.
Figure 4.8 Three-dimensional view of streamlines for side channel flow only
Chapter 4: Experimental Results & Discussions 59
Figure 4.9 Pressure measuring arrangement for the side flow experiment
This time, since there is no flow in the main channel the pressure at location 1 is the stagnation
pressure, 0P , and hence the pressure difference, P , measured by the transducer is:
20 PPP , (4.14)
where 2P is the pressure at location 2. Thus, P is a measure of the pressure downstream of the
flow from the side channels, which is the pressure of interest in this case.
The results from this experiment, conducted with the PEG solvent as well as with two
viscoelastic fluids: the 500 ppm and the 1200 ppm solutions, are shown in Figure 4.10.
Chapter 4: Experimental Results & Discussions 60
0
40
80
120
0 50 100 150 200 250 300έ (s-1) =
∆P
x (η
PE
G /
η)
(Pa
) 1200 ppm
500 ppm
PEG Solvent
QMain = 0
De (for 1200 ppm) = 1.9 3.8 5.7 7.6
Re=1
Re=11
Figure 4.10 Normalized pressure drop measurements for two viscoelastic test fluids and the Newtonian solvent with side channel flow with no flow in the main channel. For the 1200 ppm
fluid, the Deborah numbers corresponding to the extensional rates are also shown.
Like the previous graphs, the normalized pressure drop,
PEGPx is again the ordinate and this
time the abscissa extensional rate corresponding to the side channel flow rate. The results show
that, with side channel flow alone, elastic effects are observed with both solutions, with the 1200
ppm solution producing the larger effect. The onset of elastic effects takes place at a lower
extensional rate with the 1200 ppm solution than it does for the 500 ppm solution. These
observations are in accordance with elastic properties of the two test fluids. That is, the 1200
ppm fluid, being more elastic than the 500 ppm, requires a lower extensional rate to generate
elastic stresses necessary for onset.
Chapter 4: Experimental Results & Discussions 61
The maximum Reynolds number, as shown in the plot, was normally 11, while the Reynolds
number based on the change in velocity as described in Appendix B6 , is ReI = 0.2. Hence
inertial forces are only a fraction of viscous forces and thus inertia does not play a significant
role in this case either.
Since neither inertia nor viscosity has any contribution to the results, it can be deduced that the
observed phenomenon is caused by fluid elasticity. The Deborah number, defined as a ratio of
the characteristic time of the fluid to the characteristic flow time, governs the extent to which
elasticity manifests itself in response to fluid acceleration or non-homogeneous deformation
(Dealy, 2010). Specifically, Deborah number, ,De is:
TDe
, (4.15)
where , the fluid’s relaxation time, is usually taken as the characteristic time, and T is the flow
time. For extensional flows, the observation time can be taken as the reciprocal of the rate of
extension. Thus,
,1
T (4.16)
Since it was possible to determine the relaxation time for one of the fluids, the 1200 ppm
solution, the Deborah number, corresponding to the extensional rates tested for this fluid, is also
plotted on the x-axis. The extensional rates were found using equation 2.5. For this fluid, the
onset is at a Deborah number of about 1.9. Although this result fulfills the condition of
)1(ODe required for extensional effects, the data for both fluids show maxima, at 6.7De
for the 500 ppm solution and at 7.5De for the 1200 ppm. These maxima indicate that the
Chapter 4: Experimental Results & Discussions 62
observed elastic effect may not have been caused by extension because extensional effects are
expected to increase with De .
4.4.1 Elastic Effects in Extension
In order to understand whether the observed elastic effect is caused by extension, it is necessary
to identify the conditions required for an extensional effect to take place.
Boger fluids can produce large elastic effects in extension. A filament stretching rheometer can
produce purely uniaxial extension at constant extensional rates and thereby yields a true value of
extensional viscosity, E (Tirtaatdmadja and Sridhar 1993, Anna et al 2001, McKinley and
Sridhar 2002). Results from such an instrument are shown in Figure 4.11. For this typical Boger
fluid, data are plotted as the Trouton ratio, E , versus the Hencky strain, , defined as:
0
lnl
l , (4.17)
where l is the stretched length of a fluid filament with an initial length, 0l .
The Hencky strain can also be determined as:
dt , (4.18)
where is the extensional rate and t is the time with 0t corresponding to the start of
stretching. For a constant extensional rate, as produced by a filament stretching rheometer,
equation 4.18 can be simplified to give:
t . (4.19)
Chapter 4: Experimental Results & Discussions 63
As shown by Figure 4.11, for small values of the Hencky strain, the Trouton ratio remains nearly
constant at the Newtonian value of 3. When the Hencky strain exceeds about 2, elastic effects
begin and E starts to rise sharply above the Newtonian value.
Figure 4.11 Uniaxial Trouton ratio for a Boger fluid (a semidilute solution of 0.31 wt%
polyisobutylene in polybutene) stretched over a range of extensional rates, plotted as a function of Hencky strain. (Reproduced from McKinley and Sridhar, 2002)
Using 2 as the critical Hencky strain, the amount of extension, c
l
l
0
, required by the
polymer chains to produce elastic effects can then be estimated from equation 4.17:
4.7)2exp(0
cl
l (4.20)
Chapter 4: Experimental Results & Discussions 64
Thus, in order to produce extensional effects, polymer chains need to be extended at least 7 times
their length in equilibrium configuration.
In view of the above criterion, an analysis was performed, with the help of numerical simulation,
in order to determine whether, in the present experimental flowfield, the Hencky strain exceeds
the value of 2 required for producing elastic effects.
4.4.2 Numerical Analysis
Figure 4.12 shows a side view of the flowcell where the side channels meet the main channel.
For this analysis, the z -coordinate is used as the direction of flow, with 0z defined at the
edge of the side channel meeting the main channel, Sw is the width of each side channel, and the
centrelines of the main and the exit channels are as shown. Figure 4.13 shows the velocity,
obtained with COMSOL, along the main and exit channel centerlines, plotted against Swz / , the
normalized displacement in the flow direction. This centerline velocity is shown for three flow
rates covering the range of extensional rates which produced the maximum elastic effect. This
plot indicates that velocity is non-zero in the portion of the main channel approaching the side
channel intersection, even though there is no through flow in the main channel. The reason is
that, as shown by the streamlines in Figures 4.9, the fluid close to the wall of the side channel
passes through a region near the end of the main channel before entering the slanted exit channel.
This causes the velocity magnitude to be non-zero at negative values of Swz / .
Chapter 4: Experimental Results & Discussions 65
Figure 4.12 Side view of section from flowcell showing intersection of the main, side and the
slanted exit channels
0
0.1
0.2
0.3
0.4
0.5
0.6
-3 -2 -1 0 1 2 3 4 5 6 7 8
z/ws - along flow direction
Vel
oci
ty (
m/s
)[i
n t
he
flo
w-d
irec
tio
n]
Q = 1 ml/sQ = 0.7 ml/sQ = 0.6 ml/s
ws : side channel width
Figure 4.13 Numerical results for the velocity along the main and exit channel centerlines,
obtained using COMSOL.
From the slope of each curve in Figure 4.13, it is possible to determine how the extensional rate
varies along the centreline for the particular flow rate. The extensional rate, by definition, is the
Chapter 4: Experimental Results & Discussions 66
velocity gradient in the direction of flow. If iV is the velocity in the flow direction at a point i on
the curve corresponding to the location iz , the local extensional rate, i , is given by:
ii
ii
i
ii zz
VV
z
V
1
1 , (4.21)
where 1i is the point adjacent to the i th point. The slope is a measure of the extensional rate,
, which is a maximum where the slope is steepest. From equation 4.21, the maximum
extensional rate at a flow rate of 1 ml/s was found to be 120 s-1. This value is close to the
extensional rate of 150 s-1 estimated using equation 2.5 for the side flow experiment at the same
flow rate.
As mentioned above, the Hencky strain is the primary criterion for determining whether the
observed elastic effect is an extensional effect. For a variable extensional rate, the Hencky strain,
, is defined as:
dt , (4.22)
where t is the time. This integral can be evaluated numerically as:
N
iii tdt
1
, (4.23)
where N is the total number of points along a curve and the time, it , required for a fluid
element to travel from iz to 1iz , is given by:
i
iii V
zzt
1 , (4.24)
Chapter 4: Experimental Results & Discussions 67
Figure 4.14 shows the Hencky strain, calculated using equation 4.23, plotted versus the Deborah
number, De , calculated numerically as:
max De , (4.25)
where is the 38 ms relaxation time for the 1200 ppm PEO solution and max is the maximum
extensional rate calculated using equation 4.21 for each of the three flow rates shown in Figure
4.13.
1.25
1.3
1.35
1.4
1.45
2 2.5 3 3.5 4 4.5 5
Deborah Number
Hen
cky
Str
ain
Figure 4.14 Calculated values of Hencky strain in the flowfield plotted as a function
of Deborah number
The maximum Hencky strain achieved in this flowfield is about 1.38, corresponding to a
Deborah number of 4.7. This value is less than the critical value of 2 required for producing
elastic effects in extension. Moreover, the increase of Hencky strain with Deborah number is also
very slight, an observation which can be explained by the fact that, although the rate of extension
increases with the increase of Deborah number, the time of travel shortens, and hence the
Chapter 4: Experimental Results & Discussions 68
Hencky strain, being a product of the extensional rate and time, does not change much as the
Deborah number increases.
This numerical analysis was based on flow of a Newtonian fluid. If these numerical results are
valid for the viscoelastic test fluids, then the elastic effect observed in the side channel flow
experiment was not likely caused by extension. Therefore, it is necessary to investigate if the
effect was caused by shearing.
4.4.3 N1 Effect
As discussed earlier, the first normal stress difference N1 is a measure of fluid elasticity in shear.
Figure 4.15 a) shows 1N measurements obtained from an ARES rheometer for the 1200 ppm
PEO solution in PEG solvent plotted as 21
N
versus the shear rate . This plot shows that 21
N
approaches a constant value at low shear rates, as predicted by the Oldroyd-B model according to
equation 3.12. Using this constant value of 21
N
, the fluid’s relaxation time of 38 ms was found,
as outlined in section 3.4. Figure 4.15 b) shows 1N data obtained from the rheometer as well as
the Oldroyd-B model using the 38 ms relaxation time. The lowest shear rate at which 1N can be
measured by the rehometer is about 160 s-1. Below this shear rate, reliable 1N measurements
cannot be obtained from the instrument, which is unfortunate because calculations using
equation 2.4 show that, for the side flow experiment, the range of wall shear rates in the
downstream channel is much lower than the measurable range shown in this plot. However, it is
evident that at shear rates below 250 s-1, 1N for this fluid has quadratic dependence on shear
rate. Hence, by assuming that this fluid continues to behave like an Oldroyd-B fluid at shear rates
Chapter 4: Experimental Results & Discussions 69
below the measurable range, the line representing the Oldroyd-B model can be extrapolated to
obtain 1N values at shear rates corresponding to the wall shear rates in the side flow experiment.
These extrapolated values of 1N have been plotted in Figure 4.16 along with the elastic pressure
drop, obtained in the side flow experiment. The elastic pressure drop, elasticP , is calculated by
subtracting the Newtonian pressure drop, NewtonianP , at a particular flow rate from the normalized,
viscosity compensated pressure drop, icViscoelast
PEGPx
, for the polymer solution at the same
flow rate, i.e:
Newtonian
icViscoelast
PEGelastic PPxP
(4.26)
The shear rates in the elastic pressure drop data in Figure 4.16 correspond to the flow rates in the
PEG solvent’s Newtonian baseline in Figure 4.10. For each of these shear rates, the viscoelastic
pressure drop was obtained by fitting a curve through the available data for the 1200 ppm
solution, also shown in Figure 4.10. The elastic pressure drop was then obtained by subtracting
the Newtonian pressure drop from the viscoelastic pressure drop.
Chapter 4: Experimental Results & Discussions 70
Figure 4.15 Measurements of First Normal Stress Difference for the 1200 ppm PEO solution in PEG Solvent plotted a) as 2
1 / N showing a plateau for low shear rates and b) along with a slope of 2 obtained by fitting the Oldryod-B model to the first six data points (solid line)
0
10
20
30
40
50
60
0 20 40 60 80 100 120 140 160
Shear Rate (s-1)
N1
(Pa)
N1 Oldroyd-B
∆P_elastic
Figure 4.16 Comparison of elastic pressure drop for the 1200 ppm PEO solution in PEG with extrapolated values of N1 corresponding to the wall shear rate downstream of the side flows.
0.1
1
10
100
1000
10000
10 100 1000
Shear Rate (s-1)N
1 (P
a)
Data
Oldroyd - B
0.001
0.01
100 1000Shear Rate (s-1)
N1/γ
2
Data
Chapter 4: Experimental Results & Discussions 71
As is evident from Figure 4.16, the elastic pressure drop is in the same order of magnitude and
has a similar trend as the first normal stress difference corresponding to the wall shear rate at the
channel, indicating that the observed elastic effect is possibly an 1N effect. As the wall shear rate
in the exit channel increases, so do the normal stress components. As the transducer measures the
pressure plus the transverse normal stress component, an elastic effect is observed when the
shear rates are high enough to generate normal stresses that are measurable by the pressure
sensor. As 1N is a measure of these stresses, the observed elastic effect has a similar trend as
1N .
4.5 Comparison with Prior Work
Numerous works about elastic effects of dilute polymer solutions have been reported in the
literature. Some of these works relate to drag reduction and so are discussed here. Comparisons
of results are made where appropriate and relevance of the present findings in understanding
prior results is discussed.
Most works with drag-reducing fluids involved experiments with polyethylene oxide (PEO)
dissolved in water. Hence, it is difficult to make a direct comparison with the present experiment
which have been conducted with PEO dissolved in a viscous solvent containing another polymer,
namely polyethylene glycol (PEG). However, comparisons of results reveal that, to produce and
elastic effect, higher concentrations of PEO are needed in the PEG solvent than is required when
PEO is dissolved in water alone. In other words, it is easier for PEO to manifest its elasticity in
water than it is in the presence of another polymer.
Chapter 4: Experimental Results & Discussions 72
In their presheared extensional flow experiments, discussed in section 1.1 and illustrated by
Figure 1.4, James et al. (1987) obtained elastic effects with aqueous polymer solutions with
concentration as low as 20 ppm. For an upstream shear rate of 800 s-1, the pressure drop in excess
of the Newtonian baseline increased with extensional rate for extensional rates greater than 600
s-1 and reached a maximum at an extensional rate of about 1500 s-1 before starting to decrease
towards the Newtonian baseline. The same pattern, of reaching a macimum and then declining, is
shown in Figure 4.10.
For extensional flow without preshearing, i.e with no upstream shear flow, James et al. observed
no elastic effect with the 10 ppm or the 20 ppm PEO solutions and observed an elastic effect
with the 40 ppm solution but only at very high extensional rates, in excess of 3000 s-1. In the side
flow experiment of the present work, conducted in absence of any upstream shear flow, elastic
effects were obtained with both the 500 ppm and the 1200 ppm PEO solutions at much lower
extensional rates between 50 s-1 and 250 s-1. These results indicate that the extensional flowfield
in the side channel setup of the present work is more capable of generating elastic effects than
the axisymmetric contraction flowfield used by James et al. However, in the present experiment
too, the elastic pressure drop approached the Newtonian baseline at high extensional rates after
reaching a maximum, as was observed by James et al. in their presheared extensional flow
experiments.
James and Saringer (1982) observed elastic effects with extensional flow of dilute PEO solutions
in water through axisymmetric converging channels. They observed large effects with polymer
concentration as low as 10 ppm. They also observed onsets in elastic effects, which reached a
Chapter 4: Experimental Results & Discussions 73
maximum before declining towards the Newtonian baseline, as was observed in the present
work.
The above works and other prior works involving extensional flows have attributed non-
Newtonian behaviour to extensional effects. In the present work, the observed elastic effects,
although appearing to be extensional effects at first sight, appear to be caused by normal stresses
generated by shearing. Extension was ruled out because the Hencky strain was not high enough
to produce extensional effects. However, none of the prior works involving extensional channel
flows made any attempt to measure or calculate the Hencky strain in their flowfields. Therefore,
it is possible that many of the observed extensional effects reported in the literature may not
actually have been caused by extension. Instead, shear effects may have been responsible for
causing these elastic responses. For example, in a converging channel, which is a widely-used
method for creating an extensional flowfield, there is also shearing on the channel walls. As the
flow rate through the channel increases, the extensional rate increases but so does the shear rate
at the wall. Hence, it is not immediately clear whether an elastic effect that increases with flow
rate through such a channel is really caused by extension, or by shearing at the channel wall, or
by both.
74
Chapter 5: Concluding Remarks
5.1 Summary
In this work, the effect of fluid elasticity on three flows of dilute polymer solutions through a
complex geometry was studied in an attempt to understand the mechanism of drag reduction. A
flowcell was designed and fabricated as an attempt to simulate the first stage of turbulent burst in
a turbulent boundary layer. While a definitive conclusion regarding the mechanism of drag
reduction cannot be made from the present results, measurable elastic effects were recorded in
each of the flows investigated.
For each flowfield, a Newtonian baseline was established by measuring the pressure drop for a
Newtonian fluid. Pressure drop measurements for dilute polymer solutions, which exhibited
behavior of Boger fluids, were normalized to eliminate viscous effects and compared with the
Newtonian baseline to identify elastic effects.
For the primary flow, a combined flow from the main channel and the two side channels, non-
Newtonian effects were independent of extensional rate in the range of extensional rates
comparable to those in the turbulent boundary layer, indicating that the present flowcell model
does not simulate the turbulent boundary layer. However, an elastic effect which increased with
increasing shear rate was observed with one of the viscoelastic test fluids, the 500 ppm solution
of PEO in PEG solvent.
Chapter 5: Concluding Remarks 75
For main channel flow alone, without any flow from the side channels, an elastic effect that
increases with shear rate was observed with the 500 ppm polymer solution at shear rates greater
than 1000 s-1. This effect was analyzed and is attributed to the difference in the first normal stress
difference, 1N , between the upstream and the downstream sections of the channel. This
difference increases with shear rate and generates an extra normal stress which gives rise to the
extra pressure drop, i.e, one above the Newtonian baseline. Moreover, analysis showed that the
observed effect is not caused by a shear instability since wall shear rates are well below the
critical shear rate required to cause a shear instability.
For side channel flow alone, without any flow in the main channel, elastic effects were observed
with both viscoelastic fluids tested: the 500 ppm and the 1200 ppm polymer solutions. The onset
of the effect took place at a Deborah number of 1.9 for the 1200 ppm fluid, satisfying the
Deborah number condition for elastic effects in extension. However, numerical analysis showed
that the Hencky strain in the flowfield is less than 2, the value needed for extensional effects. For
the 1200 ppm solution, the elastic pressure drop and values of 1N were similar quantitatively and
qualitatively, indicating that the observed elastic effect is possibly an 1N effect.
Chapter 5: Concluding Remarks 76
5.2 Conclusions
The observed elastic effects obtained with preshearing were independent of the
extensional rate, indicating that the extensional flowfield is not a representative model of
the first stage of turbulent burst.
With main channel flow only, an elastic effect is observed at shear rates greater than
1000s-1. This elastic effect can be attributed to a higher first normal stress difference at
the upstream sections of the channel.
With side channel flow only, an elastic effect is observed at extensional rates greater
than 45s-1 for the 1200 ppm solution. Numerical analysis shows that the Hencky strain in
this flowfield is less than 2 and hence the observed effect with side flow alone is not an
extensional effect.
The elastic pressure drop is similar to extrapolated values of 1N , indicating that the
observed effect is possibly an 1N effect.
The Reynolds number based on fluid acceleration was less than 1 in all the experiments,
indicating that inertial effects can be neglected.
5.3 Future Work
The primary objective of this work was to model the first stage of a turbulent burst. However, as
the results showed, this objective was not fulfilled. Some suggestions are made in this section
regarding possible modifications to the flowcell and the experimental setup. By making these
modifications, it may be possible to make a better model of a turbulent burst.
Chapter 5: Concluding Remarks 77
In the present design, the exit channel was slanted to model lifting of fluid from the wall of a
turbulent boundary layer. As a result of this slant, the fluid bent upwards and followed the
geometry of the channel. However, in a turbulent burst, lifting of fluid takes place due to the
force exerted by the motion of the counter-rotating vortices. Therefore, if more vertical height is
provided in the region where the side channels meet the main channel, the fluid coming from the
main channel should have a greater possibility of being lifted up by the flow from the side
channels. Alternatively, the height of the side channels can be shortened to serve the same
purpose. Also, the flow rates from the side channel may need to be much higher than those used
in the present experiment in order to generate a counter-rotating motion. In fact, the required
flow rates may be so high that a pair of opposing jets may need to be created. The slant in the
exit channel made no difference to the results. Therefore, to simplify the design and fabrication,
this channel section can be made flat.
An alternative method could be to utilize flow focusing, a technique used in microfluidics to
form droplets. Having an outer flow converge to a main flow creates an extensional motion in
the direction of the flow. If the outer flow is made to attack the main flow, not only at a
horizontal angle but also at a vertical angle, it may also be possible to cause lifting of fluid.
78
Chapter 6: References
Anna SL, McKinley GH, Nguyen DA, Sridhar T, Muller SJ, Huang J, James DF (2001). An interlaboratory comparison of measurements from filament-stretching rheometers using common test fluids. J. Rheol. 45:83-114. Bird RB, Armstrong RC, Hassager O (1987). Dynamics of Polymer Liquids, Vol. 1, Fluid Dynamics. Wiley, New York. Boger DV (1977). Highly elastic constant-viscosity fluid. J. Non-Newt. Fluid Mech. 3:87-91. Chauveteau G (1981). Proceedings of the 56th Annual Fall Technical Conference and Exhibition of the SPE of AIME, San Antonio, Texas. p. 14. Collyer AA, Clegg DW (1998). Rheological Measurement. Chapman & Hall, London. Coventry KD, Mackley MR (2008). Cross-slot flow birefringence observations of polymer melts using a multi-pass rheometer. J. Rheol. 52(2):401-405. Currie IJ (2003). Fundamental mechanics of fluids. Marcel Dekker, New York. Davidson PA (2004). Turbulence – An introduction for scientists and engineers. Oxford University Press, New York. Dealy JM (2010). Weissenberg and Deborah numbers – their definitions and use. Rheology Bulletin 79(2):14-18 Donohue GL, Tiederman WG, Reischman MM (1972). Flow visualization of the near-wall region in a drag reducing channel flow. J. Fluid Mech. 56(3):559-575. Dontula P, Macosko CW, Scriven LE (1998). Model elastic liquids with water-soluble polymers. AIChE J. 44:1247-55. Graessley WW (1980). Polymer chain dimensions and the dependence of viscoelasic properties on the concentration, molecular weight and solvent power. Polymer 21:258–262. James DF (2009). Boger fluids. Annu. Rev. Fluid Mech. 41:129-142. James DF, McLean BD, Saringer JH (1987). Presheared extensional flow of dilute polymer solutions. J. Rheol. 31(6):453-481. James DF, Saringer JH (1982). Flow of dilute polymer solutions through converging channels. J. Non-Newt. Fluid Mech. 11:317-339.
Chapter 6: References 79
James DF, Saringer JH (1980). Extensional flow of dilute polymer solutions. J. Fluid Mech. 97: 655-671. James DF, Yogachandran N (2006). Filament-breaking length – a measure of elasticity in extension. Rheol. Acta 46: 171-170. Khalil MF, Kassab SZ, Elmiligui AA, Naoum FA (2002). Applications of drag-reducing polymers in sprinkler irrigation systems: sprinkler head performance. J. Irrig. Drain Eng. 128:147-152 Kim HT, Kline SJ, Reynolds WC (1971). The production of turbulence near a smooth wall in a turbulent boundary layer. J. Fluid Mech. 50:133-160. Macosko CW (1994). Rheology: Principles, Measurements, and Applications. Wiley, New York. Maxwell JC (1867). On the viscosity and internal friction of air and other gases. Phil. Trans. Roy. Soc., A157:49-88 McKinley GH, Sridhar T (2002). Filament-stretching rheometry of complex fluids. Annu. Rev. Fluid Mech. 34:375-415. Metzner AB, Metzner AP (1970). Stress levels in rapid extensional flows of polymer fluids. Rheol. Acta 9(2):174-181. Muller A, Gyr A (1986). On the vortex formation in the mixing layer behind dunes. Journal of Hydraulic Research 24:358-375 Phan-Thien (1985). Cone-and-plate flow of the Oldroyd-B fluid is unstable. J. Non-Newt. Fluid Mech. 17:37-44. Prilutski G, Gupta RK, Sridhar T, Ryan ME (1983). Model viscoelastic liquids. J. Non-Newt. Fluid Mech. 12:233-241. Schlichting (1960). Boundary layer theory. McGraw-Hill, New York. Scrivener, O (1974). A contribution on modifications of velocity profiles and turbulence structure in a drag reducing solution. Proc. Int. Conf Drag Reduction, St. Johns College, Cambridge, ed. N. G. Coles. C66-70. Sellin RHJ, Ollis M (1980). Polymer drag reduction in large pipes and sewers: results of recent field trials. J. Rheol. 24:667-684 Terrapon VE, Dubief Y, Moin P, Shaqfeh ESG (2004). Simulated polymer stretch in a turbulent flow using Brownian Dynamics. J. Fluid Mech. 504:61–71.
Chapter 6: References 80
Tirtaatmadja V, Sridhar T (1993). A filament stretching device for measurement of extensional viscosity. J. Rheol. 37:1081-1102. Toms B (1948). Some observations on the flow of linear polymer solutions through straight tubes at large Reynolds numbers. Proceedings of the 2nd International Rheological Congress. p. 135–141. Ward-Smith AJ (1980). Internal Fluid Flow - the Fluid Dynamics of Pipes & Ducts. Oxford University Press, New York. White CM, Mungal MG (2008). Mechanics and prediction of turbulent drag reduction with polymer additives. Annu. Rev. Fluid Mech. 40:235–56. Winter HH, Macosko CW, Bennett KE (1979). Orthogonal stagnation flow, a framework for steady extensional flow experiments. Rheol. Acta 18:323-334
81
APPENDIX A: Numerical Simulation
A1 Mesh
For the purpose of running the simulation, a tetrahedral mesh with a mesh refinement setting of
maximum mesh spacing of 0.04 mm and a minimum mesh spacing of 0.02 mm was used. Figure
A1 below shows the mesh in the channel geometry. The mesh at the entrance section of the
channel is shown enlarged in Figure A2.
Figure A1 Tetrahedral mesh in channel geometry
Appendix A: Numerical Simulation 82
Figure A2 Enlarged section showing mesh at the entrance to the main channel
A2 Accuracy of Numerical Results
In order to verify the accuracy of numerical simulations prior to performing computational
analysis for the flowcell geometry, results from COMSOL were obtained for a flowfield which
can be solved analytically. A rectangular channel, with width-to-height aspect ratio of 6.7, same
as that of the main channel of the flowcell, was chosen as the geometry to be tested. Figure A3
shows a comparison of the centreplane velocity obtained from this analytical solution with that
obtained from COMSOL. Agreement between the analytical and the numerical results attest to
the accuracy of the simulations for the given settings of mesh refinement.
Appendix A: Numerical Simulation 83
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2
y/d
Ve
loci
ty (
m/s
)
Rectangular channel with a/b = 6.7Flow between parallel platesCOMSOL
Figure A3 Comparison between analytical results for flow in a rectangular channel with aspect
ratio a/d=6.7 and channel height 2d, flow between two parallel plates, and numerical results from COMSOL.
84
APPENDIX B: Fluid Mechanics
B1 Comparison between Rectangular Channel Flow and Flow Between Parallel Plates
The velocity, v , in a rectangular channel of width a2 and height d2 is given by (Ward-Smith
1980):
,coscosh
cosh)1(4
2
1),(
03
22
nn
n
n
n
n
yNaN
zN
Ndyd
dx
dPyzv
(B1)
d
nNn 2
)12( , (B2)
where is the fluid viscosity anddx
dP is the pressure gradient in the direction of flow.
The velocity profile in a pressure driven flow between two parallel plates, on the other hand, can
be obtained by solving the Navier-Stokes equations for two-dimensional flow:
,2
2
dx
dP
dy
vd x (B3)
where is the dynamic viscosity of the fluid, xv is the x-component of fluid velocity, and dx
dP
once again is the pressure gradient in the direction of flow. If h is the gap between the parallel
plates, with no-slip boundary conditions at the walls, equation B3 can be solved to obtain the
velocity profile in terms of the pressure gradient in the flow:
yhydx
dPyvx
21
)( . (B4)
Appendix B: Fluid Mechanics 85
B2 Shear Rate
The shear rate, , is given by:
dy
dvx . (B5)
Substituting equation B4 in equation B5 gives:
yh
dx
dPy
2
1)(
, (B6)
where h is the height of the channel. The volumetric flow rate, MQ , in the channel is given by:
w
z
h
y
xM dydzyvQ0 0
)( . (B7)
Substituting equation B4 into equation B7 and integrating gives:
3
12
1h
dx
dPwQM
, (B8)
where w is the width of the main channel. Then, obtaining dx
dP in terms of MQ from equation
B8 and substituting in equation B6 gives:
y
h
wh
Qy M
2
12)(
3 (B9)
The wall shear rate, W , can then be obtained by setting 0y in equation A9:
,6
2wh
QMW (B10)
Equation A10 was used to calculate the wall shear rates in the experimental flowcell.
Appendix B: Fluid Mechanics 86
B3 Extensional rate
The planar extensional rate, , in a cross-slot stagnation point flow can be estimated as
(Coventry and Mackley, 2008):
S
S
w
V2 , (B11)
where SV is the average velocity in each of the in-flow side channels and Sw is the width of each
side channel. Equation B11 can be written in terms of the side channel flow rate ,SQ as
,2
2hw
Q
S
S (B12)
where SSS hVwQ , and h is the height of both the side and the main channels. Equation B12
was used to estimate the extensional rates in the flowcell.
B4 Pressure Loss
Equation A8 can be rearranged to obtain an expression for the pressure gradient in the flow in
terms of the flow rate as:
3
12
wh
Q
dx
dP M , (B13)
which can be integrated as:
2
1 03
12P
P
LM dx
wh
QdP
(B14)
to obtain the pressure drop, 21 PPP , between points 1 and 2 in the flow separated by a
distance L as below:
3
12
wh
LQP M (B15)
Appendix B: Fluid Mechanics 87
Although equation B15 gives the pressure drop in a straight channel and does not take into
consideration the complex geometry of the flowcell or the extra flow from the side channels, the
pressure drop calculated using this equation should nevertheless be a reasonable estimate of the
actual pressure drop in the actual flowcell geometry, because the side channel flow rate is only a
fraction of the main channel flowrate. Hence, this equation was used to estimate the pressure
drop between the locations of the two pressure transducers.
B5 Turbulent Boundary Layer
Turbulence is perhaps the most significant unsolved phenomenon in fluid mechanics that is
characterized by inertia-driven, unsteady, chaotic fluctuations in flow parameters such as
velocity, momentum, and kinetic energy. A turbulent flow consists of many structures among
which the turbulent boundary layer is an important one which is of particular interest to this
present study.
Figure B1 Turbulent flow over a flat plate
Appendix B: Fluid Mechanics 88
Considering a turbulent flow with free-stream velocity U over a flat surface as shown in Figure
B1, y as the vertical distance above the surface, xv as the x-component of the fluid velocity, and
as the shear viscosity of the fluid, from equation 2.3, the wall shear stress, W , is given by:
0
y
xW y
v . (B16)
Defining a friction velocity, *u , as
Wu * , (B17)
where is the fluid density.
A spatial parameter, *y , can then be defined as
yu
y ** , (B18)
where is the kinematic viscosity of the fluid defined as: / .
The turbulent boundary layer consists of three layers categorized by their distance from the wall
and the fluid dynamics that take place in each one. The first layer, closest to the wall
where 5*0 y , is called the viscous sublayer. The flow here is dominated by viscous forces
rather than inertial forces because, close to the wall, the fluid’s viscosity is the primary cause of
generating stresses. The next layer, in the region 40*5 y , is known as the buffer layer. Here
the inertia driven turbulent stresses are larger than they are in the viscous sublayer and are
comparable to the viscous stresses. Hence, the buffer layer is a transitional layer where the
viscous forces compete with inertial forces. The third and the last layer of the turbulent boundary
layer is known as the log region where 40* y . Here the inertial stresses completely outweigh
Appendix B: Fluid Mechanics 89
the viscous stresses and the flow is overwhelmingly turbulent (Schlichting, 1960., p. 465 & Kim
et al., 1971).
B6 Inertial Effects
Inertial effects are usually indicated by Reynolds number greater than 1. Reynolds number is
defined by
VL
Re , (B19)
where is the fluid density, V is the velocity, L is a length-scale, and is the viscosity. For
flows with no acceleration, eg, pipe flow, pressure loss is independent of , even for Re O(102).
To include acceleration, perhaps the Reynold number should be
VL
Re . (B20)
V represents the change in velocity in flow direction. More fundamentally,
F
F
forceviscous
forceinertial
_
_Re . (B21)
The inertial force, F , can only arise with acceleration, i.e:
xaVolmaF , (B22)
where xa is the acceleration in the flow (x) direction and Vol is the volume.
The viscous force, F , depends on the shear stress and area, i.e:
)()( AreaL
VArea
y
VAreaF
y
xxxy
, (B23)
Appendix B: Fluid Mechanics 90
where y and yL are the distances in the transverse direction over which the velocity changes
by xV .
Then,
Area
Vol
V
aL
AreaL
VaVol
F
F
x
xy
y
x
x
)(
)(Re , (B24)
where Area is a surface area, one on which the shear stress acts to resist the flow,
xLArea
Vol , (B25)
where xL is the length-scale in the flow direction.
Then
xx
yx aV
LL
Re . (B26)
Now, xa is acceleration in the flow (x) direction, which is the change in velocity xV over the
time t . The time is the length in the x-direction divided by the mean velocity, i.e,
./ x
xx
xx
xxx L
VV
VL
V
t
Va
(B27)
Then, xyxx
xxyx VLLV
VVLL
Re (B28)
This definition of Reynolds number, denoted in the text as ReI , could be used to determine whether inertial effects are important:
That is, if ,1
xy VL
inertia is not important
whereas, if ,1
xy VL
inertia likely plays a role.
[Reference: James, 2011, Private Communications]
91
APPENDIX C: Oldroyd-B Model
The Oldroyd-B equation is given by:
DD ˆ2ˆ 21 , (C1)
where is the stress tensor, D is the deformation rate tensor, 1 is the relaxation time, 2 is the
retardation time, is the viscosity, and and D are the upper convected time derivatives of the
stress and deformation rate tensors respectively. This derivative is given by (Bird et al., 1987, p.
296):
vvvt
T
, (C2)
where v is the fluid velocity vector, is the gradient of the stress tensor, v is the fluid
velocity gradient tensor, and Tv is the transpose of that tensor.
Expressing the total stress in the fluid as a sum of the stresses, S and P , generated respectively
by the solvent and the polymer, i.e:
PS . (C3)
Equation B2 can be re-written as two separate equations expressing the contribution of the
polymer and the solvent. The polymer contribution is given by:
DPPPP ˆ , (C4)
where P is the polymer relaxation time, P is the polymer viscosity, and P is the upper
convected time derivative of the polymer contribution to the stress.
Appendix C: Oldroyd – B Model 92
The stress generated by the Newtonian solvent is given by:
DSS , (C5)
where S is the solvent viscosity.
Since the solvent is Newtonian, it has no contribution to the relaxation time, and hence:
P 1 . (C6)
The fluid retardation time is a fraction of the relaxation time and is given by:
PS
S
2 , (C7)
and the overall viscosity of the fluid is given by:
SP (C8)
Response to Shear Flow
For steady shear flow, the Oldroyd-B equation can be simplified to obtain the following
expressions for the stress components (Bird et al., 1987, p347):
xyxy (C9)
22 xyPyyxx (C10)
0 zzyy , (C11)
where xy is the shear rate, xy is the shear stress, is the relaxation time, P is the polymer
viscosity, and xx , yy , and zz are respectively the normal stress components in the streamwise,
transverse, and spanwise directions. The quantity yyxx is known as the first Normal Stress
Difference, 1N , and is a measure of fluid elasticity in shear.
yyxxN 1 (C12)
Appendix C: Oldroyd – B Model 93
The quantity zzyy is known as the second Normal Stress Difference, 2N . For a Newtonian
fluid both 1N and 2N are zero because all the normal stress components in a Newtonian fluid
are identically zero.
zzyyN 2 (C13)
The Oldroyd-B model predicts the shear stress and the first Normal Stress Difference to
respectively have linear and quadratic dependence on the shear rate. It also predicts the second
normal stress difference to identically zero at all shear rates.
Figure C1 Characterization of the original Boger fluid prepared by Boger, 1977. The blue
symbols represent the shear stress and the red symbols represent the first normal stress difference. (Reproduced from James, 2009)
Appendix C: Oldroyd – B Model 94
Figure C1 shows measurements of xy and 1N under variation of xy for the original Boger fluid
prepared by Boger (Boger 1977). As shown by this plot, 1N of Boger fluids tend to follow the
Oldroyd-B model, i.e a semi-logarithmic plot of 1N vs xy has a slope of 2, in the low shear rate
range up to a certain shear rate beyond which the slope decreases and the variation is no longer
quadratic. It is therefore possible to determine the relaxation time, , of an Oldroyd-B fluid
from 1N measurements in this low-shear rate range as:
21
2 xyP
N
(C14)
For small amplitude oscillatory shear flow, the Oldroyd-B model simplifies to:
2
"
)(1
PS
G (C15)
2)(1
'
PG
(C16)
Equation 2.42 can also be used to determine a relaxation time from 'G measurements of
Oldroyd-B fluids in the range of small frequencies, i.e as 0 . In this case, the equation
reduces to:
P
G
2
' (C17)
For a viscoelastic fluid, 'G is a measure of the fluid’s elasticity while "G is a measure of its
viscosity. For a Newtonian fluid, xy is always 90° out of phase with xy and therefore, 'G is
identically zero for a Newtonian fluid.
95
APPENDIX D: Pressure Transducer
D1 Specifications
From manufacturer
From manufacturer
Appendix D: Pressure Transducer 96
D2 Components & Features
From manufacturer
Appendix D: Pressure Transducer 97
D3 Calibration
y = 0.0016x - 0.0298
2.7
2.8
2.9
3.0
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
1700 1800 1900 2000 2100 2200 2300 2400 2500
Pressure (psi)
Pre
ssu
re S
enso
r V
olt
age
Ou
tpu
t (m
V)
Honeywell Pressure Sensor
Calibration Using Column of Water
Figure D1 Low pressure calibration curve using a column of water
y = 0.0016x + 0.2269
27.8
28
28.2
28.4
28.6
28.8
29
29.2
29.4
29.6
17600 17800 18000 18200 18400 18600 18800
Pressure (Pa)
Pre
ssu
re S
ens
or
Vo
ltag
e O
utp
ut
(mV
)
Honeywell Pressure Sensor
Calibration Using Column of Water
Figure D2 High pressure calibration curve using a column of water
Appendix D: Pressure Transducer 98
D4 Engineering Drawing
From manufacturer
99
APPENDIX E: Engineering Drawings of Flowcell
E1: Snapshots from SolidWorks
Figure E1 Exterior view of flowcell
Figure E2 Interior view showing the channels inside the flowcell
Appendix E: Engineering Drawings of Flowcell 100
Figure E3 Side view of flowcell
Figure E4 Front view of flowcell
Appendix E: Engineering Drawings of Flowcell 101
E2: Engineering Drawings of Individual Parts and Assemblies
Appendix E: Engineering Drawings of Flowcell 102
Appendix E: Engineering Drawings of Flowcell 103
Appendix E: Engineering Drawings of Flowcell 104
Appendix E: Engineering Drawings of Flowcell 105
Appendix E: Engineering Drawings of Flowcell 106
Appendix E: Engineering Drawings of Flowcell 107
Appendix E: Engineering Drawings of Flowcell 108
Appendix E: Engineering Drawings of Flowcell 109
Appendix E: Engineering Drawings of Flowcell 110
Appendix E: Engineering Drawings of Flowcell 111
Appendix E: Engineering Drawings of Flowcell 112
Appendix E: Engineering Drawings of Flowcell 113
Appendix E: Engineering Drawings of Flowcell 114
Appendix E: Engineering Drawings of Flowcell 115
Appendix E: Engineering Drawings of Flowcell 116
Appendix E: Engineering Drawings of Flowcell 117
Appendix E: Engineering Drawings of Flowcell 118