Electroconvection in Sheared Annular Fluid Films ZA 2000.pdf · Electroconvection in. Sheared...

Electroconvection in Sheared Annular Fluid Film s

by

Zahir Amirali Daya

A thesis subm itted in conform ity w ith the requirem ents

for the degree of D octor of Philosophy

G raduate D epartm ent of Physics

University of Toronto

© Copyright by Zahir A m irali Daya 2000

R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.

‘ ‘Blind to a l l fa u lt , destiny can be ru th less at one's s l ig h te s t

d is t r a c t io n .’ ’

Jorge Luis Borges, The South, A Personal Anthology.

i


Electroconvection in. Sheared A nnular Fluid Film s, D octor of Philosophy, 2Q00.

Zahir Am irali Daya, G raduate D epartm ent of Physics, University of Toronto.

A bstractA nnular electroconvection in freely suspended th in fluid films undergoing circular

C ouette flow is a novel nonlinear system which, in this thesis, conies under experi

m ental and theoretical scrutiny. Its novel features, which stem from its geom etry and

its electrohydrodynam ic character, include the superposition of an azim uthal shear

flow w ith a radial electrically driven hydrodynam ic instability in a two-dimensional,

natu rally periodic system. Concentric circular electrodes support a weakly conduct

ing annular fluid film which electroconvects when a sufficiently large voltage V is

applied betw een its inner and outer edges. Dy ro ta tion of the inner edge, a C ouette

shear is imposed. T he control param eters are a Rayleigh-like num ber R ex V 2 and the

Reynolds num ber R e of the azim uthal shear. The geom etrical and m aterial properties

of the film are characterized by the radius ratio a = r , / r 0, where r ,( ru) is the radius

of the inner (outer) electrode, and a Prandtl-like num ber V . The electroconvective

flow whose onset occurs when R = R c is described by a nonaxisym m etric m ode num

ber m c and a traveling ra te 7 *. T he dependence of R c.m c,Yc on a . V .R e has been

investigated theoretically by linear stability analysis. Experim ental m easurem ents

of current-voltage d a ta were used to determ ine th e onset of electroconvection over a

broad range of a. V and 1Ze. These are com pared with the theoretical predictions.

T h e current-voltage d a ta were used to infer the am plitude of convection in the weakly

nonlinear regim e by fitting to a steady-state am plitude equation with a lowest order

cubic nonlinearity w ith coefficient g. Results for g as a function of a . V and R e

are reported . U nder various conditions, the prim ary bifurcation can be supercritical

(g > 0), tricritical (g = 0) or subcritical (g < 0). Above onset, num erous subcriti-

cal secondary bifurcations, th a t m ark transitions from one flow p a tte rn to another,

were encountered. A sam pling of bifurcation scenarios is presented an d the ir R e

dependence is studied.

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Acknowledgem entsI would like to express my g ra titu d e to Stephen M orris for all that s tu ff for which

one thanks a supervisor; to Vatche Deyirm enjian for his collaboration, encouragem ent

and many a sobering conversation; to W ayne Tokaruk for his interest, countenance

and cam araderie; to Tim othy M olteno and Kiam Choo for their inpu t and m irth;

to Fraser Code and Allan Jacobs for their criticism and counsel; to Delroy Curling,

Alvin Ffrench, K hader Khan and R aul C unha for technical assistance; to Charles De

Souza for his cheer and good will; to Malcolm G raham for his concern and hum our;

and to Derek M anchester for those not too seldom encounters rife w ith narrative,

insight and anecdote.

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C ontents

1 Introduction

1.1 In tro d u c t io n ..............................................................................................................

1.2 Previous E lectroconvection E x p e r im e n ts .....................................................

1.3 Previous Electroconvection T h e o r y ................................................................

2 Experiment

2.1 In tro d u c t io n .............................................................................................................

2.2 Experim ental D e s ig n ............................................................................................

2.3 Experim ental P r o t o c o l ........................................................................................

3 Theory

3.1 In tro d u c t io n .............................................................................................................

3.2 T he Governing Equations .................................................................................

3.3 T he Dase S t a t e ......................................................................................................

3.4 Linear S tability A n a ly s is .....................................................................................

3.5 A ssum ptions: Theory versus E x p e r im e n t .....................................................

3.6 A m plitude E q u a t i o n ............................................................................................

4 Results

4.1 In tro d u c t io n .............................................................................................................

4.2 D ata A n a ly s is ..........................................................................................................

4.3 Com parisons w ith Linear T h e o r y ...................................................................

4.4 Coefficients of th e Cubic and Q uintic N onlinearity w ithout Shear . .

iv

1

I

10

12

22

22

24

31

42

42

45

48

52

73

76

81

81

82

88

94


4.5 Coefficients of the Cubic and Quintic Nonlinearity w ith Shear . . . . 1 0 0

4.6 Secondary B ifu rc a tio n s ........................................................................................ 104

4.7 M isc e lla n y ............................................................................................................... 118

4.8 O th e r Similar S y s t e m s ...................................................................................... 122

5 Conclusions 134

5.1 In tro d u c t io n ........................................................................................................... 134

5.2 Conclusions: E x p e r im e n t................................................................................... 135

5.3 Conclusions: T h e o r y .......................................................................................... 140

6 Afterword 144

A Colourimetric Determination of the Film Thickness 145

B Cylinder Functions 151

B .l Expansion functions for the stream f u n c t i o n .............................................. 151

B .2 Expansion functions for th e p o te n t ia l .......................................................... 153

C Exact Nonlocal Solution 155

D Data at Atmospheric Pressure 159

E Data Modelling 165

F Some Future Investigations 172

F .l Electroconvection in an Eccentric A n n u lu s ................................................. 172

F .2 Electroconvection w ith Oscillatory S h e a r .................................................... 179

F.3 M easurem ent of Viscosity................ .................................................................... 182

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List o f Figures

1.1 T he geom etry for annular electroconvection w ith shear........................... 2

1.2 A schem atic of the laterally unbounded rectangular geom etry.............. 11

1.3 C harge generation at the free surface.............................................................. 14

2.1 A schem atic of the experim ental se tu p ........................................................... 23

2.2 A schem atic of the e x p e r im e n t........................................................................ 25

2.3 G eom etry of the electrodes................................................................................... 26

2.4 A schem atic of the inner electrode................................................................... 27

2.5 A schem atic of the 'film-drawing' assembly................................................... 28

2.6 A schem atic of the electrical design................................................................. 29

2.7 R epresentative current-voltage d a ta ................................................................ 34

2.8 T he drift in the curren t........................................................................................ 38

3.1 A diagram to elucidate the electroconvection instability .......................... 43

3.2 T he base s ta te surface charge density.............................................................. 51

3.3 T he relative stability a t a = 0.5 and a = 0 .8 ............................................... 52

3.4 T he surface charge density p ertu rba tion ........................................................ 58

3.5 T he m arginal stability boundary for zero shear........................................... 61

3.6 T he critical pair (72.°, m°) as a function of a ................................................ 64

3.7 T he m arginal stability boundaries a t various shears .................................. 65

3.8 T he traveling rates of the p a tte rn ..................................................................... 66

3.9 T heoretical predictions of th e suppression. P a rt I .................................. 67

3.10 T heoretical predictions of th e suppression. P a rt I I .................................. 6 8

3.11 Velocity vector field for annu lar electroconvection w ithout shear. . . . 71

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3.12 Velocity vector field for annular electroconvection with shear................. 72

4.1 A representative supercritical bifurcation....................................................... 8 6

4.2 A representative subcritical bifurcation........................................................... 87

4.3 C ritical voltages in films w ithout shear........................................................... 89

4.4 Experim ental m easurem ents of the suppression............................................ 92

4.5 Experim ental m easurem ents of the coefficient of the cubic nonlinearity

in films w ithout shear. P art 1.............................................................................. 95


in films w ithout shear. P art II............................................................................ 97

4.7 Experim ental m easurem ents of the coefficient of the quintic nonlinear

ity in films w ithout shear........................................ 98

4.8 T he size of the hysteresis Se a t various V for a = 0.33.............................. 99


in sheared films......................................................................................................... 101

4.10 Experim ental m easurem ents of the coefficient of the quintic nonlinear

ity in sheared films.................................................................................................. 103

4.11 T he size of the hysteresis 5e as a function of the Reynolds num ber. . . 105

4.12 Representative plots of secondary bifurcations in films w ithout shear. 107

4.13 Representative plots of secondary bifurcations in films with shear. . . 109

4.14 A representative plot of a sequence of bifurcations. Part I ..................... 112

4.14 A representative plot of a sequence of bifurcations. P art I I .................. 113

4.15 An exam ple of a m apping of param eter space............................................... 115

4.16 M ultiple bifurcations a t a = 0.80....................................................................... 119

4.17 Backward and forward bifurcations in the sam e experim ent.........................120

4.18 A ‘delayed1 subcritical bifurcation..................................................................... 122

A .l A chrom aticity diagram for smectic A 8 CB films......................................... 148

A .2 Colour charts for sm ectic A 8 C B ....................................................................... 149

D .l A plot of the scaled Vc° versus the onset conductance a t several a . . . 160

vii


D.2 Com parison between experim ental m easurem ents of suppression and

theoretical predictions................................................................................... 163

E .l T he conductance of the film during a current-voltage ru n .............. 166

F .l A schem atic o f the eccentric annular geom etry................................... 173

F.2 A representative plot of current-voltage d a ta from an eccentric annu lar

film....................................................................................................................... 174

F.3 T he resistance of an off-centered film..................................................... 176

F.4 R epresentative plots of the current-voltage characteristics of sheared

films in eccentric annuli................................................................................ 178

F.5 A com parison of the C ouette and Oscillatory shear profiles........... 181

F .6 A schem atic for m easuring the in-plane viscosity............................... 183

viii


List o f Tables

3.1 C ritical param eters for zero shear...................................................................... 62

3.2 C ritical param eters a t various V with constant H e ..................................... 69

3.3 C ritical param eters a t various V with constant Q ........................................ 70

4.1 Experim ental m easurem ents of the marginally stab le mode num ber, mj!. 90

4.2 Experim ental m easurem ents of the coefficient of th e cubic nonlinearity.

g w ithout shear......................................................................................................... 96

4.3 Experim ental m easurem ents of the Reynolds num ber for g = 0 .............. 102

4.4 Experim ental m easurem ents of the minimum value of g. the corre

sponding Reynolds and P randtl-like num bers................................................ 102

be


C h ap ter 1

Introduction

1.1 Introduction

M ost na tu ra l phenomena are a consequence of nonlinear and nonequilibrium pro

cesses. Yet. the physical principles th a t dictate these processes, unlike the rarer

equilibrium phenomena, are obscure. Due to the great variety of systems, there are

many approaches in trying to understand nonlinear, nonequilibrium processes. One of

the more recent advents is the s tudy of pattern form ation.[1 . 2 ] P atterns m e spatially

s truc tu red sta tes that spontaneously appear in a system when a source of nonequi

librium stress, also called a control param eter, is varied. W hereas patterns abound

in natu re , they can only be accurately studied in physical systems which can be de

fined m athem atically and whose physical properties can be measured experimentally.

These criteria are readily met by fluid dynamical system s. Consequently, most pat

te rn forming systems that are studied today are fluid dynam ical. T he system th a t

is cen tral to this thesis is m athem atically described by the century old classical field

theories of fluid and electrodynam ics and is am enable to experim ental investigation.

As is the case with this and o the r p a tte rn forming systems, the detailed classical

descriptions have given way to m odern interpretations in the context of dynam ical

system s theory.

T he system th a t is studied here is a freely suspended annular fluid film which can

be driven ou t of equilibrium by electrical forces and can be independently subjected

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a = r/ r0

Figure 1 .1 : T he geom etry for annular electroconvection w ith shear. Cylindrical coordinates are used to describe the annular film. The inner electrode is a circular disk of radius r, centered a t the origin while the outer electrode occupies the region r > r0. The film spans the annular region r, < r < ra and is two-dimensional. The rad ius ratio a = r , / r 0 where r ,( r 0) is the inner(outer) electrode radius. The positive c axis is out of th e page.

to a shear flow. A schem atic is shown in Fig. 1 .1 . The film is suspended betw een

concentric circular electrodes. T he fluid can be driven electrically by a voltage dif

ference between the electrodes and can be sheared by ro ta tin g the inner electrode

about its axis. The pa tte rn th a t emerges when the film is driven sufficiently o u t of

equilibrium is referred to as electroconvection. This s ta te is comprised of an array of

counter-ro tating vortices arranged around the annulus. How th is pattern evolves, how

it depends on the geom etry and how its properties are changed by the application of

a shear to th e film are the experim ental and theoretical questions tha t are addressed

in this thesis.

The annular geom etry in this experim ent facilitates electrical radial driv ing by

the variation of an applied voltage between the inner and o u te r edges of the annulus.

Radial driving forces are a feature th a t few experim ental system s possess and here,

originate from the interaction between the radial electric field and the surface charge

density th a t develops on the film’s free surfaces. Since the forcing in this experim ent is

2


electrical in origin, it is independent of the hydrodynam ics allowing for a superposition

of electroconvection on a variety of hydrodynam ic states. The sim plest is an azim uthal

or circular C ouette shear. T his can be im plem ented by ro ta ting the inner edge of the

annulus. T he shear flow leads to a net m ean flow in th e azim uthal direction. T he

m ean flow, by v irtue of the natura lly periodic annular geometry, is closed on itself.

T he electroconvection p a tte rn in the presence of shear travels in the direction of the

m ean flow.

T he prim ary experim ental technique involved m easuring the electrical curren t

through the film as it was driven out of equilibrium by variation of the applied voltage.

From an experim ental perspective, the annular geom etry is particularly well suited

for m aking precise m easurem ents of the electrical curren t since it does not have la teral

boundaries and so is free of leakage currents.

P a tte rn s alm ost always evolve from homogeneous s ta tes via sym m etry breaking

bifurcations.[2, 3] The prim ary concept here is th a t when an unstructured s ta te be

comes unstable to a p a tte rn sta te , the loss of stability is accom panied by a broken

sym m etry. T he transition from homogeneity to a p a tte rn is an exam ple of a bifurca

tion. T he concept of stab ility is essential to identifying bifurcations; for a bifurcation

to occur, a solution m ust becom e unstable to another solution. Linear stab ility anal

ysis is a prerequisite first s tep toward understanding any p a tte rn forming system . In

fact, it is by th e characteristics of the linear instability th a t nonequilibrium spatial

p a tte rn s are often classified.[2] The m ethod of linear stab ility is well founded and

has been used for many years in the field of hydrodynam ic stability. One generally

considers infinitesim al pertu rba tions of the variables in the system a t a known spa

tia l periodicity and exponential time-dependence, say of the form Ne‘m0+'|,t. Here the

spatia l periodicity is given by 2 n /m m easured along a coordinate 9. 7 is referred to

as the growth ra te and it is often a com plex quantity, more conveniently w ritten as

7 = 7 r + t’7 ‘, where Y ( l ‘) is the real (im aginary) com ponent of the grow th rate . W hile

7 r is m easure of the exponential growth or decay of th e pertu rba tion , 7 * describes

its oscillation. N is formally infinitesimal and is determ ined by the equations and

boundary conditions a t work in the system . Clearly the no ta tion heralds an analysis

3


in cylindrical coordinates, however, the m ethod is general.

To carry out the linear stab ility analysis, the equations th a t describe the system

are then linearized in the p ertu rbed variables. Generally, the source of nonequilibrium

stress in th e system is m easured by the value of a dimensionless param eter called the

control param eter, 71. There m ay be other dimensionless param eters th a t describe the

system which can be assumed to be held constant. T he stronger the nonequilibrium

stress, the larger the control param eter 1Z. For a given m and TZ. the m ethod of linear

stab ility analysis determines th e value of Y ■ If Y > 0(7r < 0) then the system is

linearly unstable(stable) to the perturbation . W hen Y = 0- the system is said to be

m arginally or neutrally (un)stable. In practice however, linear stability analysis is

used to determ ine the values of TZ and Y for a given m and 7 r = 0 . By varying m the

m inim um value of 7Z a t which the system is marginally unstable can be determ ined.

This m inim um value. 1Z = 1ZC corresponds to the greatest nonequilibrium stress th a t

can be applied to the system w ithou t driving it linearly unstable and is called the

critical value. The other critical param eters are m c and Yc- W hen 1Z ju s t exceeds

1ZC, the system is linearly unstab le to a pattern w ith spatia l periodicity 2 ir /m c, and

oscillates w ith a temporal period 2ir/Yc- If 7 c = 0 , the instability, the pa tte rn and the

bifurcation are termed stationary. If Yc ^ 0* th e instability and pa tte rn are term ed

oscillatory, while the bifurcation is often qualified as Hopf. The theoretical research

in th is thesis is comprised of a linear stability analysis of annular electroconvection

w ith a shear flow. The m ethod is similar to th a t outlined above and as is shown

in C h ap te r 3 the linear theory for th is system is particularly rich, displaying various

instabilities.

E xtended, three-dimensional, nonlinear systems are prone to develop com plicated

spatia l and temporal patterns even when only weakly nonequilibrium . [2] T he observed

p a tte rn s and complex dynam ics are often the result of one or more sym m etry-breaking

bifurcations. It is thus in teresting to study p a tte rn form ation in low-dim ensional

system s which are close to equilibrium but have little symmetry[4] so th a t the re is

only a very restricted set of sym m etry-breaking bifurcations available. In general,

one seeks the most complex dynam ics and patterns th a t can be realized in as sim ple

4


and restricted a system as is possible. Annular electroconvection in freely suspended

films exploits the s tric t two-dimensionality of a subm icron smectic A liquid crystal

film. The lower dim ensionality greatly reduces the variety of possible p a tte rn states

and so makes it easier to experim entally study the rich nonlinear p roperties of the

basic pattern, and sim pler to trea t the problem theoretically.

T he working fluid in th is experiment was a sm ectic A liquid crystal th a t serves

prim arily to constrain th e flow to two dimensions. Freely suspended liquid crystal

films have been often s tud ied for their various interesting equilibrium properties and

have recently been reviewed in Ref. [5]. Liquid crystalline phases or mesophases are

s ta tes of m atter w ith degrees of order interm ediate between that of th e ir norm al solid

and liquid states. In som e m aterials the liquid crystalline phases are accessible by

variation of the tem peratu re . These liquid crystals are generally organic com pounds

w ith molecular aggregates th a t are long along one axis. In fact 'c igar'-shaped would

be a surprisingly good descrip tion for these molecules. The variation in th e long range

o rder of these long molecules gives rise to the different mesophases. Sm ectic liquid

crystals consist of layers of orientationally ordered molecules. In sm ectic A. the long

axis of the molecules is norm al to the layer plane. T he layers are equid istan t from each

other. W ithin each layer, however, the distribution of molecules is isotropic. Smectic

A exhibits two-dimensional isotropic fluid properties in the layer p lane while flows

perpendicular to the layers are strongly inhibited; in fact in tha t direction, the smectic

A phase is often described as a plastic solid. Due to their layered s tru c tu re , uniform

suspended smectic films are always an integer num ber of smectic layers th ick and while

they readily flow, they seldom change thickness. As a result, a freely suspended film

of a smectic A liquid c ry sta l is robustly two-dimensional. It can su sta in rapid flows

w ithin the layer plane w ithou t flows between layers. The other m ateria l properties

in the smectic A phase, such as the conductivity and the dielectric perm eability, are

also isotropic w ithin each layer. Since smectic liquid crystals have very sm all vapour

pressures, the film can be enclosed in an evacuated environment. T h e reduction in

th e ambient pressure leads to a proportional reduction in the air d rag th a t the film

is subjected to .[6 ] A reduced am bient pressure environm ent is, for th e first tim e in

5


electroconvection experim ents, im plem ented in the research reported in this thesis.

Smectic films may be con trasted with soap films, on which m any hydrodynam ic

experim ents have been perform ed.[7, 8 . 9] Soap films cannot be produced in an envi

ronm ent w ith a pressure lower th an the vapour pressure of water, and so are always

sub ject to residual drag. In addition, ordinary soap films have a much larger electrical

conductivity than smectic films and cannot be driven to electroconvect a t reasonable

voltages. Finally, soap films always have m uch larger thickness nonuniform ities than

sm ectic films. As a result, sm ectic films are a be tte r candidate for studying electro

convection than soap films.

T he linear instability and subsequent nonlinear evolution of flows depend strongly

on the sym m etry and s tru c tu re of the unstab le base flow. One way of system atically

study ing th is dependence is to superpose sim ple flows on well-understood instabilities.

T he addition of a base s ta te flow introduces a second control param eter, alters the

sym m etries of the unstructu red sta te and can affect the bifurcation to the pa tte rn

s ta te . A nnular electroconvection possesses interesting symmetries. In the absence

of shear, the base state is invariant under azim uthal rotation and reflection in any

vertical plane containing the ro tation axis through the center of the annulus. The

electroconvection pattern s ta te appears w ith the spontaneous breaking of the az

im uthal invariance and as will be dem onstrated later, is stationary. T he application

of azim uthal shear breaks th e la tte r reflection symmetry and distinguishes between

the clockwise and counter-clockwise directions. The shear further reduces the sym

m etry of the base state and leaves it with only a single spatial sym m etry; azim uthal

invariance. The electroconvection pattern once again breaks the azim uthal sym m etry,

and since the reflection sym m etry is absent due to the shear, the p a tte rn m ust travel

azim uthally. By reducing th e accessible dim ensions, the variety of pa tte rn s th a t may

appear is reduced. On the o th e r hand, by im posing an additional control param eter

the types of physical in teractions in the system are increased. T he essence of this

s tra tegy is the notion th a t the mechanisms leading to com plexity m ay be easier to

identify and study in a lower dim ensional system than in a higher one.

O ne of the most strik ing results concerning pattern form ation is th a t hydrody-

6


naxnic instab ility is analogous to phase transitions in therm odynam ic systems. [1 0 ]

This resu lt is well established for many p a tte rn forming instabilities; in fact the

m athem atical description of bifurcations is identical to th a t of critical phenom ena

or phase transitions, [l, 2] W hether the sim ilarities are superficial or there are deep

im plications of th is coincidence, is a question open to debate. W h a t is dissatisfying

however, is th a t whereas the m athem atical description follows qu ite easily from the

m inim ization of a free energy functional in critical phenomena, th e re are no such gen

eral principles for nonequilibrium systems. [2, 11] The quest for o the r unifying physical

principles is a central goal of nonequilibrium physics.

Near th e instability, the m athem atical description of a broad range of patterns con

sists of sim ple equations of universal form often referred to as am plitude equations.[2 ,

3] These are generic equations, so-called because they describe the evolution of

the ’am plitude ' of the pattern , one of the m ost common exam ples of which is the

G inzburg-Landau equation:

r^dtA = eA -I- — </o|A|".4 . (1-1)

In Eqn. 1 .1 . A ( x . t ) is the am plitude of the stationary p a tte rn which in most fluid

dynam ic experim ents is the velocity of the convective flow. In the weakly nonlinear

regime, flows w ith length scales near the critical mode are excited and so the am pli

tude varies slowly tem porally and spatially. T he coefficients r 0. £o and gQ are real and

constant. T h e reduced control param eter, e = 'R./‘R C — 1 1 is th e small param eter

for which th e am plitude model is valid. T he universality of the am plitude equations

stems from the common sym m etries of the m ultitude of physical system s which are

prim arily controlled by the geom etry and the boundary conditions. The microscopic

details of th e individual system s are contained in the coefficients r 0, £0 and go which

define th e tem poral, spatial and am plitude scales respectively.

W hile th e re are several m ethods by w hich am plitude equations can be derived

from th e underlying microscopic or field equations th a t describe th e system, the form

of the am plitude equation can be arrived a t from the sym m etry o f th e pattern .[2 ] For

7 •


exam ple Eqn. 1.1 is invariant under the A —>■ —A operation, which is a sym m etry

of the p a tte rn th a t it describes. T he cubic nonlinearity is the lowest o rder nonlinear

te rm th a t preserves this sym m etry. O ften nonlinear terms of higher o rder are included

in the am plitude equations. W hen the spatia l derivative term in Eqn. 1 .1 is dropped,

the am plitude equation is referred to as a Landau equation and describes a pitchfork

bifurcation. It is so called because the A and — A solutions are indistinguishable and

when graphed give the appearance of the two prongs of a pitchfork.

Like phase transitions, the m ost common bifurcations are supercritical (second

order) or subcritical (first order). A supercritical or forward bifurcation evolves con

tinuously for e > 0 , while a subcritical or backward bifurcation is discontinuous at

e = 0. This distinction is m athem atically em bodied in the sign of gQ in th e am plitude

equation. Precisely how the value of this coefficient is determined depends intricately

on the microscopic model th a t describes the system. Short of calculating this co

efficient. there seems to be no way to know a priori whether a system will evolve

to a p a tte rn supercriticallv or subcritically. T he foregoing is also tru e for therm o

dynam ic system s. A system in which the nature of the bifurcation can be suitably

varied bears great promise to elucidate the physical processes th a t d ic ta te the nature

of the bifurcation. One of the im portan t results of the research reported here is to

have experim entally dem onstrated a system for which one can by varying physical

param eters choose the natu re of the bifurcation.

W hen th e p a tte rn is not stationary , the am plitude equations generally take the

form of the complex G inzburg-Landau equation:

r0d tA = e (l + ico)A + $}(1 + ic l )d%A - <7o(l - ic3)|A |2A . (1.2)

where Co, cx and c3 are real constants. Observe th a t the difference betw een Eqns. 1.1

and 1 .2 is th a t while the coefficients in the former equation are real, they are com

plex in the la tte r. W hereas the com plex Ginzburg-Landau equation describes the

am plitude of the electroconvection pa tte rn s in the sheared annular system th a t is the

sub ject of th is thesis, it is not derived here bu t simply motivated by sym m etry consid-

8


erations. In addition to periodic patterns, the solutions of Eqn. 1 .2 include localized

coherent structu res such as fronts and pulses and for some param eters, spatiotem poral

chaos. [2 ]

In sum m ary, the purpose of this project was to study pa tte rn forming instabilities

in a low dim ensional geom etrically simple system with two control param eters. T he

goal was a tta ined in th e study of annular electroconvection in sheared fluid films,

where th e control param eters are the electrical driving and the s tren g th of the shear

flow. T h e system has several novel features. The annular geom etry and electrical

natu re allow for radial driving forces and the superposition of a shear on electrocon

vection. T he inherently nonlocal interactions which stem from the electrical character

of th e system and the smectic-film-enforced two-dimensionality are not encountered

in o th e r p a tte rn forming system s. The annular geometry is natu ra lly periodic and

therefore the imposed shear comprises a closed flow. The study was both experimental

and theoretical.

T he experim ental work encompassed the design and construction of an apparatus

to perform current-voltage m easurem ents on an annular film in a reduced am bient

pressure environm ent. From the current-voltage data, one can determ ine the onset of

electroconvection, the natu re of the bifurcation and some of the nonlinear properties of

electroconvection. How these change as the geometry of the annulus and film thickness

are altered , a n d /o r when the applied shear is varied were all open to investigation.

Secondary bifurcations, which, due to the lower dimensionality o f the system could

be easily identified, were also studied.

T he theoretical p a rt of the research concerned extending th e study of surface

driven electroconvection to an annular geom etry with a sheared base flow. T he

objective was to predict the onset of electroconvection and the prevailing mode of

electroconvection a t different rates of shear. No quantitative experim ental or theo

retical work on electroconvection with or w ithout shear had ever been performed in

an annu lar geom etry prior to the research reported in this thesis.

T h e nex t two sections are devoted to a short review of previous work on electro

convection in freely suspended films. Experim ents in rectangular films are discussed

9


in Section 1.2 while the theoretical aspects are reviewed in Section 1.3.

1.2 Previous Electroconvection Experim ents

There have been several electroconvection experim ents in freely suspended liquid

crystal films.[12, 13, 14, 15, 16, 17] All the previous electroconvection experim ents

have been perform ed in rectangular geometries. For the rem ainder of this C hapter,

comments have been limited to the prototypical experiments in 2 D isotropic liquid

crystal films in the smectic A phase[l3, 14], although experim ents have also been

done in more complex anisotropic smectic phases.[16. 17]

The experim ental apparatus of Refs. [13, 14] consisted of two parallel conducting

wires separated by a distance on the m illim eter scale. The liquid crystal film spans

the region between the wires for a length between 5 and 10 tim es as broad.[13. 14]

The films were uniformly thick. Over many experim ents, a thickness range of 2 to

160 sm ectic layers were investigated. The thickness of a single layer of the sm ectic

A liquid crystal used in these experim ents is 3.16 nm .[l8 ] Thus the ratio of the film

thickness s to w idth d is s /d ~ 1 0 -4 . T he sm ectic A film was freely suspended in a

room tem pera tu re and pressure environm ent. A dc voltage drop V between the wire

electrodes is sym m etrically applied i.e. V /2 to one electrode and —V /2 to the o ther.

Particles, in some experim ents fine chalk dust while in others incense smoke, were

allowed to se ttle on the film surface. As the voltage drop V was slowly increased,

the particles which were initially s ta tionary began to move. T his particle m otion

corresponds w ith the onset of an electroconvective flow in the film. It was found th a t

the onset for each film had a threshold voltage Vc below which the film was purely

conducting and quiescent while above which, a cellular flow com prised of counter-

ro tating vortices prevailed. See the schem atic provided in Fig. 1.2 for a pictorial

description. T his array of vortices is a one-dim ensional pa tte rn w ith a repeating un it

of a pair of counter-ro tating vortices. The wavelength a t onset of the electroconvection

instability was found to be approxim ately 1.3d.[13, 14]

The focus in these early electroconvection experim ents, which preceded the the-

10


X

Figure 1.2: A schematic of th e laterally unbounded rectangular geometry.

oretical developm ent, was to s tu d y the variation of the onset or critica l voltage Vc

with changes in several param eters bu t particularly the film thickness and width. It

was found th a t Vc increased linearly w ith the film thickness and was largely indepen

dent of th e film width. [13] H ysteresis in the critical voltage was no t observed when

V was decreased leading to th e conclusion th a t th e bifurcation to electroconvection

is forward o r supercritical. From measurements of current-voltage characteristics of

the rec tan g u la r film, it was concluded th a t below Vc the film behaves as an ohmic

conductor. W hen the film is convecting, the current is increased above th a t by mere

conduction. T h e current-voltage measurements were difficult due to th e presence of

an a lte rn a tiv e current path a round the transverse edges of the film. From measuring

particle velocities, a suitable m easure for the am plitude of electroconvection, it was

11


concluded th a t the bifurcation was a supercritical pitchfork. It was argued from ve

locity m easurem ents th a t the reduced control param eter was e = (V /V c)2 — 1 . It was

by th is and sim ilar results th a t it was established th a t the electroconvection vortex

p a tte rn obeyed a simple G inzburg-Landau equation.[14]

From an experim ental perspective, it should be clear th a t quantitative m easure

m ents on electroconvecting films by means of tracer particles is a difficult procedure.

W hen smoke particles, typically from burning incense, were used, they are likely to

affect the electrical conductivity since aromatic com pounds in the smoke probably

dissolve in the film.[19] T he particles have a size th a t is com parable to the film thick

ness. and so are invasive ra the r than passive. More im portantly, using partic les to

visualize the convective flow in order to determ ine the onset of convection by eye

is a som ew hat subjective procedure. To deduce any of the properties beyond onset

required m easurem ents of flow velocities which are tedious to perform.

C urrent-voltage d a ta can be used to study the bifurcation to convection and is

the preferable technique. However, the rectangular geometry, plagued by transverse

or la teral bounds, is unsuited to current-voltage m easurem ents. The la teral edges of

the film contain a w etting layer of liquid crystal th a t allows an alternative current

pa th . [2 0 ] T he annular geom etry does not have lateral bounds and so is free o f leakage

currents. C urrent-voltage measurements are performed for the work in th is thesis.

Since the sm ectic A film was freely suspended a t atm ospheric pressure, it is likely

th a t the effects due to air drag are significant when the film is convecting.[6 ] T h e ex

perim ents perform ed in this thesis are at reduced am bient pressure and so circum vent

the consequences due to air drag, which were likely to have been an im portan t factor

in previous experim ents.

1.3 Previous Electroconvection Theory

T his Section describes the Unear and weakly nonlinear theory th a t has been developed

to elucidate the electroconvection phenomena described in Section 1.2. T he focus here

is to elucidate the m echanism th a t drives electroconvection and to s ta te th e principal

12


results of the theories. The theoretical model, which was first elucidated by the

au tho r, has been reported in Refs. [21, 22]. A subsequent nonlinear analysis has been

reported in Ref. [23]. A schem atic of the geometry for the experim ents in rectangular

films(Refs. [13, 14]) is given in Fig. 1.2.

T he voltage is applied to the film by m aintaining the lines y = ± d /2 a t po

tentials of ± V /2 respectively. This electrode geom etry is referred to as the ‘wire’

configuration and resembles the experim ental geom etry described in Section 1.2. An

a lternative electrode geom etry is the ’plate ' configuration, which requires th a t the

film be supported along its edges by two large flat sheets of m etal i.e. by plates

instead of wires. A schematic is provided in Fig. 1.3a. T he annular geom etry which

is described by the radius ratio a = r , / r 0 where r ,( r0) is the inner(outer) electrode

radius approaches the ’plate' geom etry as a -» 1. In the theory, all m aterial and flow

properties were assum ed to be constant over the thickness of the film. s. T he film is

trea ted as a two-dimensional sheet since s /d <gi 1 .

T h a t an electrically driven instability results in th is configuration can be under

stood by considering the d istribution of charges on the film's free surfaces. W hen

the potentials ± V /2 are applied, a current flows in the film. Assuming th a t the film

behaves ohmically, the potential decreases linearly from V /2 to —V/2. Consequently,

the electric field inside the film is constant and has no com ponent perpendicular to

the film 's free surfaces. However, it is obvious tha t outside the film, the electric field

is not purely horizontal. As a result, there is a t the filnrs free surfaces a discontinuity

in the vertical com ponent of th e electric field. It is this discontinuity in the electric

field th a t supports a charge density q a t the free surface.[24] See Fig. 1.3b for a picto

rial explanation. The electrical forces th a t arise due to the interaction of th is surface

charge density w ith the electric field in the film’s plane drive the electroconvective

instability.

In suspended films, one finds th a t the electrical boundary conditions enforce a

poten tially unstable surface charge density configuration on the film w ith positive

charge close to the positive electrode and negative charge close to the negative elec

trode. Since th e surface charge density depends on th e electric field in th e free space

13


convectiveflow plate

electrode+V/2

plateelectrode-V/2

outside

outside

outside

inside

Figure 1.3: C harge generation a t th e free surface. In (a) is shown a schem atic of the film in the ‘p la te ’ geometry. In (b) is a schem atic illustration of th e electric fields in the small box in p a rt (a), q is the surface charge density th a t resu lts due to the discontinuity in th e norm al com ponent of the electric field a t the free surface.

14


surrounding the film, it m ust depend on the precise electrical boundary conditions

th a t are prescribed by the geometry. W h at is im portan t to realize is th a t the elec

tric field anyw here is determ ined by charge densities everywhere i.e. the relationship

determ ining the charge density is inherently nonlocal. As a result, the electrical

boundary conditions have to be specified everywhere on the boundary to properly

specify the surface charge density.

T he theoretical model required th a t the flow was two-dimensional, th a t the fluid

was isotropic in two dimensions, incompressible and was a weak electrical conductor.

It assum ed th a t the fluid can be described by the Navier-Stokes equation subjected to

an electrical body force. A charge continuity equation th a t allows for the conduction

and convection of electrical charges supplem ented the Navier-Stokes equation and the

incom pressiblity requirem ent. Finally, the usual electrostatic relation between the

charge density and electric fields com pleted the system of equations in the theoretical

model. T he im plem entation of this final relation is somewhat subtle. Whereas the

Navier-Stokes equation, the incompressibility requirem ent and the charge continuity

equation describe the fluid velocity, the charge density and the electric field in the two-

dim ensional film plane, the Maxwell relation couples the three-dim ensional electric

field exterior to the film to the surface charge density a t the film's free surfaces.

T he theoretical model was electrohydrodynam ic in character in th a t it treats the

lim it of a poorly conducting fluid so th a t electric fields are prom inent and electric cur

ren ts are weak. In this lim it the m agnetic fields and forces th a t arise from them are

negligible. T he opposite lim it th a t is often explored is th a t of a strongly conducting

fluid such th a t electric fields are negligible bu t electric currents and consequently mag

netic effects are dom inant. This is the m agnetohydrodynam ic lim it.[25, 26] Another

approxim ation is the assum ption th a t electrical polarization effects were insignificant.

T he film is also assum ed to be freely suspended in a vacuum. N ote th a t none of the

anisotropic properties of liquid crystals are included in the m odel. In fact, the sur

face driven electroconvection mechanism is very general and would occur in any freely

suspended fluid film subjected to the appropria te electrical boundary conditions.

W hen su itab ly non-dimensionalized, two non-dim ensional param eters describe the

15


film. A dimensionless control param eter 1Z and a param eter V are given by

where a, q and p are the bulk electrical conductivity, viscosity and mass density

respectively. e0 is the dielectric constant of free space while s and d, as m entioned

earlier, are the film thickness and w idth. All lengths are non-dim ensionalized by the

T he stab ility analysis tests when a mode th a t describes a pertu rba tion proportional

value 1Z™ = 76.77 and w ith a non-dim ensional critical wavenumber K* = 4.744 in

occurred a t 1Z£ = 91.84 and = 4.223. Clearly, the precise electrical boundary eon-

7Z given in Eqn. 1.3, the 'w ire' case loses stability a t lower voltages than the 'p la te '

case, all else being equal.

Several com parisons between the experim ent and theory were made. Since the

conductiv ity a and the viscosity q are poorly known[l3, 14], little em phasis can be

placed on some quan tita tive comparisons. However, other predictions and trends can

be com pared. The wavelength A of the one-dimensional vortex pa tte rn corresponds

to a non-dim ensional wavenumber /« = 2ird/X. Experim ental m easurem ents of this

quan tity a t onset give xjfP* = 4.94 ± 0.25. This is in good agreem ent with th e value

of kc = 4.74 for ’wire’ electrodes. From the expression for the control param eter TZ

in Eqn. 1.3, it is evident th a t the critical voltage Vc is given by

N ote th a t Vc is independent of the film w idth d and varies linearly w ith th e film

thickness s. B oth trends were observed experimentally. Vc was accurately linear w ith

(1.3)

film w idth d. T he results of a linear stability analysis about the quiescent and con

ducting s ta te leading to an electroconvecting s ta te were found to be independent of V .

to etKX becomes marginally stable. Following standard procedures, it was found th a t

for th e ’wire" case, the conducting sta te becomes marginally unstable a t a critical

units of d l . For the 'p la te ’ case, likewise, it was determ ined th a t marginal stability

ditions have a significant effect on the critical values of 7Z and k . By the definition of

(1.4)

16


s for films w ith thicknesses less than a b o u t 25 sm ectic layers.[14] For th icker films Vc

increased w ith s sublinearly. This is a clear indication th a t three-dim ensional flow

effects, in p a rticu la r the relative m otion of sm ectic layers, is more pronounced in

thicker films. T h is is to be expected since the electrical forcing is a t the free surfaces

of the film. A greem ent between the p roportionality constant between Vc an d s was

not unreasonable, given that the m ateria l param eters were poorly known.

Experim entally K was independent of d for relatively ‘broad’ films a n d showed

a weak dependence as the film becam e narrower, [13] while Eqn. 1.4 im plies th a t

Vc is independent of d. This disagreem ent m ay be explained by three-dim ensional

electrode effects which are expected to be im portan t when the film is ‘n a rro w ’. In

dimensionless term s, the electrode size scales w ith d~ l . For a detailed num erical

comparison betw een the theory and th e experim ents see Refs. [13], [14]. [17] and [22].

At present, experim ental data are not available for comparison to the p red ic tions for

the ‘plate ' e lec trode geometry.

W ith the linear stability analysis accom plished, subsequent theoretical work fo

cussed on a w eakly nonlinear analysis near the onset of electroconvection.[23] The

system of equations of the theoretical m odel introduced in Ref. [21] w ere system

atically p ertu rb e d near the critical p aram eters (kc,7^c). The procedure used was a

standard m ultiple-scales perturbation th eo ry th a t allows a convenient sp littin g of the

fast and slow scales. A slowly-varying, real am plitude A (x .t) was used to describe

the envelope o f th e electroconvection p a tte rn . It was found that the am p litu d e a t

the lowest o rd er nonlinearity obeyed th e G inzburg-Landau Eqn. 1 .1 . W hereas £0 is

independent o f V . r 0 is not. The weakly nonlinear theory was developed for V — ex.

Of im portance are the calculated values for the ‘w ire’ case of £0 = 0.284. r 0 = 0.351

and <7o = 1.746. For comparison, £0 = 0.297, r 0 = 0.372 and go = 2.842 for th e ‘p la te ’

geometry. Since gQ > 0, the pitchfork b ifu rcation to electroconvection is forw ard or

supercritical. R ecall th a t in Section 1.2 it was em phasized that from curren t-vo ltage

observations th e onset of electroconvection in freely suspended fluid films is super

critical. This is in agreement with th e non linear theory.

The p red ic ted correlation length from the linear and nonlinear th e o ry for the

17


‘w ire’ case. £0 = 0.284, is about 20% smaller th an the experim ental value of ^ xpt =

0.36 ± 0.02 of Ref. [14]. This is fair, bu t not completely satisfactory, agreem ent.

Experim entally, determ ining the correlation length requires th e film to be convecting

a t some m easurable velocity. To arrive a t b o th kecxpt and , the experim ental

m easurem ents were made nondim ensional by dividing by the m easured film w idth d,

which was known to w ithin about 5%.[14]

Since experim ents in rectangular films have m aterial and geom etric param eters

such th a t V 2> 1, the V = oo value of g0 is used to com pare w ith experim ental m ea

surem ents of g0. T he agreem ent was unsatisfactory.[14] The quan tita tive agreem ent

between the theoretical values and experim ental values of r 0 were also unsatisfactory.

Unlike kc and £u. which only scale w ith d. the scaling of gQ and r 0 depends on the

poorly known conductivity. Also, both of the above m easurem ents involved films

moving a t substan tia l velocity above onset. Since the experim ents were conducted a t

atm ospheric pressure, it is likely th a t air drag effects were significant.

T he experim ents th a t are reported in this thesis have taken into account some of

the non-ideal features of the previous electroconvection experim ents. Velocity m ea

surem ents of the flow by means of tracer particles has been replaced by m easurem ents

of th e curren t through the film. T he annular films are enclosed in a reduced am bient

pressure environm ent so th a t air drag is negligible.

18


Bibliography

[1] J.P . Gollub and J.S. Langer, '‘P a tte rn formation in nonequilibrium physics.”

Rev. M od. Phys., 71. S396. (Centenary 1999)

[2] M. C. Cross and P. C. Hohenberg. "Pattern formation outside of equilibrium ,”

Rev. M od. Phys. 65. 851 (1993).

[3] A.C. Newell, T . Passot and J. Lega, "Order param eter equations for pa tte rn s.”

Ann. Rev. Fluid Mech., 25, 399 (1993).

[4] G. Ahlers. in Lectures in the Sciences o f Complexity, edited by D. Stein, Addison,

Reading, MA (1989), p. 175.

[5] A.A. Sonin, Freely suspended liquid crystalline film s , John W iley (1998).

[6 ] D. D ash and X.L. Wu, "A Shear-Induced Instability in Freely Suspended

Smectic-A Liquid C rystal Film s.” Phys. Rev. Lett. 79, 1483 (1997).

[7] Y. Couder, J.M . Chom az and M. Rabaud, “On the hydrodynam ics of soap films,”

Physica D 37. 384 (1989).

[8 ] X-l. Wu, B. M artin, H. Kellay, and W. I. Goldburg, “H ydrodynam ic Convection

in a Two-Dim ensional C ouette cell,” Phys. Rev. Lett. 75, 236 (1995), B. M artin

and X-l. Wu, “Double-Diffusive Convection in Freely Suspended Soap Film s,”

Phys. Rev. L ett. 80, 1892 (1998).

[9] M. Rivera, P. Vorobieff, and R. E. Ecke, “Turbulence in Flowing Soap Films:

Velocity, Vorticity, and Thickness Fields,” Phys. Rev. L ett. 81 , 1417 (1998).

19


[10 ] V.M. Zaitsev and M.I. Shliomis, ‘"Hydrodynamic fluctuations near the convection

threshold ,” Soviet Phys. JE T P , 32, 8 6 6 (1971).

[1 1 ] H.S. Greenside, “Spatiotem poral chaos in large systems: T he scaling of com plex

ity w ith size,” workshop of the C entre de Recherche en M athem atiques, (1995).

[12] S. Faetti, L. Fronzoni, and P. Rolla, ‘‘S tatic and dynam ic behavior of the vortex-

electrohydrodynam ic instability in freely suspended layers of nem atic liquid crys

ta l,” J . Chem . Phys. 79. 5054 (1983).

[13] S. VV. Morris, J. R. de Bruyn, and A. D. May, “Electroconvection and P a tte rn

Form ation in a Suspended Smectic Film .” Phys. Rev. L ett. 65. 2378 (1990).

“P atte rn s a t the onset of electroconvection in freely suspended smectic films,” J.

S ta t. Phys. 64 , 1025 (1991), "Velocity and current m easurem ents in electrocon-

vecting sm ectic films.” Phys. Rev. A 44, 8146 (1991).

[14] S. S. Mao, J. R. de Bruyn. and S. VV. Morris, “Electroconvection p a tte rn s in

sm ectic films a t and above onset.” Physica A 239. 189 (1997).

[15] S. S. Mao, J. R. de Bruyn. Z. A. Daya, and S. VV. Morris, “Boundary-induced

wavelength selection in a one-dimensional pattern-form ing system ,” Phys. Rev.

E 54. R1048 (1996).

[16] A. Becker, S. Ried, R. S tannarius, and H. Stegemeyer. "Electroconvection in

sm ectic C liquid crystal films visualized by optical anisotropy,” Europhys. L ett.

39 , 257 (1997).

[17] C. Langer and R. S tannarius, “Electroconvection in freely suspended sm ectic C

and sm ectic C* films,” Phys. Rev. E 58, 650, (1998).

[18] A .J. L eadbetter, J.C . Frost, J.P . G aughan, G.W. Gray, and A. Mosly, “T he

s tru c tu re of sm ectic A phases of com pounds with cyano end groups,” J . Phys.

(Paris) 40, 375 (1979).

20


[19] Large fluctuations of the current were observed in experim ents in annu lar geom

etry when incense particles were introduced on the film.

[20] S. VV. Morris. "Electroconvection in a freely suspended sm ectic film.” PhD . The

sis, unpublished (1991).

[21] Z. A. Daya, S. VV. Morris, and J. R. de Bruyn, "Electroconvection in a suspended

fluid film: A linear stability analysis.” Phys. Rev. E 55. 2682 (1997).

[22] Z. A. Daya, "Electroconvection in suspended fluid films.” MSc. Thesis, unpub

lished (1996).

[23] V. B. Deyirm enjian. Z. A. Daya, and S. VV. Morris, "Weakly nonlinear analysis

of electroconvection in a suspended fluid film.” Phys. Rev. E. 56. 1706 (1997).

[24] J.D. Jackson. Classical Electrodynamics. Wiley (1975).

[25] A. Castellanos. "Coulomb-driven Convection in Electrohydrodynam ics.” IEEE

Transactions on Electrical Insulation, 26. 1201 (1991).

[26] R.V. Polovin and V.P. Demutskii, Fundamentals o f Magnetohydrodynamics. Con

sultants Bureau. New York (1990).

21


C hapter 2

Experim ent

2.1 Introduction

T his C hap ter describes the experim ental setup and protocol. T he purpose is to

perform precise current-voltage m easurem ents of electroconvection in annular films

under im posed shear. In overall structure, th e experim ent consists of an annulus,

constructed ou t of circular electrodes, th a t is housed in a vacuum cham ber and is

controlled by two com puter interfaced devices. A schem atic of the overall design is

shown in Fig. 2 .1 . T he vacuum cham ber is a box of approxim ate dim ension 16 x 14 x 8

inches. It was constructed out of 3 /4 inch thick alum inum plates, equipped with four

feed-through ports and was evacuated by a ro tary vacuum pum p. A CCD colour

video cam era assembled w ith a microscope is used to view the film through an optical

window in the vacuum cham ber. A Keithley electrom eter controlled by a com puter

was used to ob ta in the current-voltage characteristic of the film. A high precision

s tepper m otor, triggered by the C om pum otorP lus drive was opera ted by the com puter

interfaced H ew lett-Packard frequency generator. T he m otor was used to apply a shear

to the film by ro ta ting th e inner edge of the annulus. A detailed description of th e

experim ental design follows in Section 2.2 while the experim ental m ethodology is

recounted in Section 2.3.

22


RS-232 Bus Text InserterIntel/QNX Color

M onitor

IEEE GPIB

HP frequency generator

Keithley Electrom eter

M icroscope

Source

Splitter

m s m s s fv m m m s

Electrodes

VacuumCham ber

Rotary Vacuum

Pump

Figure 2.1: A schem atic of the experim ental setup.

23


2.2 Experim ental Design

This Section describes the experim ental apparatus. Specific a tten tio n is given to the

experim ental design. A schem atic of the experim ent is provided in Fig. 2.2. T h e

annular film is housed in a vacuum chamber. T he vacuum cham ber has a glass

window in its upper surface and an O-ring feed-through in its lower surface. T h e

glass window perm its visual observation of the annulus while the O-ring feed-through

allows the ro ta tio n of the inner edge of the annulus. The ro ta tion is effected by a

stepper m otor. A n electrical feed-through perm its electrical signals to be transm itted

to and from the annulus and a slip ring assembly perm its conducting electrical signals

to the ro ta ting parts of the annulus. Also housed in the vacuum cham ber is the •film-

drawing’ assem bly - an appara tu s which facilitates the drawing of the annular film.

The annulus was constructed out of two stainless steel electrodes. The inner

electrode was a circular disk of diam eter 2 r ; mm. T he outer electrode was a larger

circular p la te of diam eter 9.00 cm with a central hole of d iam eter 2ra mm. T h e

outer electrode was 0.73 ± 0.01 mm thick. A diagram of the annulus is shown in

Fig. 2.3. W ith the exception of the experim ents reported in A ppendix D. several

inner electrodes w ith radii between = 3.60 ± 0.01 mm and r, = 5.26 ± 0.01 m m

were used. T h e radii of the ou ter electrodes were between ra = 5.57 ± 0 .0 1 mm and

r0 = 11.25 ± 0 .0 1 mm. W hen concentrically placed, the space between the inner

and outer electrodes defines an annulus of w idth d = ra — r, and radius ratio a =

r i/r0. E xperim ents were conducted a t six different radii ratios betw een a = 0.33 and

a = 0.80 by use of several pairs of inner and ou te r electrodes. T he edge of the o u te r

electrode had th e form of a wedge of angle 20 — 30°. The lower surface of the wedge

was polished. T h is was done by means of a soft drem el tool and a micropolishing

powder while the outer electrode was on a lathe; the lathe and th e dremel tool were

rotated in opposite senses. T he resulting outer electrode had a sharp edge which

was required so as to reduce the wetting layer of the film. B urrs on the u pper

surface of th e o u te r electrode were lapped away. W hile the sam e trea tm en t applies to

the inclined surface of the inner electrode, it is by com parison fairly com plicated in

24


Aluminum frameMicroscope eyepiece

White light source

Beam splitter

Glass Window Vacuum vessel

Film-drawing’ assembly

OuterElectrode

Electricalfeedthrough

W .V .W J/

Slip Ring ^Assembly

gears to rotary pump

Ring feedthrough

Stepper motorXY Translation Stage

Figure 2.2: A schem atic of the experim ent. W hile no t to scale, th is d iagram is indicative of th e rough proportions of the various elem ents. A few dim ensions are given to suggest th e size of the experim ent.

25


(a)

T ~

9.00cm

(b) r 0.73mm

Figure 2.3: G eom etry of the electrodes. T he electrodes were m ade of stainless steel and are shown here in top view (a) in cross section (b). T he surface of the inclined edges were polished.

construction. An assembled inner electrode is shown in Fig. 2.4a. It consists of four

separate cylindrical pieces th a t are put together along their com m on axis. T he inner

electrode, w ith dimensions, is shown in Fig. 2.4b. It is constructed out of stainless

steel and is in the form of a top hat. T he inner electrode fits on a stainless steel

shaft, shown w ith dim ensions in Fig. 2.4c. T he inner d iam eter of the inner electrode

is ju s t right to allow a tight fit on the neck of the shaft. I t is held in place by a

m inute screw. T he shaft has a set screw which is used as an electrical connection. An

im portan t consideration in designing the assembled inner electrode is the electrical

isolation of the inner electrode. This is achieved by m eans of an insulating sleeve

m ade ou t of G l0 (a fibre glass epoxy lam inate characterized by high s treng th and

good electrical insulation). The shaft makes a tight fit in to the sleeve as shown in

Fig. 2.4a. W hile this fit is tight, it does allow for ro tating th e shaft in the sleeve. This

freedom facilitates th e final ad justm ent to ensure th a t the inner electrode ro ta tes tru ly

abou t its axis. T he sleeve is press fit into the slip ring assem bly which is described

26


(a) (b) In

0.502'

0.249'

10 B.A. screw

Steep Counter Sink

0.250’

10.375" I L -9-I3QL - J

0 . 195"

(C) O<N

P

o<ncs©

£•■ji■Sia.

i

=i

A * O

♦ ^- - r - o

®lidd|

Msi

- 4 i

©CNI

____ t

Figure 2.4: A schem atic of the inner electrode. An assembled inner electrode (a) consists of the electrode (b), a shaft (c) an insulating sleeve and a sm ooth rod.

27


(a)O

Plastic StageRubber Coupling Stepper

MotorSpring

Razor Blade Translation Stage

(b)

o oo oPlastic Stage

Razor Blade

Figure 2.5: A schem atic of the ‘film-drawing’ assembly. T he razor blade is firmly held against the ou te r electrode by a stretched spring.

shown in Fig. 2.4a. The inner electrode was made to ro ta te about its axis by means

T he stainless rod provides a sm ooth surface for an O -ring vacuum seal. See Fig. 2.6

for a diagram . T he O-ring was regularly ‘m oistened’ w ith vacuum grease to reduce

wear of the rubber. To ensure sufficiently sm ooth ro ta tion , a high precision stepper

m otor was used. T he Compum o tor P lus stepper m otor was operated a t 25600 steps

per revolution.

T he ‘film -drawing’ assembly is shown in Figs. 2.5a and b. The principal compo

nent is a stainless steel razor blade which was inclined a t approxim ately 25° to the

electrodes, held ta u t against the electrodes by a tensioned spring and was operated

by a m otorized translation stage assembly. The step p er m otor could be operated

to push or pull the razor blade a t variable drawing ra te s between 2 — 30 mm /m in.

T he entire assembly, including the s tepper m otor, were inside the vacuum chamber

below. A sm ooth stainless steel rod is press fit into th e lower end of the sleeve as

of a gear a ttached to the rod which coupled to a s tepper m otor by m eans of a belt.

28


Triax Cable

EquivalentElectrometerCircuit

Slip Ring Assembly

Noise Shield Vacuum Chamber

O Ring Seal

Figure 2.6: A schem atic of the electrical design. The slip ring assembly facilitates the application of an electrical voltage to th e inner electrode while it rotates.

and could be operated via electrical feed-throughs in reduced am bient pressure. In

experim ents w ith the larger outer electrodes, it was necessary to remove the part of

the ‘film-drawing’ assembly shown in Fig. 2.5b from the vacuum cham ber. T his was

because, even a t its farthest, the razor blade was sufficiently close to th e film th a t

the electrical pertu rba tion it caused was not insignificant.

T he prim ary experim ental probe is th e measurement of th e current-voltage char

acteristics of the freely suspended annu lar fluid film. Since th e current transported

through the film is picoamperes in m agnitude, particular care has to be exercised to

avoid s tray currents. A schematic of th e electrical com ponent of the experim ent is

given in Fig. 2 .6 . T he inner electrode is electrically isolated from all o ther com po

nents of the appara tus by the aforem entioned insulating sleeve. The assembled inner

electrode is press fit into the slip rings, the area of contact being the insulating sleeve.

T he slip rings are an electromechanical Silver G raphite b rush device th a t facilitates

the transm ission of electrical signals to and from the ro ta tin g inner electrode. A

K eithley electrom eter is used both as a voltage source and a picoam m eter. T he ‘high’

of the variable dc voltage source of th e K eithley is connected to the inner electrode,

29


which is the only part of the entire apparatus th a t is not a t ground potential. The

outer annulus though at ground potential is electrically disconnected from ground by

teflon washers. T he rest of the appara tus is grounded. Electrical noise is reduced by

shielding the electrodes and m ost of the experim ental appendages in a large Faraday

cage which doubles as the vacuum chamber. A low noise triaxial lead was used to

collect the cu rren t from the ou ter electrode and was measured by th e Keithley elec

trom eter. A low noise triaxial feed-through was used to carry the signal out of the

vacuum cham ber.

The op tica l appendages to the experiment are simple and consist primarily of

a microscope and a colour CCD video camera. A Tungsten-H alogen lamp was the

light source. W hite light was d irected by a beam sp litte r into the vacuum chamber

through th e glass window in the lid. The image of the film under reflection was

viewed by th e CCD cam era a ttached to a microscope assembly and recorded by a

VCR. T he system magnification was between 5 — 15 x w ith the corresponding field of

view 58 — 2 0 mm . The purpose of the optical assembly is to allow one to glean the

thickness and uniform ity inform ation contained in th e interference colour of the film.

A final design consideration concerns the two-dimensionality of th e flow. Experi

ments in freely suspended films have principally em ployed soap so lu tions and liquid

crystals. It is well known th a t soap films and liquid crystal films in th e nem atic phase

are prone to thickness variations[l, 2]. Films of liquid crystals in the sm ectic phase are,

however, fundam entally two-dimensional and are resistan t to thickness variations[3].

Furtherm ore, unlike soap solutions, the liquid crystal films are not susceptible to evap

oration and can be m aintained in much lower am bient pressures. Since th is study calls

for a tw o-dim ensional isotropic fluid for experim ents a t reduced am bien t pressure, it

necessitated th e use of a smectic A liquid crystal. A successful can d id a te previously

established in earlier experim ents is smectic A octylcyanobiphenyl[3. 4].

30


2.3 Experim ental Protocol

T his Section recounts the m ethods employed in perform ing the experim ents. The

first step is to p repare the liquid crystal sam ple. The liquid crystal used in the ex

perim ents was octylcyanobiphenyl (8 CB). T he 8 CB liquid crystal has the smectic

A phase between 21°C and 33.5°C. Since the electrical conductivity of the smectic

A liquid crystal (8 CB) is due to several ionic impurities[5] of varied and unknown

concentrations, the na tu re of the ionic species can be controlled by doping 8 CB with

tetracyanoquinodim ethane (TCN Q ), so th a t the dom inant species contributing to

the electrical conductivity is the dopant. T he procedure for doping 8 CB was simple.

TC N Q was dissolved in acetonitrile (ACN) and added to the 8 CB sample. T he ACN

was evaporated in a vacuum oven while warm ing the m ixture so th a t the liquid crystal

was in its isotropic phase. T he liquid crystal samples used had dopant concentrations,

by weight of dopant to liquid crystal, of 7.62 x 10-5 . 1.11 x 10- 4 and 2.96 x 10-4 .

Experim ents w ith significantly higher or lower dopant concentrations have been less

reproducible due to non-ohmic effects. Samples w ith a high concentration of TCN Q

have, left to themselves, changed colour from an original off white to orange to green.

T he same has been noticed for a solution of TCN Q in ACN independent of concen

tra tion . T he du ra tion of tim e during which these changes occur is weeks to months.

T he samples were always an off white colour for the dopant concentrations used in

the experim ents.

T he experim ental appara tus was assembled w ith the exception of the electrodes.

T he inner and o u te r electrodes were cleaned, first w ith m ethanol then w ith de-ionized

w ater and dried. T h e teflon washers th a t electrically isolate the ou ter electrode were

also cleaned w ith m ethanol and de-ionized water. D irt, grease in particu lar, can

greatly effect the electrical conductivity and therefore th e function of electrical isola

tion th a t the teflon serves. T he assembled inner electrode less the top h a t electrode

was ad justed to ro ta te true to its axis to w ithin 0.002 inches. T he top h a t electrode

was screwed down to the shaft of the inner electrode and the ou ter electrode was, by

use of recently cleaned plastic screws, assembled. T he ou te r electrode can be moved

31


by an XY transla tion stage (see Fig. 2.2) and was adjusted so th a t the inner electrode

was concentrically placed.

T he ‘film -drawing’ assembly was then used to form th e liquid crystal film. T he

sharpened and polished edges of the inner and outer electrodes were first gently wet

w ith the liquid crystal sample. T hen the ‘film-drawing’ assembly is used to slowly drag

the stainless steel razor blade, also wetted w ith liquid crystal, across the annulus to

form the film. Usually the first few attem pts in drawing the film were unsuccessful,

w ith the film breaking during the drawing process. W hile drawing, a low power

microscope, colour CCD video cam era and a colour m onitor were used to view the

film. W hen viewed in reflected w hite light, the films display several interference

colours. By moving the razor blade back and forth during the drawing, a film w ith

uniform colour can be selected and drawn. Films of uniform colour could be obtained

when the razor blade was drawn slowly ~ 2 — 3 m m /inin. Once a film w ith a uniform

colour over its entire area is draw n, the razor blade of the ’film-drawing’ assembly

is draw n as far back from the edge of the annulus as is mechanically possible. T he

distance is often several film w idths. W hen such a distance could not be reached as

when using large ou ter electrodes, the razor blade was disassembled and removed from

the vacuum cham ber. This procedure ensures th a t electrical pertu rba tions because

of the m etal blade to the film are negligible.

T he thickness of the film was determ ined from the interference colour of the film

under reflected w hite light. This m ethod works well for in term ediate film thicknesses

where the interference colour can be unambiguously m atched to a colour chart. Very

th in films appear black and were not used, while thicker films are a pale off-white and

were also avoided in the experim ents. By using standard colourim etric tables and

procedures[6 , 7, 8 ] a colour-thickness chart has been m apped ou t for this experim ent.

T he calculation and the colour chart are given in A ppendix A. Since smectic films are

an integer num ber of smectic molecular layers, w ith each layer of sm ectic A 8CB[9]

being 3.16 nm thick, the film colour is used to identify th e film thickness m easured

in sm ectic layers. M ost of the experim ents were perform ed w ith films between 25

and 85 layers thick. Over m ost of the middle of this range, th e film thickness can

32


be determ ined to w ithin ± 2 layers, while close to the ends of th e range a more

conservative determ ination of w ithin ± 5 layers was used. During the course of an

experim ent, th e films were visually m onitored to ascertain th a t they rem ained uniform

in thickness to w ithin ± 1 layer. Experim ents in which the film spontaneously became

non-uniform were abandoned. A t times, films w ith several colours were draw n. These

films, left to themselves, anneal to a film with uniform thickness and hence colour.

The annealing process can be accelerated by electroconvecting and shearing the films.

T he plane of the film often deviates from the horizontal. T he tilt of the outer

electrode was adjusted until the film is horizontal. T he outer electrode was then

adjusted until the inner electrode was again centered. Excess liquid crystal on the

inner electrode was cleaned by a Q tip moistened w ith m ethanol. T he lid of the

vacuum cham ber was put in place and the air was then evacuated. W hile enclosed,

w hether in an evacuated environm ent or otherwise, the film left to itself is robust in

the sense th a t it rarely ruptures or changes thickness. T he air was evacuated slowly so

as to prevent vigorous air flows th a t may cause the film to rupture. Failing to remove

excess liquid crystal from the inner electrode would cause the trapped air to form a

bubble of liquid crystal th a t would alm ost certainly cause the film to rup tu re . T he air

surrounding the film was pum ped down to an am bient pressure of 0.1 — 1.0 torr. At

these pressures the m ean free paths of nitrogen and oxygen are between 0 .5 —0.05 mm,

com parable to the film w idth d. It is expected th a t the drag on the film due to the air

a t reduced pressure is negligible.[10] T he pressure was not actively controlled during

an experim ental run, allowing the vacuum to slowly decay. Between experim ental

runs, the vacuum cham ber was evacuated. As a result, the pressure never exceeded

5 to rr during the course of an experim ent.

U tilizing the 25600 steps per revolution stepper motor, the inner electrode was

ro ta ted abou t its axis a t angular frequencies up to u = rad /s . T he tem pera tu re was

not actively controlled; nevertheless, all experim ents were perform ed a t the am bient

room tem pera tu re of 23 ± 1°C, well below the sm ectic A-nematic tran sitio n a t 33.5°C

for undoped 8 CB. T he com puter interfaced K eithley electrom eter served as bo th a

voltage source and a picoam m eter.

33


16

14

12

ILio

:(b)

Increasing voltage Decreasing voltage

10 20 30Voltage (volts)

40

/a

a »a »

4 Increasing voltage * Decreasing voltage


80

Figure 2.7: Representative current-voltage data. C urrent-voltage characteristics for com plete experim ental runs a t rad ius ratio a = 0.467 in th e absence of shear (a) and w hen strongly sheared (b).

34


An exam ple of a current-voltage characteristic in a film w ithout shear is shown

in Fig. 2.7a. It consists of d a ta obtained for increm ental and decrem ental voltages.

T he current-voltage characteristic clearly shows two regions; one for voltages smaller

th an a critical voltage Vc and one for voltages greater th an Vc. T he critica l voltage

th a t separates these two regions is in the vicinity of th e kink or elbow in th e current-

voltage characteristic. In the regime V < Vc, the curren t is linearly dependent on the

voltage and therefore the film is ohmic. Experim ents in 8 CB with significantly higher

and lower concentrations of TCN Q show, a t least initially, non-ohmic current-voltage

characteristics even for V < Vc. Prior to obtaining the d a ta shown in this figure, a

prelim inary te st run was performed. The test run consisted of m easuring a current-

voltage characteristic of the film fairly rapidly and w ith large voltage-separation be

tween d a ta points. The purpose of the test run is to quickly estim ate Vc. A plot of

the d a ta from test run results in a by-eye determ ination of Vc denoted Vcapprax' . With

this approxim ation, a current-voltage characteristic is obtained as follows.

A reduced control param eter e is defined as

y a p p ro x .

C urrent-voltage d a ta is then obtained by se tting the voltage as a function of the

reduced control param eter e given by inverting Eqn. 2.1

K(e) = v T T e Vcapprox- . (2.2)

Beginning a t e = —1 i.e. V = 0 , the applied voltage on the inner electrode is

increm ented. T he voltage increm ents are effected by increm enting the reduced control

param eter e. Two constan t increm ents axe used e3tep and (large step- Between e = - 1

and typically ( = —0.25, the increm ents were in (large step■ Between e = —0.25 and

a m axim um value of e usually between 0.25 and 2 .0 , th e increm ents were in (3tep. In

these experim ents 0.001 < eJtep < 0.0125 with (targe step between 6 and 24 tim es larger.

As a result the voltage increm ents are not constant and get progressively smaller as

35


V —>■ Vcapprox-. The increm ents are typically less th an 0.1 volts when V ~ Vcapprox\

A fter each voltage increm ent the film was allowed to relax for between 10 — 18

seconds. T he lower bound ensures th a t the capacitive transien ts have sufficiently

decayed. This relaxation tim e £ is not constant a t each applied voltage bu t gets longer

th e closer the V is to V/cappror' . For the d a ta obtained w ith increm ental voltages the

relaxation time when longer than 10 seconds, was as a function of e, determ ined from

at)=ceil(r r ^ i) - (23)

where ceil(x) is the sm allest integer not sm aller than x. This technique allows for

longer relaxation tim es closer to the critical voltage where critical slowing down is

expected. The relaxation tim e is a t its m axim um close to e = 0 or equivalently when

V = \£,‘W oz . Relaxation times longer th an about 18 seconds per d a ta point were not

feasible due to the drift of the electrical conductivity, which will be discussed later.

A fter the relaxation period, between 100 — 200 m easurem ents of the curren t, each

separated by 25 milliseconds, were averaged. The standard deviation of the m easure

m ents or 1% of the average, whichever was greater, was taken as the error in the

m easurem ent. The reading error for m easurem ents in the picoam m eter range is 1%.

T he voltage was increm ented up to a predeterm ined m axim um e. T he applied voltage

was then offset by half an incremental s tep and then decrem ented. The decrem ents

were of size estep until typically e = —0.3, bu t certainly e < —0.25, so th a t large hys

teresis in the critical voltage may be detected. In Fig. 2.7b is shown a representative

current-voltage characteristic in a film under shear. Note the hysteresis. Since the

test run only provides an approxim ation to the critical voltage for the increm ental

rim , the decrem ental run m ust explore in sm all decrem ents a larger portion of regime

for e < 0 so as to cap tu re the hysteresis. Between e = —0.3 and e = — 1 , th e decre

m ent was e/arge step- T he relaxation tim e £ a t each voltage on the decrem ental run

was, when greater th a n 10 seconds, determ ined from determ ined from

<(£)=ceil( r r w ) ' (2-4)

36


T he difference between the functional forms in Eqns. 2.3 and 2.4 is m otivated by the

need to have long relaxation times a t larger |e| on the decrem ental run so as to detect

any large hysteresis effects in critical voltage. W hen the film was sheared by ro ta tion

of the inner electrode, it was allowed a t least 30 seconds after a change of shear ra te

to a tta in a steady s ta te before the current voltage characteristics, as described earlier,

were obtained.

T he ohm ic portions of the current-voltage characteristics of the increm ental and

decrem ental runs in Figs. 2.7a and b do not coincide. Since the film 's geom etrical

properties are unaltered during the course of the run, it is the electrical p roperties th a t

have changed. It is well known th a t liquid crystals degrade upon dc excita tion [ll].

The changes in the electrical conductivity are probably due to electrochem ical reac

tions w ith the electrodes. The presence of suitable dopants is known to arrest the

degradation process, nonetheless the electrochemical reactions w ith the electrodes

slowly change the film conductivity. The drift in the conductivity was usually ~ 2%

but ranged up to 10% during the course of an experim ental run of 30 — 120 m inutes

duration. In Fig. 2 .8 are p lotted the current as a function of time a t two voltages, one

below the critical voltage and one above the critical voltage. The tim e dependence of

the drift cannot be described simply. It is clearly dependent on the applied voltage

and on th e s ta te of flow in the film. Further, from qualitative observations it has been

noticed th a t the drift depends on the shear, the am bient pressure and even on the

physical electrode. Different electrodes, even though they are of th e sam e m aterial,

had different rates of drift. Perhaps, this was because of the different circumferences

th a t were in contact w ith the liquid crystal. Films a t atm ospheric pressure had less

drift th a n those a t reduced am bient pressure.

T he d rift of the electrical conductivity which is a non-ideal featu re of this ex

perim ent has two im portan t consequences. F irst, if the dimensionless param eters of

the film are to be approxim ately constant over the course of an experim ent, then

the am ount of drift should be minimal. T his means th a t the experim ent has to be

conducted as quickly as possible, hence the relaxation times were never in excess of

18 seconds. Second, the unavoidable drift during the rim has to be accounted for in

37


13.4

13.22.8

O 2.712.8

12.6

12.4

12.2

2.4

200 3001000

<u>oX>3

£3u

time (minutes)

Figure 2.8: T he drift in the current. P lot of current versus tim e for a uniform film w ith a = 0.467 when subjected to fixed voltages Vx < Vc and V2 > Vc.

some reasonable m anner so as to collectively understand d a ta from a m ultitude of

films of different geom etries and under various shears. This has resulted in a stra igh t

forward bu t tedious d a ta analysis procedure which is introduced in Section 4.2 and

discussed in detail in A ppendix E. W hile the drift in the conductivity is inconvenient,

it can be corrected for in th e data. Surprisingly however, the drift in the conductivity

facilitates th e exploration of a much broader range of the param eter space of the

experim ent. O ne of th e param eters introduced earlier was the P randtl-like param eter

V . T he drift in the conductivity makes accessible a wide range of V w ithout having

to draw a film of different thickness or w ith a different dopan t concentration.

A large num ber of experim ents had initially been perform ed in an environm ent a t

atm ospheric pressure and room hum idity. T he protocol, though slightly different from

th a t discussed earlier, was similar. T he results from these experim ents are presented

38


and discussed in Appendix D. A sm all num ber of experiments were performed on

films which h ad a nonuniformity in thickness of about ± 2 smectic layers. Most often

the nonuniform films tha t were used had two thicknesses and in appearance they

have two colours. Current-voltage m easurem ents were not obtained from these films

but they were instead used to visualize the flow pattern . A few experim ents were

performed in annular films in an eccentric or off-centered geometry. This geometry

was accessible by moving the outer electrode so th a t the inner electrode was no

longer concentrically placed. T he procedure was sim ilar to th a t described above and

is further discussed in Appendix F, Section F .l . T he extensive experim ental results

of electroconvection with and w ithout C ouette shear in a reduced am bient pressure

environment are presented and in terpreted in C hap ter 4.

39


B ibliography

[1] Y. Couder, J.M . Chom az and M. R abaud, "On the hydrodynam ics of soap films,”

Physica D 37. 384 (1989).

[2] S. Faetti, L. Fronzoni, and P. Rolla, "S tatic and dynam ic behavior of the vortex-


ta l,” J. Chem. Phys. 79. 5054 (1983).

[3] S. W . Morris. J. R. de Bruyn. and A. D. May, "Electroconvection and P atte rn

Form ation in a Suspended Smectic F ilm ,” Phys. Rev. Lett. 65. 2378 (1990),

"P atte rns a t the onset of electroconvection in freely suspended smectic films.” J.

S tat. Phys. 64, 1025 (1991), “Velocity and current m easurem ents in electrocon-

vecting smectic films.” Phys.Rev. A 44, 8146 (1991).

[4] S. S. Mao. J . R. de Bruyn, Z. A. Daya, and S. W. Morris. “B oundary-induced

wavelength selection in a one-dimensional pattern-form ing system ,” Phys. Rev.

E 54, R1048 (1996), S. S. Mao, J. R. de Bruyn, and S. VV. Morris, “Electro

convection patterns in smectic films a t and above onset,” Physica A 239. 189

(1997).

[5] P. G. de Gennes and J. P rost, The Physics o f Liquid Crystals 2 nd ed., C larendon,

Oxford (1993).

[6 ] G. W yszecki, “Colorimetry,” in Handbook o f Optics, McGraw-Hill (1978).

[7] A. Nemcsics, Colour D ynam ics, Ellis Horwood (1993).

40


[8 ] E .B . Sirota, P.S. Pershan, L.B. Sorenson and J. Collett. "X-ray and optical

stud ies of the thickness dependence of the phase diagram of liquid-crystal films,”

Phys. Rev. A, 36, 2890-2901 (1987).

[9] A .J . L eadbetter, J.C . Frost, J.P . G aughan, G .W . Gray, and A. Mosly, "The

s tru c tu re of sm ectic A phases of com pounds w ith cyano end groups.” J. Phys.

(Paris) 40. 375 (1979).

[10] D. D ash and X.L. Wu, ,lA Shear-Induced Instability in Freely Suspended

Smectic-A Liquid C rystal Film s,” Phys. Rev. L ett. 79. 1483 (1997).

[11] S. B arret, F. G aspard. R. Herino, and F. M ondon, "Dynamic scattering in ne

m atic liquid crystals under dc conditions. 1. Basic electrochem ical analysis.” J.

Appl. Phys. 47. 2375 (1976).

41


C hapter 3

T heory

3.1 Introduction

T his C hapter provides a detailed description of the model tha t has been developed

to explain the phenom enon of electroconvection in freely suspended annular fluid

films under Couette shear, [l] A schematic of the geom etry is shown in Fig. 1 .1 . In

th is geometry, the film occupies the annular region r; < r < ra between th e inner

and outer electrodes. In th e experiments, the film has a thickness ,s which is much

sm aller than the film w idth d = r0 — ri: such th a t the ratio s /d ~ 1 0 -4 . In the theory,

justifiably, the film is trea ted as a two-dimensional sheet which lies in the plane z = 0 .

T he inner electrode, which occupies the circular region 0 < r < r,, z = 0 is held a t a

variable potential V while the outer electrode which spans the region r > ra, z = 0

is held a t ground or zero poten tial. In general bo th the inner and ou ter electrodes

can be allowed to ro tate and from a theoretical perspective, it is no more costly than

sim ply allowing for the ro ta tio n of a single electrode. However, as will be explained in

Section 3.3, it is necessary to consider the ro ta tion of only a single electrode. Hence

it is only the inner electrode th a t will be allowed to ro ta te about the z axis.

T he instability th a t leads to electroconvection can be physically understood by

a trad itional ‘exchange of parcels![2] argum ent. Consider figure 3.1. For th e present

the film is not sheared. Tw o elements of fluid of equal area (also of equal m ass since

the fluid density is constant) are centered abou t radial positions r x and r 2, an d are of

42


Figure 3.1: A diagram to elucidate the electroconvectiou instability. T he fluid elements shown have the same area. The arrows indicate the m agnitude and direction of the electric field in the plane of the film.

equal angular ex ten t d6 and o f radial dimensions d r x and dr2. T he dimensions and

hence the elem ents or parcels are infinitesimal w ith area dA = r^dr^dQ = rodr^dO.

Since an electrical potential V is applied to r < r, and the region r > r a is held a t zero

potential, it shall a t this point be assumed th a t th e potential decreases monotonically

from V to zero as r increases from to r„. T h a t this is so will be dem onstrated in

Section 3.3. W ithou t elaborating on the origin and the radial dependence of the charge

density, it shall be assumed and proven later (Section 3.3) th a t th e charge density

is a m onotonically decreasing function for r* < r < ra. In calculating the energy of

the configuration illustrated in Fig. 3.1, only th e electrical energy of assembling the

charge d istribu tion in the fluid elements before and after the exchange is considered.

Let q(r) be the charge density per unit area and 'P(r) be the electrical potential.

Because the parcels are infinitesimal, the energy before parcel exchange is

Ebefore = [ g f a W r ! ) + g (r2 ) ^ ( r 2)]dA . (3.1)

43


T h e energy after exchanging the parcels a t r! and r2 in tact w ith their charge densities,

i.e. after the physical exchange of th e fluid in the two parcels, is

EaJ ter + q t o W i r i f l d A . (3.2)

T he deformation required to place th e parcel a t into the void left by th e parcel at

r 2 and vice versa is of no consequence. The change in th e energy of the configuration

due to the exchange, given by A E = E af ter - Ebefare is

A £ = - g (r2)]['I'(r2) - ^ ( r ^ d A . (3.3)

Based on the assum ptions made earlier, g (rt ) — q{r2) > 0 and ^ ( r 2) — ^ ( r j < 0

imply that A E < 0. i.e. the system is potentially unstable. This c riterion can be

s ta ted somewhat more succinctly as E • Vq < 0 , where E is the electric field. In

the presence of a shear nothing changes, except th a t it is necessary to account for

the additional cost in energy to decelerate one of the parcels and to accelerate the

o ther when m aking the parcel exchange argument. W hen A E is sufficiently negative

th a t it exceeds th e energy cost to accelerate and decelerate parcels then the system

is potentially unstab le . The preceding argum ents are placed on firmer footing in the

remainder of th is C hapter.

Section 3.2 describes the physical model, its governing equations and th e approx

imations th a t are inherent in it. Section 3.3 presents the solution for th e base state,

th e state of the system when it is no t convecting. Following this, a linear stabil

ity analysis of th e system about the base state is presented in Section 3.4. In this

Section are discussed the many predictions of linear theory. The assum ptions of the

theoretical m odel Eire, in Section 3.5, contrasted with the experim ental realities. Fi

nally, Section 3.6 presents simple sym m etry arguments for the form of th e appropriate

am plitude equation th a t describes th e weakly nonlinear regime in this system .

44


3.2 The Governing Equations

T his Section presents the physical model th a t is used to describe electroconvection in a

suspended th in film. T he analysis here is sim ilar to th a t of Ref. [4]. Unless otherw ise

s ta ted , all operators, field variables, and m aterial param eters are two-dim ensional

(2D) quantities. T he film is trea ted as a 2D conducting fluid in th e z = 0 plane, w ith

areal m ass density p, molecular viscosity q, and conductivity a. T he fluid is assumed

incom pressible so th a t the 2D velocity field, u , is divergence free,

V • u = 0 . (3.4)

T he Navier-Stokes equation w ith an electrical body force,

<9u . _ 'L- + (u . V )u = — V P + 17V 2u + r/E . (3.5)

governs the fluid flow, where V , P , q, and E are the 2 D gradient operator, pressure

field, surface charge density, and electric field in the film plane, respectively. The term

pE is the electric force acting on the surface charge density. T he charge continuity

equation

= - V - (gu + ffE), (3.6)

takes into account the convective and conductive curren t densities. <711 and crE re

spectively.

Subscrip t three will be used to denote three-dim ensional (3D) differential opera

tors, m aterial param eters and field variables. T he 3D electric po ten tia l ^ 3 is governed

by the 3D Laplace equation,

V * * 3 = 0 , (3.7)

w here V 3 is the 3D gradient operator. The coupling of w ith the 2 D charge density q

is specified by requiring ^ 3 to satisfy certain boundary conditions on the 2 = 0 plane.

T h e surface charge density q derives from the discontinuity in the 2 —derivative of $ 3

across th e two surfaces of th e film:

45


5 ^ 3q = —e0

— — 2 e0

d z

0*3

^ 0 * 3 + €0~^—

,=0+ d z

dz

z=0-

(3.8): = 0+

where e0 is the perm itivity of free space. If q is known. Eqns. 3.8 co n stitu te Neumann

conditions on $ 3 011 the film, while Dirichlet conditions, described below, hold on the

electrodes. If instead Dirichlet conditions are specified on the film, Eqns. 3.8 can be

used to determ ine q.

T he 2D and 3D potentials are related via * = ^ 3 |i=o- Equations 3.8 relate the

surface charge density to the discontinuity in the z —component of th e 3D electric

field E 3 = — V 3 'I' 3 across the film plane. On the o th e r hand, the x and y com ponents

of E 3 which form the 2D electric field E = - V 'l ', are continuous across th e film. This

continuity is required by the usual m atching conditions for electric fields across the

surfaces of dielectrics. Note th a t it is the 2D quan tity E and not E 3 th a t appears in

Eqns 3.5 and 3.6. One cannot sim ply use a Maxwell equation to elim inate the charge

density in favor of the field because the 2D quan tities in question are confined to a

plane em bedded in a 3D. otherw ise empty, space an d in general V • E ^ q/eo-

Equations 3.4 - 3.8. together w ith the appropria te boundary conditions on the

electrodes, m odel the system. T he model assumes the electrohydrodynam ic limit

where m agnetic fields and the resu ltan t Lorentz forces are negligible. O ne can also

show th a t dielectric polarization effects are negligible in the limit of a th in film.[4. 5]

In the subsequent analysis, the stream function 0 is defined by

u = V x 0 , (3.9)

where 0 = 0 z. Using Eqn. 3.9, E = —V * and elim inating the pressure field by

applying the curl operator, Eqns. 3.5 and 3.6 reduce to

L S + ‘V x ^ - V (V x V x 0 ) — tjV2(V x V x 0 ) + (Vg x V 'l/) = 0(3.10)

46


These equations are rendered dimensionless by rescaling th e length, time, and

electric po ten tia l by d , e0d / a , and V. respectively, where d and V are the cross-film

width and po ten tia l difference. It follows th a t th e stream function and charge density

are nondim ensionalized by a d /e 0 and e0 Vjd. Applying this rescaling to Eqns. 3.10,

3.11, 3.7, and 3.8 gives

V 2 - —— V d t

(V x V x x V</) = ^ ( V x ^ v ] ( V x V x ^ ) , (3.12)

^ + (V x 0 ) • V q - V 2'!' = 0 . (3.13)at

<7

where the dimensionless param eters

O r r '7 *7 r / * ?

n = m i l =an a-i q3s2

are antilogous to the Rayleigh and P randtl num bers in the Rayleigh-Benard problem .

1Z will be known as the Rayleigh-like or control param eter. V will be referred to as

the Prandtl-like param eter. In these param eters, s is the thickness of the film. In

this 2D trea tm en t it is assum ed th a t K i T he 2D m aterial param eters are rela ted

to their three-dim ensional counterparts by a = cr3s, q = '13s - and p = p^s. T he

control param eter TZ is proportional to the square of the applied voltage difference,

but independent of the film w idth d. V is the ratio of the charge relaxation tim e

scale in a film e0d / a 3s to th e viscous relaxation tim e scale p^dr/q^. T hat these are

the relevant tim e scales can be seen by considering separately charge relaxation in

the familiar charge continuity equation restric ted to conduction in 2 D and viscous

= 0 ,

= - 0 .. chi' 3

O Z c = 0 +

(3.14)

(3.15)

and V = e°ri e0'?3

pad p^a^sd(3.16)

4 7


relaxation in the Navier-Stokes equation. Since V appears in Eqn. 3.12 as 1 j V , it

should be clear th a t the dependence on th e Prandti-like param eter grows w eaker as

V increases and for all cases V ~ 10 is considered large.

Equations 3.12 - 3.15. together w ith appropriate boundary conditions, describe

electroconvection in a th in conducting film suspended in otherwise em pty space, for

any 2 D arrangem ent of the film and electrodes.

3.3 The Base State

This Section applies th e governing equations introduced in Section 3.2 to the case of

an annular film. These have been solved for the case of a general Couette shear flow

which forms the potentially unstable base sta te .

The film is suspended between two circular electrodes which cover the rem ainder

of the z = 0 plane as shown in Fig. 1.1. Cylindrical coordinates (r, 0, z) are employed.

The inner electrode has a radius r, and is a t potential 1 in dimensionless units. T he

outer electrode, which occupies the 2 = 0 plane for r > r0, is a t zero potential. T he

cross-film w idth is rQ — r, = 1 in dimensionless units and the radius ratio, a is defined

as

a = r t / r 0, (3.17)

so th a tq 1

rt = ------- , ra = -------- . (3.18)1 — a 1 — a

R otation of the inner electrode abou t the central r = 0 axis produces a C ouette

shear in the base s ta te . The base sta te variables are denoted by the superscrip t zero.

Under shear, the rad ia l derivative of base s ta te stream function is given by

a^ l0,(r) = (r - ' (3-19>

where is the dimensionless angular ro ta tio n frequency of the inner electrode. If the

fluid is not sheared, th e base s ta te velocity field is zero and 0 (o)(r) = 0 . T he s tren g th

48


of the shear is described by a Reynolds number, w ith the velocity determ ined by the

motion of th e inner edge and the length by the film width,

U e = r- £ , (3.20)

where V is th e Prandtl-like num ber given by Eqn. 3.16.

T here is no loss of generality by trea ting only ro tations of the inner electrode. Since

the system is 2D. one can always reduce independent rotations of bo th electrodes to

this case by transform ing to a ro ta ting reference frame in which the ou ter edge is

stationary. T h e transform ation introduces a Coriolis term into Eqn. 3.5, which can

simply be absorbed into the pressure.[6 ] As a result there me im portan t differences

in the stab ility of 2 D and 3D system s, and in particu lar for rigid ro ta tion which is

discussed fu rth e r in Sections 3.4 and 4.8.

The base s ta te potential and charge density q(U)(r) are independent of

the base s ta te shear flow. T hey are determ ined by the electrostatic boundary value

problem given by Eqns. 3.14 and 3.15, with Dirichlet boundary conditions on the

z = 0 plane

#<°>(r ) = ^ o)(r , 0 ) = <

1 for 0 < r < r,

j^ y [Z n (l - a) + ln(r)J for r, < r < ra (3.21)

0 for r > r0 .

The boundary condition for < r < ra is found by treating the annular film as

a 2 D ohmic conductor subject to a dimensionless potential of 1 a t th e inner edge

and 0 a t th e outer, and requiring the continuity of the 2D current density. The

logarithm ic form follows from th e cylindrical geometry. It is clear from Eqn. 3.21

th a t th e po ten tia l decreases m ontonically from r, to ra as was assum ed in Section 3.1.

T he Laplace Eqn. 3.14 for th e potential in the half-space z > 0 is solved by the

ansatz

z ) = r dk e - hzJ0{kr )A {k ) , (3.22)Jo

49


where JQ is the zeroth order Bessel function. Inversion of the above equation results

m

A{k) = k I™dr 0)JQ{kr) . (3.23)Jo

Hence, th e base s ta te surface charge density is given by

qM{r) = 2 H d k k2Ja(kr) f ° ° d ( { C . Q ) M K )Jo Jo

(3.24)

where C is a dum m y integration variable. Evaluation of the integrals[7, 8 ]. results in

<7(<V ) = p ~In a

l F ( l i . 1 - J±Vr \ 2 ! 2 ’ r 2 / r, V2 ’ 2 ’ ' r ? )

(3.25)

where F is the hypergeom etric function T his function is p lo tted in Fig. 3.2 for two

values of a. As a -> 1, q ^ approaches the base s ta te charge density for the laterally

unbounded rectangular film introduced in Section 1.3. which is odd-sym m etric about

the m idline of the film. This sym m etry helped simplify the analysis in th a t case.[4]

However, for 0 < a < 1 , the annular base s ta te charge density <7(0) is neither even nor

odd abou t the midline, so the analysis here is m ore com plicated. T his deviation from

odd sym m etry is larger for sm aller a.

T he surface charge density shown in Fig. 3.2 is 'inverted ' in the sense th a t the

positive charges lie close to the positive, inner electrode and negative charges are

near th e ou ter electrode. This unstable surface charge configuration gives rise to an

electroconvective instability, much like the unstab ly stratified density configuration

in Rayleigh-Benard convection.[4] The divergences of a t the edges of the film

are a consequence of the idealized geometry in which the electrodes have zero thick

ness. T h is idealization leads to the boundary conditions in Eqn. 3.21. whose radial

derivative is discontinuous a t r = r, and r = ra. This is reflected in Eqn. 3.25 where

1 ; l ) is indeterm inate i.e. when r = r , and r = ra, so th a t q ^ diverges a t the

edges of the film.

A h in t a t the relative s tab ility of films of different a lies in the com parative shape

and m agnitude of the surface charge density. I t may be suggestive b u t perhaps not

50


r (a = 0.8)4 4.25 4.5 4.75 5

4

3

2

a = 0.5I

a = 0.8

0

-I

21.25 1.5 1.75 9

r (a = 0.5)

Figure 3.2: T he base s ta te surface charge density. q ^ ( r ) versus r at a = 0.5 (lower scale) and a = 0 .8 .(upper scale)

com pletely convincing from Fig. 3.2 th a t a = 0.50 is the more unstable of the two.

by virtue of its larger surface charge density. The picture becomes a little clearer

if instead the quan tity p lo tted is E(0) • V q(0). which was shown in Section 3.1 to

have some bearing on the potentiality of an instability. The electric field is given by

E(°)(r ) = _ V ^ (0)(r) = — l / r / n ( a ) r for < r < r0. Fig. 3.3 shows the quan tity

E (0)dqw / d r for the two a and it is a little clearer th a t a = 0.50 is the more unstable

of the two. Such a conclusion is guessed a t by com paring the areas betw een the

curves and the E ^ d q ^ / d r = 0 axis. T he larger the area the more unstable; bu t one

m ust exercise caution in such an argum ent. Com parative stab ility based on ‘energy’

argum ents of th is sort can be inaccurate if there are com peting or degenerate stable

configurations. In such cases one m ust pay atten tion to th e s ta te to which instability

tends. A conclusion of th is so rt of com parative stability was easily ob ta inab le for a

51


r (a = 0.80)4.2 4.4 4.6 4.8

Urn

' u '

©^ c rT3

'w 'oT5j

3

-4

5

6

a = 0.80*a = 0.507

8

-9

101.4 1.6 1.81

r (a = 0.50)

Figure 3.3: The relative stability a t a = 0.5 and a = 0 .8 . P lo tted is E t0 )( r ) ^ ' - versus r a t a = 0.5 (lower scale) and a = 0.8.(upper scale)

laterally unbounded geometry. In th a t case, the relative stab ility of films in ‘w ire’ or

■plate' geom etry was quite clear from comparing the shapes of d q ^ / d r for the two

cases. [5]

T he base sta te is fully described by the functional forms for the stream function

po ten tial and surface charge density given by Eqns. 3.19, 3.21 and 3.25

respectively. It is im portan t to observe th a t the base s ta te is always axisym metric.

3.4 Linear Stability Analysis

In th is Section the axisym m etric base s ta te is tested for stab ility to non-axisym m etric

pertu rba tions. T he pertu rbed quantities will be denoted by the superscript one, and

axe defined by

52


0(r,0) = 0 (o)(r) + <p(1)(r,8),

q{r,8) = qW { r ) + q[1]{ r ,8 ) ,

#(r,0) = ^ (0)(r) + ^ (l)(r,0), (3.26)

¥ 3 M , 2 ) = ^ 0)( r , 2 ) + ^ l)( r ,f l ,2 ) .

Substitu tion of the pertu rbed field variables into Eqns. 3.12 - 3.15 and retaining only

the term s which are linear in the pertu rbed quantities yields

y 2 _ i ( £ _ \ i a (,, _V \ d t r d r 88 J r \ d r 88

1 dd>W 8 ( d2 _ 1£ \ 0(O)< r dr ) ’

d q W d q w ' ~08 8r~

r V 88 drd<?(l) 1 ( 8<t>W dqw _ dqw 8<f)m \ _ _ 2 . ( l )

8 t r \ 08 8 r 08 d r )

7(l) = - 2dz

(3.27)

(3.28)

(3.29)

(3.30)2= 0 +

where

d 2 I d i d 2 V ‘ = 701 + Z~ZZ +d r 2 r d r r 2 88-

and V I = V 2 +d z 2 '

T he variables 0 (1). 'Id1). a n d s a t i s f y th e following boundary conditions:

0 (1)(ri,0) = dr4>w (ri, 8) = 0 (1)(ro,0) = <9r0 (1)(ro, 0) =

& l){ri, 8 ) = * W ( r 0,d) =

* {3l)( r , 8 , z ) - +

0 , (3.31)

0 , (3.32)

0 for z —► ± oo .(3.33)

53


E quation 3.31 is a consequence of rigid boundary conditions on the velocity of the

fluid. T he Dirichlet boundary conditions for the pertu rbed po ten tia l on the 3 = 0

plane are

0 for 0 < r < ri

'Id1' ( r . 8) for i~i < r < ra (3.34)

0 for r > ra .

T he pertu rba tions can be conveniently decom posed into p roducts of axisym m etric

and non-axisym m etric term s of the following form,

1 0 (1)( r . 0 ) N

qw {r,0)

4'(1)(r \0)

0m (0

<lm{r)

4'm{r)

\ 4?3m(r,z) )

(3.35)

w here the azim uthal mode num ber m is an integer which corresponds to the number

of vortex pairs in the pattern . The functions 0 m. and satisfy the same

boundary conditions as the perturbations. Eqns. 3.31 - 3.34. and are axisym metric.

T he growth ra te 7 may be complex.

S ubstitu tion of Eqn. 3.35 into Eqns. 3.27 - 3.30 gives

f n m - \ - , I f im D 0 (o)\ / m - \ ,( B . B - - ) - - ( 7 - — — j [ D . D - - ) v

irriR.{ [ D ¥ Q))qm - (£>g(0,) ¥ m) = ^ 2 . D(D.D<j>W ) ,

/ rn2\ T ( im D qw \ . / zm£)0 (o) \\ D * D f j j ^ ~ J(pm - ( 7 “ J q m — 0 •

/ m 2 d 2 \^ ^3m = 0 ?

qm = -2 -d * 3m

dz

(3.36)

(3.37)

(3.38)

(3.39): = 0+

w here D = dr and D , = D + 1 / r . For the rem ainder of th is thesis w ith the exception

o f A ppendix F, Section F.2, the discussion will be restricted to a base s ta te flow which

is e ither quiescent or C ouette. In these cases D ( D mI?0(o)) = 0 and the right hand

54


side of Eqn. 3.36 is identically zero.[9]

In the lim it a —> 1 , and for zero shear 0 (o) = 0. Eqns. 3.36 - 3.39 reduce to the

linear s tab ility equations for electroconvection in a laterally unbounded s tr ip .[4] The

narrow gap or a —► 1 limit is im plem ented by the transform ation D . D —► D 2 =

d 2/ d y - and rn /r —> k in Eqns. 3.36 - 3.39. k is the wavenumber of the unstab le mode

th a t describes th e one dim ensional array of electroconvection vortices.

The com plex grow th exponent 7 is w ritten as Y -F *7 *. In order to find the

conditions for m arginal stability, the real part vanishes i.e. Y = 0 . T he task is

then to solve Eqns. 3.36 - 3.39 for a given a, V and TZe, by determ ining consistent

values of V, and 7 * for each m. The rate of azim uthal travel or angular velocity of

each m arginally stab le mode around the annulus is 7 1 / m. The m arginal stability

boundary, which is defined only a t discrete m. has a minimum a t the critical values

m c, TZC, while the critical mode travels at 7 */m c.

The axisym m etric term s of the perturbations in Eqn. 3.35 are expanded as follows:

< M r ) = 5 I^ n 0 m ;n ( r ) ! (3.40)n

*m (r) = J ^ A n'Zrn;n{r)< (3.41)n

*3m(r,z) = Y i A n<b3m,n( r , z ) . (3.42)n

<7m(r) = ^ 2 A nQ m-n(r) • (3.43)rt

where the A n are am plitudes. T he expansion eigenfunctions 0 m;„(r). (r)? and

$ 3ra;n( r , : ) satisfy the following boundary conditions:

0 m ;n (r i) = 3 r 0 m ;n( r i) = 0 m;n( r o) = dT<f>m.n(r0) = 0 , (3 .4 4 )

= * m;„ ( ro ) = 0 , (3 .45 )

^3 m ;n (r , 2 ) 0 for 2 ->• ± 0 0 , (3 .4 6 )

55


0 for 0 < r < r,

^ 3m;n(^~ = 0 ) = < for r, < r < ra • (3.47)

0 for r > r0

T h e functions 0 m;n(r), which satisfy th e rigid boundary conditions given in Eqn. 3.44.

can be identified w ith the C handrasekhar cylinder functions[2 ],

T h e boundary conditions Eqn. 3.45. imply tha t the 2 D poten tial expansion function

'Prn;n can be further expanded in a series of functions of the form.

T h e functions Crn;n and tpm;p, along w ith their associated constants 3mn, Dmn, Cmn,

D mn, Xmpi and brnp are described in detail in Appendix B.

T he m ain barrier to solving Eqns. 3.36 - 3.39 lies in th e difficult nonlocal coupling

betw een and qm in Eqns. 3.36 and 3.37 which is required by the 3D electrostatic

in which the 3D Eqns. 3.38 and 3.39 are replaced by th e following simple 2 D closure

relation:

In the above expression. f m is a closure factor which is independent of r. and is to

be specified. As shown below, a consequence of this approxim ation is tha t the charge

density and the 2 D potential are rela ted pointwise, or locally.

T his approxim ation was m otivated by the following physical reasoning. If instead

of an annular film, one considers an annu lar column, w ith a height much larger th an its

w id th , then there is a straightforw ard Poisson relation betw een a bulk charge density

an d the 3D potential inside the colum n. If the 3D po ten tia l is independent of z, and

is equal to 'I'm(r)e ‘mS, there are no free surfaces to consider and one has in place of

(r) = Jm{P inn^) T b3nln 1 m\3rnrit') T CmnI m{j3mnl ) ~r D lnn A m {,3mnl )

(3.48)

— ^ m ( X m p ^ ) 4" b,npY'rn( \ tnpr) . (3.49)

Eqns. 3.38 and 3.39. This problem can be circumvented by making an approxim ation

(3.50)

56


Eqns. 3.38 and 3.39 the relation ( D . D — m 2/ r 2) 'I 'm = — qm. If one now hypothesizes

th a t the charge density reta ins its radial profile when the bulk is squeezed down'

to a 2D film, then one m ust include only an r independent scaling factor f m. as in

Eqn. 3.50.

F u rther expanding each 'Pmjn, using Eqns. 3.41 and 3.49, 'Pm can be expressed as

w here the B m.np are constants. It then follows from Eqns. 3.50 and Eqn. B.9 th a t

and its surface charge density in this approxim ation. Subsequent use of Eqn. 3.53

yields expressions which give some insight into the general linear stab ility problem .

A choice for / m;p can be m ade by considering the em pty upper half space c > 0

w ith homogeneous boundary conditions a t infinity and Dirichlet boundary conditions

such th a t the potential is equal to everywhere on the c = 0 plane. This

boundary condition is sm ooth, unlike the piecewise sm ooth conditions in Eqn. 3.47,

which respect the sharp edges of the annulus. T hen the 3D potential for z > 0 satisfies

th e equation.

From the eigenvalue relation Eqn. B.9. it follows th a t kmp = y mp. T he corresponding

surface charge density is thus

C om paring Eqn. 3.55 w ith Eqn. 3.53 leads to th e following choice for th e closure

^ r n — ^ ' - d n ' f r n ; n ) ’ ■dn B m<nplpm -p , (3.51)f t n p

u p n p

where

(3.53)

Eqn. 3.53 dem onstrates the pointwise. local relation between the po ten tial on th e film

(3.54)

z=0+(3.55)

57


12

10

8

6

$ 4

2

0

.7

•41.4 1.6 1.8 7

r

Figure 3.4: The surface charge density pertu rba tion . 96;i(r ) versus r a t a — 0.5. The dashed line shows th e result of the local approxim ation while the solid line shows the result of the exact nonlocal calculation.

factor:

f m-,P = — • (3.56)X m p

For the rem ainder of this Chapter, Eqns. 3.53 and 3.56 are chosen to close the

system of equations. T his approxim ation is referred to as the 'local theo ry ’. Fig. 3.4

shows a plot of the approxim ate charge density, <76;i a t a = 0.5, corresponding to

a po ten tia l 06;1 on the film. It is com pared to a more accurate num erical solution,

referred to as the ‘nonlocal theory’ which is discussed in A ppendix C. As m ight be

expected, the approxim ation is accurate except close to the edges of th e film.

T he rem aining 2 D equations are solved using the expansions in Eqns. 3.40, 3.48, 3.51

and 3.52. Only the sim plest case is considered here: a single expansion mode, so th a t

Ax = 1 and A n = 0 for n > 1 . Similarly, th e expansions of the po ten tia l and charge

density are trunca ted a t p = 1 so th a t j3m;lp = 0 for p > 1 . T he expansion coefficient

58


B m = B m.<n is complex. Thus, using Eqn. 3.56, Eqns. 3.40, 3.51 and 3.52 reduce to

$m — Cm-1,

= {Brm + iB'm)lpm;1,

<7m = 2{Brm + i B lm)Xmli>ma-

(3.57)

Substitu tion of Eqn. 3.57 into Eqn. 3.37, w ith 7 = v f results in an equation, the real

and im aginary parts of which are

= 2 ( 7 ' - ,

O ' + 2 ( y - - 0 .

respectively. E lim inating B rm from the above pair of equations results in

(3.58)

(3.59)

1 + (3.60)

M ultiplying Eqn. 3.60 by i/>m ; integrating to form inner products denoted by

rr„( . . . ) = / ...Jri

r d r , (3.61)

and solving for the expansion constant,

B l — m T (3.62)

In Eqn. 3.62, is a norm alization factor given in A ppendix B and th e m atrix

elementDqM

Em — ( Crrt; i •r 4 / : m D ^0)\ 2 1 1 \ ,

, 1 + ^ ( 7 “ — ) 1 * - » } < 3 ' 6 3 )

A similar pro jection of Eqn. 3.58 and some simplification results in

2 B i (7 ' -

m / w *1 <Kr \ W m \I 'Ipm; I y

•Ntfmu x r »(3.64)

59


E quations 3.62 - 3.64 determ ine the expansion coefficients of the po ten tia l and charge

density for a given stream function. Substitu tion of Eqn. 3.57 into Eqn. 3.36. with

7 = i y and a C ouette shear gives

a dmi ' ( m D S ° ) \/?mi4Cm;1 - i --------^ — J (Vm;1 - ^ t;l) (3.65)

: ( ( D rm + i B ‘J (2 X,ni D ^ 0) - Z V u>)0 m;1) = 0.rnTZ,

— i -

w here Um-i and V’m;i are defined in A ppendix D. Projecting the above equation with

C m;l and solving for TZ:

i^mi* J Cm-A + 1 $fn j ( fn F m — 7 ‘G m)TZ(a,P,TZe,m,y) = ------------------------- -----------------------r—— . (3.66)

X m tJm ~ K rriJ

where

F m = <C„;1 D®'01 (vm;, (3.67)

G m — (3.68)

JfTt -- r ) (3.69)

K m = ( c y ^ ) vy . (3.70)

T he norm alization factor A/cm., is given in A ppendix B.

To determ ine the linear stab ility boundary for a given radius ratio a, P rand tl

num ber V and Reynolds num ber TZe, Eqn. 3.66 is solved for a sequence of azim uthal

m ode num bers m, using M athem atica for all integrations. Consider the following

special cases.

For zero shear, <f> = y = 0 and Eqn. 3.66 reduces to

= — 1 Y l2jSfc,na^ pl . (3.71)f2 XmlJm KmJ

N ote th a t Eqn. 3.71 is independent of the P ran d tl num ber and is always real. Fig. 3 .5

60


1C2 2 .5 3 3.5 4 4 .5 5 5.5 6 6 .5

200

180

160

& 140

120

100

3 54 6 7 8 9 10

m

Figure 3.5: T he m arginal stab ility boundary for zero shear. P lo t of 71 versus in a t q = 0.5 for TZe = 0 . T he open(solid) symbols are the results of the local(nonlocal) theory. The dashed line is the m arginal stability boundary in a rectangular geometry. 71 versus k , where ac(upper scale) is the dimensionless wavenumber of the pattern.

shows the resulting stab ility boundary for a = 0.5. Also shown is the marginal

s tab ility boundary calculated using the nonlocal theory developed in Appendix C.

Except for being discretized in integer values of m by the annular geometry, neither

of the boundaries is significantly different from th a t of th e infinite rectangular case[4],

which is defined a t a continuum of wavevectors k . T he discrete curve approaches the

continuum one in the lim it a —> 1 . For the nonlocal solution, it was found that as

a —> 1, increases such th a t m ° / f approaches th e correct lim iting value, kc = 4.223.

Here, f = (r, + r„ ) /2 is the midline radius of the annulus and kc is the critical

wavenum ber for an infinite rectangular film in th e ‘p la te ’ electrode geom etry.[4] The

m inim a of the m arginal s tab ility boundaries define the critical param eters ra°(a) =

m c(a ,7 le = 0 ) and 7Z^.(a) = 7lc(ot, TZe = 0 ). Some critical values, as determ ined by

b o th th e local and nonlocal solution schemes, for various a are collected in Table 3.1.

61


rad ius ratio local(n = l , p = 1 )

local(n = 3 ,p = 1)

nonlocal (n = 1 ,Z = 2 0 )

nonlocal (n = 3 ,/ = 20)

Q n\i TZ°C nm c n°c m */r TZ°C0.33 4 102.58 4 77.43 4 91.62 4 4.03 82.15

0.467 5 94.23 6 81.47 6 92.62 6 4.36 88.730.56 7 91.25 7 84.23 7 92.47 7 3.95 89.640.60 8 90.53 8 85.82 8 92.63 8 4.00 90.38

0.6446 9 90.04 10 93.00 10 4.33 91.120.80 18 88.84 18 8 8 .2 0 19 93.59 19 4.22 93.10

Table 3.1: C ritical param eters for zero shear. T he critical param eters for the onset of electroconvection a t TZe = 0 as determ ined by the local approxim ation and the nonlocal (A ppendix C) linear stability analysis. T he integer n (p or I) is the num ber of modes (expansion functions) used in the series representation of the field variables. T he critical wavevector for electroconvection in a laterally unbounded geom etry is kc = 4.223 (see Ref. [4]). The ratio m ° /f —> kc as a —► 1 , where r is the mean radius (ri + r 0) / 2 .

T here is generally good agreement between the two m ethods, except a t small a , where

more expansion modes are needed.

Figure 3.6a shows the zero shear critical control param eter TZ as a function of

the radius ratio a. T here are discrete values of a where TZ° {a, m°) = TZ° (cv, m ° + 1 ),

so th a t two adjacent modes become unstable simultaneously a t a co-dimension two

point. These points occur a t the cusps in Fig. 3.6a, while between the cusps a single

value of m is critical. It is interesting to note th a t the trend in TZ° is increasing

overall, as function of a. This is opposite to w hat is found for radially driven Rayleigh-

Benard convection (RBC) in a rotating annulus.[6 ] This difference is a ttrib u ted to the

differences between the na tu re of the charge density 'inversion5 and thereby electrical

forcing w ith the buoyancy inversion of RBC.

A lternatively, Fig. 3.6b shows a plot of the zero shear critical mode num ber m°

as a function of the radius ratio a . The sam e characteristic of constant between

co-dim ension two points is evident. As a —► 1 , th e co-dimension two points become

closely spaced and the value of TZ° approaches a lim iting value1 while the m ° / f -»■

l In the nonlocal solution, the numerical results were not extended to the limit of an infinitely

62


kc = 4.223, where kc is the critical wavevector for electroconvection in a laterally

unbounded 'p la te ' geometry[4|. r is the m ean radius (t* + va) j 2.

For non-zero shear. Eqn. 3.66 simplifies som ew hat in the lim it th a t the P rand tl

num ber V —> oc. A word of caution is perhaps appropriate here. As V -> oo, it is

required th a t ft —► oo such th a t Q / V and therefore H e (see Eqn. 3.20) are finite. In

th is lim it, w ith the proviso th a t H be real, Eqn. 3.66 becomes

is defined by Eqn. 3.63. For zero shear. L m = K m, while for non-zero shear. L m is

bounded above by K ,„. Hence, H ( a . H e . m , Y ) is bounded below by H ( a . H e =

Q .r n . y = 0 ), the zero shear value. Thus, one expects suppression of the onset of

convection for non-zero shear for every non-axisym m etric mode m. This feature is

also present for finite V.

For arb itra ry a , V and He, one can solve Eqn. 3.66 for various m by a one-

diinensional search procedure. At each m, Y is varied to find a H th a t is real and

a m inim um . Exam ple neutral curves are shown in Fig. 3.7 for a = 0 .8 . V = 10 and

several He. T he suppression of convective onset is evident, as well as a tendency

for the critical mode num ber mc to decrease with He. It was found th a t H (He) is

a m onotonically increasing function of He. T he nonlocal analysis produces neutral

curves which resemble those shown in Fig. 3.7 and differ from them only slightly.

T h a t the critical mode num ber m c decreases w ith increasing shear is also shown

in Fig. 3.8, which displays the angular traveling ra te of the critical m ode, 7 ' / m c, as

a function of H e for several a . For ft > 0 , 7 * < 0 . which indicates th a t the critical

m ode travels around the annulus in the same sense as the inner electrode. 7 ‘/m c is

a very nearly linearly decreasing function of He, w ith very sm all discontinuities a t

dense relaxation grid, as was done for the rectangular case in Ref. [4]. This leads to a small difference between the numerical results for H" in the limit a —► 1 and those for the rectangular case, for which

d m l X m l(3.72)

T he only shear dependence in Eqn. 3.72 occurs through the m atrix elem ent L m, which

Rc = 91.84.

63


94

92

90

88

86

o u84

82

80

78

76

740.2 0.4 0.50.1 0.3 0.6 0.7

a

12

10

3 U8

6

4

2

0.2 0.30.1 0.4 0.5 0.6 0.7

a

Figure 3.6: T h e critical pair (71°,m °) as a function of a . (a) T he critical control param eter for zero shear 72° versus radius ratio a using the nonlocal theory. Special rad ius ratios, a t which two adjacent modes and m° + 1 are sim ultaneously m arginally unstable, occur a t th e cusps of the curve, (b) Between th e cusps, the critica l m ode num ber m° rem ains constant.

64


320

300

280

260

240

220

& 200

180

160

140

120

100

80

'DnJD n n n n D

1• • • • • • • • •

; ° 0 ° O o o k o o o O ° °...................................... ■ I ..................................I ■

,o°:

10 12 14 16 18 20 22 24 26 28 30

m

Figure 3.7: The m arginal stab ility boundaries a t various shears. P lot of 7Z versus m (local theory) for a = 0.80 and V = 10 for several H e numbers. At H e = 0(open circles) the linearly unstab le mode is m c° = 18, whereas a t H e = 0.25(filled circles), m c = 17, and at 1le = 0.75(squares), rnc = 16.

points where the critical azim uthal mode num ber m c changes, as shown in Fig. 3.8.

Each of these discontinuities, which are too sm all to resolve on the scale o f Fig. 3.8,

is a co-dimension two point, where two adjacent m modes w ith very slightly different

traveling rates are sim ultaneously unstable a t onset. These special points are also

slightly V dependent, as well as being a dependent in a m anner sim ilar to the zero

shear case discussed above.

T he suppression due to the shear is m easured by

e {He) =H c(He)

H°c- 1 (3.73)

T h e dependence of the suppression on the radius ratio a and the P randtl-like num

b er V is briefly explored in Figs. 3.9 and 3.10 respectively. Figure 3.9a shows the

65


a = 0.80

o - 2

a = 0.6446

a = 0.56

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14Re

Figure 3.8: T he traveling ra te s of the pattern . R atio of im aginary p a rt of the grow th rate to critical mode num ber, 7 ‘c/rnc versus Reynolds number H e for a =0.56, 0.6446, and 0.80 and V = 123 using the nonlocal linear stability calculation. The breaks in the curves for each a show the intervals over which the critical azim uthal m ode number m c has the value indicated.

suppression curves a t four different a, all a t V = 5. It appears th a t the suppression

is overall a decreasing function of a a t all He. A t a larger V — 50. this trend is no t

observed as is shown in Fig. 3.9b. From these plots, it is fairly convincing th a t for

small enough Reynolds num ber, H e , the suppression decreases w ith increasing a a t

all V . F igure 3.10a shows th e suppression curves a t three different V a t a = 0.33. It

is evident th a t the suppression is an increasing function of V. T he sam e conclusion is

furnished a t a = 0.80 as is clear from Fig. 3.10b. Even though no t manifest from th e

Fig. 3.10, it is expected th a t th e suppression has, a t each He, a finite limiting value

as V 0 0 .

T he dependence of H c for non zero He on V is treated in two ways. First m ain tain

constant th e Reynolds num ber He, change V and consequently let Q vary. In th is

66


0 .0 4

0.035

0.03

0.025

kU 0.02

0.015

0.01 a = 0.33 a = 0.47

a = 0.64 a = 0.80

0.005

0.02 0.040 0.06 0.08 0.1

Re

1.2

0.8

0.4

- a = 0.33- a = 0.47

a = 0.64- a = 0.80

0.2

00 0.02 0.04 0.06 0.08 0.1

Re

Figure 3.9: T heoretical predictions o f the suppression. P lotted are th e com puted values (local theory) of the suppression i versus Tie for selected a a t (a) V = 5 and (b) V = 50.

67


0 .025

0.02

0.015

0.01

0.005

0 0.01 0.02 0.03 0.04 0.05 0.06Re

0.8

0.6

0.4

0.2

0 0.01 0.02 0.03 0.04 0.05 0.06Re

Figure 3.10: Theoretical predictions of th e suppression. P lo tted are the com puted values (local theory) of the suppression i versus H e for selected V . (a) a = 0.33. T he d o tted (dashed) [solid] lines denote V of 10(5) [l] respectively, (b) a = 0.80. The dotted(dashed)[solid] lines denote V of 125(50) [15] respectively.

68


P ran d tl R otation ra te local (n = 1 , •e II nonlocal (n == 1 , Z = 2 0 )V n m c nc 7 ‘ m c nc 7c

100 5.5135 8 319.50 -14.949 8 282.61 -16.38310 0.5513 9 99.65 -1.655 9 98.52 -1.8611 0.0551 9 90.16 -0.166 10 93.06 -0.205

0 .1 0.0055 9 90.05 -0.017 10 93.00 -0 .0 2 0

0 .0 1 0.0006 9 90.04 -0 .0 0 2 10 93.00 -0 .0 0 2

T able 3.2: C ritical param eters a t various V w ith constant He. C ritical Param eters for a range of P rand tl num bers V a t radius ratio a = 0.6446 and Reynolds num ber H e = 0.1. T he integer n (p or /) is the num ber of modes (expansion functions) used in the series representation of the field variables.

case, the V dependence of these results is not strong, except a t large V . Because H e

is proportional to V ~ l in Eqn. 3.20, this lim it corresponds to large values of Q and

a large suppression effect. Some results for various V are sum m arized in Table 3.2.

In th e second protocol m aintain constant the angular ro ta tion rate Q. change V and

consequently let H e vary. In this case, the V dependence is very weak a t all V

investigated. Results are sum m arized in Table 3.3.

W hereas linear analysis cannot provide the m agnitude of the fields above onset,

it is nevertheless interesting to exam ine the spatial s tru c tu re of the linearly unstable

modes. Figure 3.11 shows th e velocity vector field a t a = 0.35 in the absence of shear.

T he critical mode num ber for this flow is 4; there are 4 counter-rotating vortex

pairs in the annulus. The vortices in each pair are sym m etric. Figure 3.12 displays

the velocity vector field of a critical m ode for a = 0.56, p lo tted w ith an arb itra ry

am plitude. In Fig. 3.12a is shown the sta tionary vortex p a tte rn a t H e = 0 . Here

= 7, so 7 sym m etric vortex pairs are arranged around the annulus. This p a tte rn

is purely non-axisym m etric or ‘colum nar’. I t is an exact solution to the governing

equations and has been observed experim entally. In Fig. 3.12b is shown the typical

flow p a tte rn for a large H e > 0 , as viewed in the laboratory frame. T he periodicity

of th e p a tte rn is reduced (i.e. m c < m °), and the traveling p a tte rn appears as a

m eandering wave in the laboratory frame. Figure 3.12c shows th e velocity field for

th e sam e H e as in p a rt b, b u t as seen in the frame in which the p a tte rn is stationary.

69


Prandtl Reynolds local (n = 1 ,P = 1) nonlocal (n == 1 . / = 2 0 )V Re m c nc 7 ‘. m c nc 7c

100 0.0181 9 115.20 -2.995 9 109.03 -3.37210 0.1814 9 115.29 -3.002 9 109.02 -3.3651 1.8137 9 115.27 -3.007 9 108.99 -3.307

0 .1 18.1373 9 115.24 -3.039 9 109.14 -3.1650 .0 1 181.3731 9 115.16 -3.086 9 109.31 -3.108

Table 3.3: Critical param eters a t various V w ith constant H. C ritical Param eters for a range of P rand tl num bers V a t radius ra tio a = 0.6446 and ro ta tion rate Q = 1 .0 . T he integer n (p or I) is th e number of modes (expansion functions) used in the series representation of the field variables.

This frame rotates in the sam e sense as the inner electrode bu t w ith an angular speed

of t ‘c/ t n c, which is less th a n Q. In this frame, each vortex pair consists of a larger and

a smaller member and so breaks the sym m etry between vortices in each pair.

Two special situa tions in annular electroconvection w arrant fu rther comments.

T he first, which has been referred to before, is the case of electroconvection in a

rigidly rotating annulus. T he other concerns the stability of circu lar C ouette flow in

the absence of electroconvection. Rigid body ro tation of the annulus can be attained

by rotating both th e inner and outer electrodes a t the same angu lar ro tation rate. In

such a situation, the base s ta te has an azim uthal flow which is described by D 0 ^°'(r) =

—fir . Alternatively, in the co-rotating fram e th e 'shear1 vanishes and the system is

apparently no different from the zero shear non-rotating system . As noted earlier,

the stability of the rigidly ro tating annulus is the same as th a t of zero shear case.

However, one has to be clear th a t in the zero shear case m arginal stability occurs

a t 1ZC° with critical m ode m° and traveling ra te Yc/m ° = 0 . while in the rigidly

ro tating case m arginal stability occurs a t 'R.c° w ith critical m ode m “ and traveling

ra te 7 lc/m ° £ 0 . In fact 7 ‘/m ° = —Q, so th a t th e pattern travels a t th e same angular

velocity as th a t of th e rigid rotation. As a resu lt the electroconvection flow p atte rn

in the co-rotating fram e of the rigidly ro ta ting annulus is identical to th a t of the zero

shear non-rotating annulus, an example of which is illustrated in Fig. 3.11. T h a t

7 c /m c = can be observed, as a case in poin t, from the expressions for TZ in the

70


Figure 3.11: Velocity vector field for annular electroconvection w ithout shear. P lotted is the velocity vector field of a rb itrary am plitude a t a = 0.35 w ith m c° = 4.

lim it V -> oo: see Eqn. 3.72. T he only dependence on th e base s ta te flow profile

D 0 (o)(r) occurs through the m atrix elem ent Lm. which is defined by Eqn. 3.63. The

question of stab ility requires determ ination of the lowest value of 1Z for a given m and

variable 7 '. From Eqn. 3.72 it is clear th a t the question of s tab ility is tan tam oun t to

maxim izing the value of Lm. From Eqn. 3.63 it is easy to see th a t Lm assum es its

m axim um value when 7 * = mD|f (°) . Further this value is K m, the zero shear upper

bound for Lm. Hence the rigidly ro ta tin g annulus, when driven to electroconvect, can

be seen in the case V —► 0 0 , to becom e m arginally stable a t a value 7ZC° corresponding

to th e zero shear non-rotating case. The unstable mode is m ° and the traveling rate

is determ ined from 7 * = = —m°Q. T he sam e conclusion holds for finite V. It

is interesting th a t it is the shear th a t has stabilizing properties and not the rotation;

a point which is revisited in Section 4.8.

P lane parallel C ouette flow had been assum ed, rightly so, to be linearly stable

71


R e > 0

co-rotating frame

Figure 3.12: Velocity vector field for annular electroconvection w ith shear. P lo tted is th e velocity vector field of a rb itra ry am plitude a t a = 0.56 w ith m c = 7 a t (a) H e = 0, (b) H e > 0 when viewed in th e laboratory frame, and (c) as in (b) b u t when viewed in the fram e th a t co-rotates a t 7 c‘/ m c-

72


for m any years[9] un til a proof of stability was finally provided by Romanov[lO],

Likewise, due to the sim ilarity in the form of equations for the stab ility of circular

C ouette shear flow in two dimensions to those of plane parallel C ouette flow, it too

is assum ed to be linearly stab le .[9] Experim entally, in the range of H e investigated,

there is no evidence of instability.[1 1 ] T he form alism derived in this C h ap te r tests for

the loss of stability to electroconvection in the presence of a circular C ouette flow. It

is easy in this formalism to simply ‘tu rn off’ the electrical term s in th e appropria te

equations and thereby question the stability of the 2D shear flow. By se tting H and

the o ther electrical term s in Eqn. 3.66 to zero and reintroducing the real p a rt of the

grow th ra te Y by the transform ation Y ~► Y + 7 r /L it follows th a t the stab ility of

2D circular C ouette flow is given by

Linear stab ility occurs when Y defined in Eqn. 3.74 is negative. Since th e num erator

in Eqn. 3.74 is positive definite, linear stab ility dem ands th a t G m < 0 . In all the

calculations performed w ith the local approxim ation, over several a and m any m . it

was numerically found th a t G m < 0 always. W hereas the foregoing by no means

constitu tes a proof of stability, it does however provide further indication th a t the

conjectured stability of 2D circular C ouette flow m ay indeed be true. N ote th a t

m athem atically proving th a t G m < 0 is yet no t a proof of stability; it would only

m ean th a t a t the lowest approxim ation in the expansion, the 2D C ouette flow is

linearly stable. A proof of stability is much m ore rigorous and difficult undertaking.

3.5 Assum ptions: Theory versus Experim ent

This Section addresses the experim ental relevance of the assum ptions in the theoret

ical model. Three assum ptions of the theoretical model which cannot be precisely

realized experim entally are discussed below. These are the condition o f exact two-

dim ensionality, of constan t electrical conductivity and of infinitesimally th in elec-

73


trodes. Each of these is a possible source of system atic disagreem ent between the

experim ent and th e theory.

All physical fields are, by the assum ption of two-dimensionality, taken to be con

s ta n t through the thickness of the film. A film of thickness s has vertical dim ension

—s /2 < z < s /2 , bu t is trea ted as a sheet a t z = 0. The discontinuity of th e norm al

com ponent of the electric field a t z = ± s / 2 supports the surface charge densities

there. Suppose as an extrem e exam ple th a t these surface charge densities are com

pletely localized a t the free surfaces z = ± s / 2 . T hen the electric force acting on the

charges is also localized a t z = ± s / 2. Consequently, the surfaces are preferentially

driven while the bulk of the fluid w ithin —s / 2 < z < s /2 is only driven by viscous

coupling. Hence th e velocity of the fluid will depend on the r coordinate. It is by the

prem ise of two-dim ensionality th a t this dependence is neglected. T he approxim ation

is not severe. In fact the surface charge densities are not localized sheets a t z = ± s / 2

b u t have some thickness, a ’skin dep th ', which extends into the bulk of the film.

Diffusion sm ears the surface charge over a thickness known as the Debye screening

length. Ad- given by

w here D is the diffusion constant and e is the dielectric perm ittiv ity .[1 2 ] For the

sm ectic A m aterial used, estim ates of D give Ad ~ 10 smectic layers.[12] Therefore

the assum ption of two-dim ensionality will begin to break down for films th a t are sig

nificantly thicker th an 2 0 layers corresponding to one Debye length near each free

surface. It has been dem onstrated in previous electroconvection experim ents (see

Section 1.3) th a t the critical voltage Vc was accurately linear w ith s for films w ith

thicknesses less th a n about 25 sm ectic layers.[14] For thicker films, the critical volt

age was sublinear, suggesting th a t the surfaces were being preferentially driven. Since

m ost of the experim ents reported here were perform ed on films w ith thicknesses be

tween 25 and 85 sm ectic layers, it is likely th a t three-dim ensional effects were present

to some degree. T he prim ary effect is th a t th e velocity field becom es dependent

on the z coordinate. N ote th a t the velocity field does not develop a z com ponent

74


b u t the r and 9 com ponents of th e velocity field become --dependent. As a result,

a layer-over-layer shear flow occurs which was neglected in the theory. This weak

three-dim ensional effect may be a source of som e system atic disagreem ent between

experim ental m easurem ents and theoretical predictions.

T he m aterial properties of the fluid such as th e electrical conductivity, the density,

and the viscosity, were assumed to be constant in the theoretical model. It has been

dem onstrated by the experim ents (see Section 2.3) th a t the electrical conductvity

is not constan t during the experim ent. T he conductivity drifts slowly due to elec

trochem ical reactions between the electrodes and the ionic species in th e fluid. T he

conductiv ity change is weakly tim e-dependent. It is likely th a t the conductiv ity drift

is spatially dependent as well. T he uniform ity of the conductivity is enforced by the

various charge transport mechanisms, including diffusion. For exam ple, the conduc

tiv ity a t an electrodes may be different from th a t a t the centre of a vortex. T he drift

of the electrical conductivity is affected by several factors (see Section 2.3). It is not

clear how the drift would alter th e predictions of th e theory. W hile the effect of the

drift can be partially corrected for (see Section 4.2 and A ppendix E), the residual

drift is a source of system atic disagreem ent w ith theory.

Finally, the geom etric assum ptions in the theory are ideal. T he electrodes were

trea ted as sheets of zero thickness in the theory, while the physical electrodes have a

nonzero thickness. There are two consequences of this assum ption. In the first place,

the actual surface charge density a t the electrodes cannot diverge as it does in the

theory. [13] T hus the stability analysis considered pertu rba tions abou t an idealized

base s ta te th a t is not precisely realized in the experim ent. However, the difference is

probably sm all except a t the electrodes. Second, th e film is a ttached to th e electrodes

v ia a th in w etting layer of bulk liquid crystal. As a result, th e boundary conditions

between the electrodes and the film may not be perfectly rigid as was assum ed in the

theory. More im portantly, the w etting layers differ from film to film and so the exact

na tu re of th e boundary conditions may be slightly different from film to film.

75


3.6 A m plitude Equation

T he weakly nonlinear regimes of pa tte rn s are often well described by a class of equa

tions generally referred to as am plitude equations. The am plitude in these equations

is the m agnitude of the underlying physical field, for exam ple the velocity of the

convecting fluid. P art of their appeal derives from their sim ple form and the ir ap

plicability to num erous and diverse system s.[14, 15] W hereas th e process of deriving

am plitude equations from the prim itive, microscopic or field equations of the system

is often arduous and tedious, the form of the am plitude equation can often be in tu ited

from the sym m etries of the unstable so lu tion[l6 |. The specific details of the individ

ual system s are contained in the constan ts th a t set the scales of space, tim e and

am plitude in the am plitude equations. G enerally the am plitude is a complex variable

which can be related to the real physical quantities in the experim ent. O ften, the

real and im aginary parts of the complex am plitude are trea ted separately.

In the absence of shear, th e base s ta te solution of the annular electroconvection

system is invariant under azim uthal ro tations and reflections through planes which

are perpendicular to the fluid layer and contain the centre of the annulus. This

requires th a t the am plitude equation, for a complex am plitude .4 m corresponding to

a nonaxisym m etric mode m, be unchanged under the transform ations

9 -> 9 + 9 ' , ,4m —¥ A meimB' . (3.76)

and

6 —> —9 , 4 m —> A m . (3.77)

In the above, 9 is an azim uthal angle, and the overbar denotes com plex conjugation.

T he m ost general am plitude equation th a t is invariant under th e sym m etry operations

Eqs. 3.76 and 3.77 is

TodtAm = eAm — go\Am\2A m + — ... , (3.78)

where r0, go. and h0 are real coefficients, c is a param eter th a t is small, and is the

76


reduced control param eter defined by e = (1Z/TZC) — 1. The am plitude equation

describes a bifurcation from the A m = 0 s ta te which prevails for e < 0 to th e A m ^ 0

s ta te th a t prevails for e > 0. A sharp bifurcation occurs a t e = 0. If the fluid is

subjected to a circular C ouette shear, the base s ta te is only invariant under azim uthal

rotations, Eq. 3.76. In this case,

To(0 t — ^(1 ~F f/o(l "1" |d m |”d m

+ hQ( 1 + ic3 )|.4 rri|4Am — ... . (3.79)

is the general am plitude equation where a/„, is the im aginary part of the eigenvalue

of the unstable m ode m a t onset. The term s c0. c2, c3 are real. W hereas Eqn. 3.79 is

here m otivated by sym m etry considerations, it has been rigorously derived for annular

electroconvection w ith shear from the basic equations.[17]

T he m easurable quan tity in this experim ent is the global charge tran sp o rt mea

sured by the reduced Nusselt num ber, n. Since n is real, it is useful to solve for th e real

and imaginary parts of the am plitude equation. By letting A,n[t) =

where A n ( 0 is a real am plitude and is the phase, one gets by substitu tio n in

Eq. 3.79. equations for the m agnitude and phase:

TadtAtn = (Am — + ^0A^n ~ ••• • (3.80)

T o { d t $ m - a l m ) = ecQ- g 0c2*4 ^ + . . . . (3.81)

T he current-voltage m easurem ents, as will be explained in Section 4.2 and detailed

in A ppendix E, can be transform ed into m easurem ents of (e, n). Furtherm ore, the

reduced Nusselt num ber n which measures the ratio of curren t transpo rt by convec

tion to th a t by conduction is related to the am plitude by n = |A m |2 = .4 ^ . [17] Hence

the raw experim ental d a ta can be transform ed into m easurem ent of the reduced con

trol param eter e and th e real am plitude Am. This am plitude is proportional to the

m agnitude of the rad ia l com ponent of the velocity of the convecting flow.

Since the m easurem ents are steady s ta te determ inations of the current, th e time-

77


independent amplitude equation is

eA - g A 3 - h A 5 + f = 0 . (3.82)

In Eqn. 3.82. the subscripts on the coefficients g and h have been dropped and A =

A m . T he field term / is conveniently added to model an imperfect b ifurcation.[14] Due

to th e geom etrical aberrations in the experim ent, the s ta ted sym m etries which vali

d a te the am plitude equation m odel are slightly inexact. As a results th e bifurcation is

no longer sharp a t e = 0 bu t is ra th e r smeared or imperfect. This sym m etry-breaking

field term models an im perfect bifurcation. The current-voltage d a ta is interpreted by

fitting to Eqn. 3.82. This equation describes an im perfect pitchfork bifurcation. The

sign of g determ ines w hether th e bifurcation to electroconvection is forward [g > 0 )

or backward (g < 0). A tricritical bifurcation occurs when g = 0.

78


Bibliography

[1] Z.A. Daya, V. D. Deyirmenjian, and S. W. Morris, "‘Electrically driven convection

in a thin annular film undergoing circular Couette flow.'’ Phys. Fluids. 1 1 . 3613

(1999).

[2] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Dover Publica

tions Inc. (1961).

[3] D.C. Jolly and J.R . Melcher. "Electroconvective instability in a fluid layer.” Proc.

Roy. Soc. Lond. A.. 314. 269-283 (1970).

[4] Z. A. Daya. S. W. Morris, and J. R. de Bruyn. "Electroconvection in a suspended

fluid film: A linear stability analysis.” Phys. Rev. E 55. 2682 (1997).

[5] Z. A. Daya, "Electroconvection in suspended fluid films,’’ MSc. Thesis, unpub

lished (1996).

[6 ] A. Alonso, M. Net, and E. Knobloch, “On the transition to colum nar convection.”

Phys. Fluids,7, 935 (1995).

[7] A. P. Prudnikov, Yu. A. Brychkov, and O. I. M arichev. Integrals and Series, Vol.

2, Special Functions, Gordon and Breach, New York (1992), p 207.

[8 ] I. S. G radshetyn and I. M. Ryzhik, Integrals, Series and Products, Academic

Press, New York (1980), p 6 6 6 .

[9] P. G. Drazin and VV. H. Reid, Hydrodynamic Stability, C am bridge University

Press, Cam bridge (1989).

79


[10] V. A. Romanov. ‘'S tability of plane-parallel C ouette flow.” Functional Anal. &

Its Applies, 7 137 (1973).

[11] X-l. Wu, B. M artin, H. Kellay, and VV. I. G oldburg, “H ydrodynam ic Convection

in a Two-Dimensional C ouette cell.” Phys. Rev. Lett. 75, 236 (1995).

[12] S. W . Morris, "Electroconvection in a freely suspended sm ectic film,” PhD . The

sis. unpublished (1991).

[13] J.D . Jackson, Classical Electrodynamics, W iley (1975).

[14] M. C. Cross and P. C. Hohenberg, “P atte rn form ation outside of equilibrium ,”

Rev. Mod. Phys. 65. 851 (1993).

[15] A.C. Newell, T. Passot and J. Lega, “O rder param eter equations for p a tte rn s ,”

Ann. Rev. Fluid Mech., 25. 399 (1993).

[16] E. Knobloch, “B ifurcations in R otating System s.” in Lectures on Solar and Plan

etary Dynamos edited by M. R. E. Proctor and A. D. G ilbert. C am bridge Uni

versity Press, New York (1994), p. 331.

[17] V. B. D eyirm enjian, Z. A. Daya, and S. W. M orris, “Weakly nonlinear analysis

of electroconvection in a suspended fluid film,” Phys. Rev. E. 56, 1706 (1997).

"W eakly nonlinear theory of annular electroconvection with shear." in progress,

to be subm itted to Phys. Rev.E.

80


C hapter 4

R esu lts

4.1 Introduction

This C hap ter presents and discusses the results from the experim ental investigation

described in C hap ter 2. T he results are prim arily obtained by analyzing the current-

voltage characteristics of the film. The first ta sk is to specify the descriptors of the

film. These are the relevant dimensionless num bers tha t describe the geometry, the

m aterial param eters and the state of flow of th e film at the point ju s t before it is

driven to electroconvect. These descriptors are the radius ratio a. a Prandtl-like

num ber V and the Reynolds number He. W ith the exception of a . which is merely

geom etrical and independent of the film, the values of V and 1Ze require, additionally,

knowledge of the m ateria l properties of the film. Section 4.2 describes how current-

voltage d a ta can be used to determine up to a constant the values of V and He.

T he current-voltage d a ta when the film electroconvects contains inform ation about

the am plitude of electroconvection. The procedure to extract this inform ation is

described in Section 4.2. Also discussed in th is Section is the appropria te am plitude

equation model, in troduced in Section 3.6, as it pertains to analyzing the current-

voltage d a ta . T he coefficient of the cubic nonlinearity, g in the am plitude equation,

is o f prim ary in terest. Some of the results from th e d a ta analysis are concerned with

th e d is tribu tion of critical voltages for films o f different radius ratio a , thickness s,

P randtl-like num ber V and Reynolds num ber H e. These results do no t concern the

81


am plitude of electroconvection but the onset of electroconvection. They are precisely

rela ted to th e linear theory th a t determ ines the onset and no t nonlinear theory th a t

prim arily determ ines the am plitude. In Section 4.3, these results are com pared w ith

predictions from the linear stability theory developed in C hap ter 3. By means of

Levenberg-M arquardt nonlinear fitting procedures and M onte-Carlo methods, the

current-voltage d a ta are fit to the am plitude equation m odel.[2 ] In Section 4 .4 are

reported m easurem ents of g a t various a and ranges of P for electroconvection in

the absence of shear or TZe = 0 . T he results of fits to th e am plitude equation for

sheared films are presented in Section 4.5. Some of the current-voltage characteristics

indicate th e occurrence of secondary bifurcations tha t m ark the transition from one

flow p a tte rn to another. Secondary bifurcations and the ir dependence on TZe are

discussed in Section 4.6. In Section 4.7 are presented a collection of bifurcations

w ith miscellaneous properties. O ther nonlinear systems th a t are sim ilar to annular

electroconvection w ith shear are discussed in Section 4.8.

4.2 D ata Analysis

Below the onset of electroconvection. the film is ohmic. As a result, the current-

voltage characteristics furnish inform ation about the film's resistance and, w ith knowl

edge of the film's thickness, the conductivity. Due to the d rift in the electrical con

ductivity. the film's resistance, or equivalently its conductance, is changing. Each

voltage-current m easurem ent. (V. I ), constitu tes an experim ental determ ination of

the film’s conductance, c = I / V . On the other hand, the conductance of the film can

be expressed in term s of its geom etry and the conductivity of the film. For a film of

radius ratio a , thickness s and uniform electrical conductivity cr3, the conductance is

given by

27TQ3S

° ln ( l / a ) ‘ ( ^

Intriguingly, the conductance is independent of the size of th e film, i.e. independent

of or r 0. A general derivation of the resistance of an annular film and hence of

82


Eqn. 4.1(see Eqn. F .8 ) is given in Appendix F, Section F .l. Eqn. 3.16 defined a

Prandtl-like num ber, which w ith d = r0 — r, and Eqn. 4.1, can be w ritten as

V ^ €0*73 = 27T6QT73 1

p3a 3sd p3(ra - n ) In ( 1 / a ) c '

Note tha t V can be determ ined from the m easured and dim ensional values of r,, ra

and c and m ateria l param eters, th e density p3 and viscosity q3. Eqn. 4.2 is used

to determine th e value of V for every current-voltage characteristic. Drift of the

electrical conductiv ity results in, by way of the conductance, a corresponding drift of

the Prandtl-like number.

The Reynolds num ber TZe of th e circular C ouette flow was defined in term s of di-

mensionless param eters in Eqn. 3.20. By rewriting in term s of dim ensional param eters

and m aterial properties, TZe can be calculated from

TZe = p3u r t(r0 - r , ) / / ; 3 , (4.3)

where u is the m easured angular frequency of the inner electrode in ra d /s . The 3 D

density p3 of 8 CB a t room tem perature[l] is 1.0 x 1 0 3 kg /m 3; the viscosity q3 has

not been m easured and will be estim ated from a com bination of d a ta and theory in

Section 4.3.

The next ta sk is to deduce, from the data, m ore inform ation regarding the onset

and am plitude of electroconvection. T he current-voltage characteristics, examples of

which were p lo tted in Figs. 2.7a and b, show th a t a critical voltage dem arcates the

conduction and convection regimes. However the critica l voltage Vc cannot simply be

chosen by-eye b u t m ust be determ ined by some ‘m ost probable’ or ‘best fit’ criterion.

For the present, assume th a t a Vc and the corresponding conductance c have been

determined; th e subtleties will be introduced la ter an d recounted in A ppendix E.

Each instance of d a ta on the current-voltage characteristic is com prised of a triple of

numbers ( V , I , A I ) where A / is th e measurem ent erro r in the current. T he reduced

83


control param eter e can be calculated from

(4 .4 )

T he reduced Nusselt num ber n which m easures the electrical curren t due to convection

relative to tha t due to conduction can be calculated from

( V , I , A I ) can be transform ed to the equivalent triple ( e . A, AA) . In Section 3.6,

it was justified by sym m etry argum ents th a t the relevant am plitude equation th a t

describes the weakly nonlinear state is given by Eqn. 3.79. T h is equation provides a

tim e-dependent description but the d a ta are tim e-independent. As a result th e d a ta

w ith an ‘im perfection1 te rm / was given in Eqn. 3.82 and is repeated below.

A Levenberg-M arquardt least squares nonlinear fit routine is employed to m odel the

d a ta triples (e, A, AA) obtained from the current-voltage characteristic. Best fit pa

ram eters (g , h , / ) are ob ta ined from this chi-square m inim ization procedure. A M onte-

C arlo bootstrap m ethod is utilized to o b ta in the statistical uncertainties in th e best

fit param eters [2 ].

W hile Vc is m arked by a relatively obvious feature in th e raw ( / , V) d a ta , the

am plitude A is indirectly deduced via th e pair of transform ations Eqns. 4.4 a n d 4.5

w hich are quite nonlinear. This am plitude is in turn fit using Eqn. 4.6. which is again

highly nonlinear. Thus, the param eters (g , h, f ) are rather d is tan tly related to th e raw

(I , V) data . Also, th e n a tu re of the m odel necessarily involves several fit param eters

which are not independent. Consequently, the determ ination of these p aram eters is

m uch more difficult th a n Vc. As discussed in Sections 4.4 an d 4.5, (g , h, f ) can be

n = (4.5)

Further, by identifying n = A2, where A is the am plitude o f convection, th e triple

are modelled by the real and steady s ta te p a rt of Eqn. 3.79. which when augm ented

eA - gA 3 - h A 5 + / = 0. (4.6)

84


influenced by sm all system atic effects in the d a ta leading to sca tte r which is larger

th a n the statistical uncertainties in the fits. Nevertheless, the general trends are clear.

Figure 4.1a shows a current-voltage characteristic in the absence of shear. The

transition from conduction to convection is continuous and so the bifurcation is su

percritical or forward. Figure 4.1b shows the result of transform ing, and fitting the

d a ta in Fig. 4.1a to Eqn. 4.6. Doth the (e, .4 .A A ) and (e. n. A n ) d a ta are shown.

T h e lines are calculated from the best fit param eters. For this case g > 0 . h > 0 and

0 < / < S l . For all the d a ta analysis in this thesis / • C l .

Figure 4.2a shows a current-voltage characteristic for a film under shear. The

transition from conduction to convection is discontinuous and so the bifurcation is

subcritical or backward. Figure 4.2b shows the result of transform ing and fitting the

d a ta in Fig. 4.2a to Eqn. 4.6. Both the (e..4 ,A .4) and ( e . n .An) d a ta are shown.

T he lines are calculated from the best fit param eters. For this case g < 0 . h > 0 ,

0 < / C l . The dependence of g on a , V and TZe is reported in Sections 4.4 and 4.5.

In developing the theory in C hapter 3. it was explained tha t onset of convection

occurs when the control param eter H equals or exceeds a critical value 7Z,. given by

A consequence of the drift of the electrical conductivity is th a t the critical voltage

slowly changes during the course of an experim ent. Since the detailed physical pro

cesses responsible for the drift are poorly characterized, it was thought th a t ra ther

current-voltage d a ta . T he methodology employed in diagnosing th e drift in the con

ductance. in determ ining a conductance adjusted critical voltage, and in M onte-Carlo

procedures to ascertain a best fit Vc is detailed in Appendix E.

°V/3S(4.7)

th a n trying to m odel the effect of the drift it would be better to correct for it in the

85


<Q.

C22u3

u

5.5

4.5

:(a)£i

S 'sssssss: y

y

^ Increasing voltagej T v Decreasing voltage

16 17 18 19 20 21 22 23Voltage (volts)

0.5T 0.25

0.40.2

0.3

0.2

Z 0.05

n, increasing E A, increasing E ■ o"O

-0.25 0 0.25 0.5 0.75 1 1.25

<uT33O.E<

Figure 4.1: A representative supercritical bifurcation. In (a) is plo tted a po rtion of the current-voltage characteristic for a film a t a = 0.64 and Tie = 0 . In (b) is plotted th e result of analyzing the da ta p lo tted in (a). It consists of a plot of th e reduced Nusselt number (n) and am plitude (A) versus the reduced control param ete r e. The lines are nonlinear least squares fits to the data .

8 6


6.5

U 5.5

Increasing voltage

1716 18 19 20Voltage (volts)

0.50.25 ■(b)

0.40.2

-CE 0.15 0.3

0.2

0.10.05

n, increasing e A, incresing e

- 0.2 0 0.2 0.4 0.6

<u-a

E<

Figure 4.2: A representative subcritical bifurcation. In (a) is p lo tted a portion of the current-voltage characteristic for a film a t a = 0.80 and 1Ze > 0 . In (b) is plotted the result of analyzing the d a ta p lo tted in (a). I t consists of a p lo t o f the reduced Nusselt num ber (n) and am plitude (A) vs the reduced control param eter e. The lines are nonlinear least squares fits to the data .

87


4.3 Comparisons with Linear Theory

T his Section presents th e comparisons betw een the experim ental m easurem ents of

the onset of electroconvection and the relevant theoretical predictions of C hap ter 3.

T he prim ary theoretical result in the absence of shear concerns the prediction of the

onset of electroconvection. The critical voltage, which has been previously expressed

for the general case in E qn. 4.7, is squared and w ritten for th e zero shear case below.

03*73 S '

£q272°(a). (4.8)

N ote th a t the critical voltage depends on th e film geometry th rough the radius ratio

a and the film thickness s but not the Prandtl-like num ber V . Dy using Eqn. 4.1,

Eqn. 4.8 can be expressed more conveniently as

4 7 r e 02<73(V;0 ) 2 ( « ) 2 / , m------------- s---------- = n3c . (4.9)( ln ( l /o ) )272°(a)

W ritten as such, Eqn. 4.9 expresses a relationship th a t allows determ ination of the

viscosity r\3 which is th e only unknown param eter. There is one caveat: th is de

term ination is not en tirely experim ental b u t requires the theoretical prediction of

72°(a). The quantity on the left hand side of Eqn. 4.9 w hich is referred to as the

‘scaled critical voltage' is com puted as follows. The d a ta fitting procedure outlined

in Section 4.2 was used to deduce from a current-voltage characteristic the critical

voltage V® of electroconvection and the conductance c a t onse t. From th e colour of

th e film, a thickness s was determ ined and w ith knowledge of th e radius ratio a and

Eqn. 4.1, the conductivity <r3 was calculated. The value of 72.°(a ) was the highest

order numerical result o f the nonlocal theory; see Table 3.1.

Figure 4.3 plots the left hand side of Eqn. 4.9 versus the sq u are of the conductance.

T he scaled critical voltages were obtained from 228 current-voltage characteristics a t

six different a . The d a ta encompassed a b road range of conductivities: 5.9 x 1 0 - 8 <

<x3 < 8.4 x 10_7n _1m _ 1 . Consequently, th e range of P rand tl-like num ber V is equally

broad. Despite the diversity in param eters i.e. the different a , s, cr3 and V , Fig. 4.3

88


C/JS*00

(NICi

sOCM

l £CM

a

DCMO u >

{N•*t

♦ a = 0.33o a = 0.47• a = 0.56o a = 0.60X a = 0.64■ a = 0.80

r| = 0.18 +/- 0.03 kg/ms

2 4 6(conductance)2, (10‘26 Q '2)

Figure 4.3: C ritical voltages in films w ithout shear. A plot of the square of the scaled critical voltage versus the square of the conductance a t several a. T he line is a least squares fit to the data. The fit was for 0 < c2 < 5 x 10- 24fl~2, however, only the interval m ost dense with da ta is plotted.

corroborates th e linear relationship th a t is theoretically predicted in Eqn. 4.9. One

is confident th a t the geometric scaling with respect to the film thickness s and the

radius ratio a is properly accounted for. A single variable linear fit to the scaled

critical voltage as a function of the square of the conductance provides a measure

of the viscosity of the film; the only unknown param eter. A weighted least squares

m inim ization leads to rj3 = 0.18 ± 0.03 kg /m s.[2 ]. This value for the viscosity is

reasonable, w hile it has not been independently m easured, it is expected to be of

order 0.1 k g /m s .[3]

The theore tical model assumes stric t two-dim ensionality which is an assum ption

that in th ick films may be som ew hat invalid. T he 2D assum ption im plies th a t the

89


radius ratio a

Experim entalnm c

Local theory (n = 3, p = 1)

N onlocal theory m° (n = 3. 1 = 20)

0.33 4 4 40.47 6 6 60.56 8 7 70.60 8 8 8

0.64 10 10 10

0.80 20 ►—1

00 19

Table 4.1: Experim ental m easurem ents of the marginally s tab le mode num ber, m°.

velocity of the film is independent of the film ’s thickness. Since the electrical forcing

is a t the free surfaces, it is likely th a t in th ick films the surface layers are preferentially

driven and so the motion is not accurately 2 D. From an experim ental perspective, the

geom etry is imperfect in th a t there are 3D w etting layers on the circumferences of the

inner and ou ter electrodes. These produce boundary conditions th a t are som ewhat

unlike those assumed in the theory. Both th e foregoing aspects differ from experim ent

to experim ent and it is thus not surprising th a t the d a ta in Fig. 4.3 have system atic

deviations from linearity w ithout any overall trend. See Section 3.5 for a discussion

of the assum ptions in the theory. It is rem arkable tha t the d a ta , despite the scat

ter, dem onstrates the linear trend considering the broad range of a and c th a t are

represented.

T he second feature of linear theory th a t can be com pared w ith experim ent con

cerns the unstab le mode number. As m entioned in Section 2.3, some experim ents were

perform ed b o th a t reduced ambient pressure and a t atm ospheric pressure in films w ith

slight thickness nonuniformity. This perm itted flow visualization and therefore it pro

vided qualita tive confirm ation of the flow field and a q uan tita tive measure of the m ode

num ber of the flow. Qualitatively, the flow th a t was observed in the absence of shear

com prised of counter-rotating pairs of vortices much like those depicted in Fig. 3.11.

The m ode number, corresponding to th e num ber of vortex pairs, was in excellent

agreem ent w ith predictions of linear theory. Table 4.1 sum m arizes the results.

T he final feature of linear theory th a t was tested by th e experim ental d a ta was

90


th e degree of suppression of electroconvection th a t is im posed by the C ouette shear.

T he suppression is m easured by Eqn. 3.73, which is w ritten below w ith im portan t

functional dependencies:

i (a, He, V ) =7Zc(a, H e , V )

rc ° (a )

f VJct, c. He) \ "- 1 = - k ^ r - 1 - «■ «»

where c is th e film conductance. T h e two equivalent expressions in Eqn. 4.10 are

used to calculate the suppression theoretically and experim entally. The theoretical

calculation is described in Section 3.4. The experim ental determ ination of i is as

follows. At each radius ratio a. th e critical voltage for a sheared film Vc(c ,H e) is

determ ined by the d a ta fitting procedures outlined in Section 4.2 and A ppendix E.

Since the conductance c drifts, it is unlikely tha t V^{c) can be determ ined from a

single current-voltage characteristic. Instead, several current-voltage characteristics

for films in the absence of shear are fit to provide a set o f d a ta consisting of (c. I /0).

These are then modelled by a linear function, and it is from this fit th a t the critical

voltage a t zero shear- and a t the same conductance as th e sheared film is determ ined.

Hence, the experim ental value of e can be com puted. T h e uncertain ty in th is value is

due to the uncertain ties in Vc(c. He) and V®{c). W hereas th e uncertainty in the former

is sm all as it is from a single m easurem ent, the uncertain ty in the la tte r is larger due

to the sca tte r in the (c, V.0) data . T h is uncertainty dom inates the uncertain ty in the

suppression.

Each experim ental m easurem ent of the suppression is a t a given radius ra tio a, a

m easured conductance and therefore Prandtl-like num ber V and a m easured Reynolds

num ber H e. W hereas the dependence of the suppression on these param eters was

studied theoretically in Section 3.4, it is experim entally convenient to vary the radius

ratio a and th e Reynolds num ber H e. Due to the drift in the electrical conductivity,

the Prandtl-like num ber V is not exactly constant in th e experim ent. Consequently

the suppression i is m easured, a t several a , as a function o f H e while the V is simply

m easured and noted. Figures 4 .4a and b show experim ental m easurem ents of the

suppression a t a = 0.47 and a = 0.64 respectively. For th e d a ta a t a = 0.47, the

91


14 : (a)

12

10

8

0

4

— Local theory N onlocal theory

0

0 0.5 2.51.5Re

4

3

Local theory N onlocal theory

00 0.250.05 0.1 0.15 0.2

Re

Figure 4.4: Experim ental m easurements of the suppression. P lo ts of the comparison betw een the experim ental m easurements of the suppression e versus Tie and the predictions from local and nonlocal theory. In (a) a = 0.47 and th e different symbols denote th e P -quartiles: 13.3 < o < 15.4 < • < 17.5 < □ < 19.6 < A < 21.7 T he theoretical lines are for the mean V = 16.3 of the d a ta . Likewise in (b) a = 0.64, 29.1 < o < 37.1 < • < 45.2 < □ < 53.2 < A < 61.2 and m ean V = 45.2.

92


film had a variable conductance such th a t, by use of Eqn. 4.2 and the measured value

of 773, 13.3 < V < 2 1 . 7 with a mean value V = 16.3. The d a ta in each quartile in

this range are p lo tted w ith different sym bols while a theoretical curve is calculated

for V = 16.3 by the m ethods of the local and nonlocal theories. In a similar m anner

the d a ta a t a = 0.64 had 29.1 < V < 61.2 with mean V = 45.2. Note th a t th e

ranges of H e for these two a are different by a factor of ten and th a t the suppressions

are also very different. It is quite astonishing tha t the shear suppresses the onset of

electroconvection to i ~ 14 i.e. H c( a .H e , V ) = 1572.°(a)! In term s of agreem ent

w ith theory, it is clear from both Figs. 4 .4a and b tha t the d a ta are in reasonable

agreem ent w ith both the local and nonlocal theories, perhaps som ewhat better w ith

the local theory. The reason for this is probably the divergence of the pertu rbed

charge density a t the electrodes; see Fig. 3.4. The experim ental system has electrodes

th a t despite having sharpened edges, m ust have a finite size and therefore would not

have a divergent charge density.[4] W hereas the nonlocal theory has divergences in

the charge density a t the electrodes, th e local theory has vanishing charge density a t

the electrodes. Everywhere else on the film the charge densities calculated by either

m ethod agree rem arkably well. In reality the perturbed charge density is finite and

therefore somewhere between these two extremes.

The suppression has also been studied a t a = 0.33,0.56, 0.60, and 0.80. The results

are sim ilar to those plotted in Figs. 4.4a and b. Overall, it appears th a t the linear

theory fares particularly well in predicting the suppressions a t various a and V for

a range of "Re. T he d a ta is constrained by m aterial param eters to 1 < V < 130 and

0 < H e < 3. T he upper bound on H e can be exceeded by higher ro tation rates bu t

the suppression is not expected to be different.

T he good agreem ent shown in Figs. 4 .4a and b is essentially independent of the

value of 773. Recall th a t 773 was determ ined by a single param eter fit to Eqn. 4.9.

Since the rj3 dependence in the H e scaling of both the theory (via V in Eqn. 3.20)

and the experim ent (according to Eqn. 4.3) are proportional to I / 773, any change in

773 m ultiplies bo th by the same factor. T his simply results in a rescaling of the H e

axis in Figs. 4.4a and b, w ith no change in the quality of th e agreement.

93


4.4 Coefficients of the Cubic and Quintic Nonlin

earity without Shear

W ith this Section begins the enum eration of experim ental results th a t perta in to the

weakly nonlinear properties of electroconvection in the absence of shear. T he variables

are th e radius ratio a and the Prandtl-like number V; however, experim entally V is

passive since it is not chosen beforehand but is simply m easured from the data.

I t varies proportionally to the conductivity. Since numerous experim ents have been

perform ed a significant range of V has been investigated. The prim ary purpose here is

to model the current-voltage characteristics by an am plitude equation and to deduce

the nature and 'streng th ' of the bifurcation from conduction to convection. The

basic procedure is to transform the current-voltage d a ta into e-A d a ta where e is

the reduced control param eter defined in Eqn. 4.4 and .4 = s / n is the am plitude of

electroconvection defined in term s of the reduced Nusselt num ber n in Eqn. 4.5. The

d a ta are then modelled by the am plitude equation given in Eqn. 4.6. T he am plitude

equation describes a pitchfork bifurcation, and the coefficient of the cubic nonlinearity

g is of param ount interest. The m agnitude and sign of g determ ine the 's treng th '

and nature of the bifurcation i.e. for g < 0 and |f/| <£ 1 the bifurcation is weakly

subcritical. A fuller sketch of how g is determ ined is given in Section 4.2 while details

o f the d a ta analysis procedure are given in Appendix E.

Figures 4.5a through c show the coefficient g as a function of the Prandtl-like num

ber V . The sca tte r th a t is manifest in these plots exceeds th e sta tistica l uncertainty

of the fit. As discussed in Sections 3.5 and 4.2, the scatter originates from system atic

effects due to the non-ideal features of the experiment. Nevertheless, since the scatter

appears to be w ithout trend, the gross features in Fig. 4.5 can still be ex tracted .

A t a = 0.33, th e measurements explored the range 2 < V < 8 . It is only a t

a = 0.33 th a t d a ta was obtained for V < 1 0 ; this 'constra in t' derives from the

P randtl-like num ber (see Eqn. 4.2) w herein the w idth of th e film d — ra — r, affects

V . It happens th a t for these experim ents d is greatest a t sm allest a and since V oc d~l ,

the Prandtl-like num bers are lowest there . As is clear from Fig. 4.5a, g is dependent

94


7

1 3 4 6 7

3

->

CO

0

p

30 35 40 45 50 55 60 65

Figure 4.5: Experim ental m easurem ents of the coefficient of the cubic nonlinearity in films w ithout shear. P lots of the coefficient of the cubic nonlinearity g versus V for annu lar electroconvection w ithout shear a t a = 0.33 (a), a = 0.47 (b), and a = 0.64 (c). T he solid line in (a) is obtained from a linear fit to th e d a ta while in (b) an d (c) th e line represents a weighted average.

95


radius ratio a Experim ental g Prandtl-like number V range Theoretical g0.33 -0.74 ± 0.23 2.1 < V < 4.40.47 1.64 ± 0.06 13.5 < V < 20.70.56 0.73 ± 0.15 59.4 < V < 100 .8

0.60 2.72 ± 0.34 31.3 < V < 38.90.64 1.87 ± 0.10 25.2 < V < 63.00.80 2.21 ± 0.29 15.3 < V < 142.8

1 .0 0 (‘p la te ’) V — oo 2.842

Table 4.2: Experim ental m easurem ents of the coefficient of the cubic nonlinearity, g w ithout shear.

on the V in the range shown and increases w ith increasing V . N ote th a t g = 0 at

V ~ 5. Hence a t a = 0.33. the bifurcation to electroconvection is subcritical (g < 0 )

for V ~ 5 and supercritical (g > 0 ) for V ~ 5. It is tricritical {g = 0 ) a t V = 5. For all

o ther a investigated, g was found to be independent of V. However, for each of these

cases. V > 1 0 . Exam ples of th is independence are plotted for a = 0.47 in Fig. 4.5b

and for a = 0.64 in Fig. 4.5c. T he preceding is also true for a = 0.56, 0.60 and 0.80.

In order to exam ine how g depends on a , the V dependence is removed by aver

aging. For large a, g is roughly independent of V over broad ranges of V . A weighted

average of g was obtained for these a . For a = 0.33. where some V dependence was

found, only d a ta in the narrow range 2 .1 < V < 4.4 was averaged. In doing this, the

system atic sca tte r is treated as random error. It is likely th a t the tru e uncertainty

in the average values of g will be much larger than the standard deviation of the

mean. T he u ltim ate justification of this procedure lies in the com parison of the aver

aged values of g and the theoretical predictions. As is shown below, th is comparison

is favourable. T he lines in Figs. 4.5b and c indicate the average value of g for the

p lo tted d a ta . These results are tab u la ted in Table 4.2 and are p lo tted in Fig. 4.6.

It is clear from Fig. 4.6 th a t, overall, g increases with a. A t present, no direct

com parison betw een theory and experim ental values of g is available for arb itrary

a. T he weakly nonlinear theory th a t would calculate g is in progress. [5] O n general

principles it is expected th a t g approaches a lim iting value as a —► 1 . T he lim it a —► 1

was discussed briefly in Section 3.4. This lim it corresponds to an unbounded lateral

96


4

3a = 2 . 0 , ' p l a c e '

1 5 . 3 < P < 1 4 2 . 8 g=2.842 .2

0

0.2 0 .4 0.6 0.8

a

Figure 4.6: Experim ental m easurem ents of the coefficient of the cubic nonlinearity in films w ithout shear. A plot of the coefficient of the cubic nonlinearity g versus o for annular electroconvection w ithout shear. T he value of g is averaged over a range in V in which it is independent. T he error bars show one standard deviation of the mean, trea ting the scatter in Figs. 4.5a, b and c as random . On the plot are annotated the range of V over which g was averaged a t each a.

geom etry in which the film is a strip of fluid suspended a t its long parallel edges by

two semi-infinite p late electrodes; consequently it was called the ‘p la te ' geometry. A

weakly nonlinear analysis of electroconvection in th is geom etry introduced in Ref. [6 ]

was successfully com pleted for the limiting case V = oo.[7] The result of th a t analysis,

g = 2.842 is encouragingly close to the experim ental value of g a t a = 0.80, g =

2.21 ± 0 .2 9 .

From linear theory th e critical param eters 72.° and m ° / f a t a = 0.80 is very close

to th e lim iting value for a = 1 . See Fig. 3.6 and Table 3.1. T he aspect ratio A

97


60 -

40

20

03 5 64 7 8

12

10

8

6

4

15 17 1814 16 19 20 21

oooo<£tP°o q, o

Figure 4.7: Experimented m easurem ents of the coefficient of the quin tic nonlinearity in films without shear. P lo ts of the coefficient of th e quintic nonlinearity h versus V for annular electroconvection w ithout shear a t a = 0.33 (a), a = 0.47 (b), and a = 0.64 (c).

98


0.05

0.04

0.01

CQ>

3 6 7 84 5P

Figure 4.8: T he size of the hysteresis 5e a t various V for a = 0.33. Since 0 < / 1 ,the size of the hysteresis Se = g2/4h when g < 0 .

is a m easure of the size of the system and is often the ratio of the longest to the

shortest length in the geometry. For annular electroconvection A is the length of

th e film divided by the w idth of the film. T h e length of the film is taken as the

circum ference of a circle w ith radius (r, + r a) / 2. T he aspect ratio A can be expressed

as A = 7t (1 + q ) / ( 1 —q). A t q = 0.80, A = 9ir. A lm ost all the experim ents performed

in the rectangular geom etry had A ~ 10 [8 . 9, 10, 11] and were thought to be well

m odelled by the theory for an unbounded la tera l s trip which has A = oo. W ith the

two foregoing reasons in m ind, it is valid to tre a t a = 0.80 as large and suitable for

com parison to theory a t q = 1 . From the d a ta it has been dem onstrated th a t g is

independent of V for V ~ 1 0 . See Fig. 4.5. N ote th a t from the governing equations

th e P randtl-like param eter appears as V ~ l so th a t the dependence on V dim inishes

rap id ly as V increases. Hence it has been justified th a t d a ta for V ~ 10 can be

com pared to theoretical results for V = oo.

99


C urrently experim ental d a ta a t a > 0.33 and V < 10 is unavailable; the regime

while not inaccessible would require larger electrodes. It would be interesting to de

term ine w hether g becomes negative a t large a as V decreases. All electroconvection

experim ents on freely suspended films in rectangular geom etry have reported super

critical bifurcations[8, 9], These experim ents were a t large V and so the possibility

of a subcritical bifurcation in rectangular films rem ains largely unexplored.

T he coefficient, h of the quintic nonlinearity in Eqn. 4.6 obtained from the mod

elling the d a ta is p lo tted in Fig. 4.7. T he plots are for the sam e param eters as those

given for g in Fig. 4.5. These plots are included for com pleteness and h by itself is

not in terpreted . Since 0 < / < 1 an approxim ate size of the hysteresis when g < 0

is given by die = g21 Ah. W hen g > 0. Se = 0. T he quantity die is p lo tted for various

V for a = 0.33 in Fig. 4.8. Note th a t the hysteresis vanishes for V ~ 5 and when

non-zero it is always sm all i.e. Se < 0.05.

4.5 Coefficients of the Cubic and Quintic Nonlin

earity w ith Shear

In the presence of shear, the coefficient of the cubic nonlinearity is strongly dependent

on the Reynolds num ber H e of the imposed C ouette shear. T he m ethods used to

determ ine g are as described in Section 4.4. As a representative exam ple of the He

dependence of g, consider Fig. 4.9 which shows the m easurem ents a t a = 0.47. At

H e = 0, the value of g is found, as described in Section 4.4, by averaging over a range

o f V . In the case shown in Fig. 4.9, g = 1.64 ± 0.06 for a m ean V = 16.3. The data

for H e > 0 had 13.3 < V < '21.7. This range is divided into quartiles and different

sym bols denote d a ta obtained w ithin the different quartiles. T h e m ajor result here

is th a t the n a tu re and streng th of the prim ary bifurcation to electroconvection can

be ‘d ialed’ by adjusting th e shear. The bifurcation is supercritical a t H e = 0 and

weakens as th e Reynolds num ber increases. T he bifurcation is tricritical a t H e ~ 0.2

and is subcritical for H e ~ 0.2. T he bifurcation becomes strong ly subcritical with g

100


2

1

0

2

3

a = 0.474

0 0.5 1 1.5 2 2.5

Re

Figure 4.9: Experim ental measurements of the coefficient of the cubic nonlinearity in sheared films. A plot of the coefficient of the cubic nonlinearity g versus TZe a t a = 0.47. T he different symbols denote the appropriate quartiles of V: 13.3 < o <15.4 < • < 17.5 < □ < 19.6 < A < 21.7 The value of g a t TZe — 0 is for d a ta with mean V = 16.3.

decreasing between 0.2 ~ TZe ~ 0.85 to a minimum value g = —3.7. For TZe > 0.85,

the bifurcation rem ains subcritical b u t here g is overall an increasing function of the

Reynolds num ber. For the range of Reynolds num bers investigated the bifurcation

does not become supercritical again.

There is some sca tte r in the data; nonetheless, the overall trends axe clear. The

sources of these system atic deviations are as described in Section 3.5. Note th a t the

system atic deviations are comparable to th a t in Figs. 4.5a, b and c. T he results are

sim ilar for a = 0.56,0.60,0.64 and 0.80. T he value of TZe a t which g = 0 is different a t

101


radius ratio a Reynolds num ber TZetricritical Prandtl-like num ber V range0.47 0.18 ± 0.02 15.8 < V < 16.60.56 0.03 ± 0.02 75.4 < V < 85.40.60 0.03 ± 0.01 30.3 < V < 31.70.64 0.08 ± 0.06 29.1 < V < 61.20.80 0.01 ± 0.01 65.5 < V < 70.6

Table 4.3: Experim ental m easurements of the Reynolds num ber for g = 0.

different a. This Reynolds num ber will be denoted TZetricntical. W hether the variation

of TZetricritical with a is purely a consequence of changing the radius ratio a o r is in

p a rt due to the different P randtl-like numbers is not clear from the data . Even though

in the absence of shear g was found to be independent of TP for V ~ 10, there is lim ited

experim ental data to draw any conclusions about the V dependence of g for TZe # 0.

If linear theory is to be any guide, one expects th a t in the sheared case there should

in general be a greater V dependence. Recall th a t in Section 3.4 it was established

th a t th e linear theory for TZe = 0 was independent of V while the presence of shear

in troduced a V dependence even a t the level of linear theory. Table 4.3 lists the

values of TZetrtcrttlcat. T heoretical work on the weakly nonlinear analysis of annular

electroconvection with shear is currently work in progress. [5] Consequently, there can

be no comparison with theory here. It appears from the tabu lated results th a t as V

and a decrease, the TZe for g = 0 increases.

radius ratio a

Minimum value of g

Reynolds num ber TZe

Prandtl-like number V

M axim um value of TZe

0.47 -3.68 ± 0.19 0.83 ± 0.18 15.3 2.590.56 -5.15 ± 1.04 0.11 ± 0.05 63.3 0.220.60 -1.74 ± 0.04 0.05 ± 0.02 32.1 0.130.64 -4.34 ± 0.79 0.23 ± 0.02 53.4 0.250.80 -9.17 ± 0.56 0.04 ± 0.01 12.0 0.10

Table 4.4: Experimental m easurem ents of the minim um value of g, the corresponding Reynolds and Prandtl-like num bers. The uncertain ty in TZe is related to the density of d a ta for each a.

102


a = 0.47

0 0.5 1 1.5 2 2.5Re

Figure 4.10: Experim ental m easurem ents of the coefficient of the quintic nonlinearity in sheared films. A plot of the coefficient of the quintic nonlinearity h versus TZe a t a = 0.47. T h e different symbols denote the appropriate "P-quartiles: 13.3 < o <15.4 < • < 17.5 < □ < 19.6 < A < 21.7.

T he m inim um values of g = g{TZe) are also different a t different a . Table 4.4

lists th e m inim um values assumed by g as a function of th e Reynolds num ber TZe.

A look a t Fig. 4.9 shows th a t the m inim um value is only approxim ately known. It

must also be noted th a t the minimum repo rted is a local minim um and investigation

is for ranges of TZe th a t are different for each a . The m axim um value of th e Reynolds

num ber investigated is also tabulated. Since the param eter space for th e d a ta is

defined by several param eters it becom es difficult to make m eaningful com parisons

of the experim ental results when more th a n one param eter changes. T h e experim ent

w arrants a greater degree of control so th a t the variables in the p aram eter space

103


can be changed one a t a tim e and more direct com parisons can be made. To w hat

ex ten t th is is experim entally feasible and to whether it is profitable will be discussed

in C hap ter 5. However, one fortunate comparison can be gleaned from Table 4.4. At

a = 0.47, th e film had V = 15.3 while a t a = 0.80, V — 12.0. Since the rad ius ratios

are very different and the V are not, it is not unreasonable to directly com pare the

m inim um values of g and the TZe for these cases. It is evident tha t the bifurcation

is much m ore strongly subcritical a t a; = 0.80 than a t a = 0.47. Also note th e very

different TZe for these minima.

A representative plot of the coefficient of the quintic nonlinearity h as function

of TZe is given in Fig. 4.10. In this plot a = 0.47 and 13.3 < V < 21.7. As before

different sym bols indicate d a ta obtained in different quartiles of the range in V . In

com paring Figs. 4.9 and 4.10 it appears that as g decreases, h increases and vice

versa.

As always / 1, so th a t when g < 0, the size of the hysteresis 5e is approxim ated

by g2/Ah. For g > 0. Se = 0. Figure 4.11 plots the size of the hysteresis 6e as a

function of TZe. Note th a t 8e is vanishing for TZe ~ 0.2 and non-zero for TZe ~ 0.3.

Unlike the variation of 5e as a function of V for TZe = 0 studied in the previous

section, the values of 8e here, are considerably larger. G iven the differences between

these two examples, one may loosely say that the shear-induced-hysteresis is larger

th an the P-induced-hysteresis in the absence of shear. W ith this Section te rm inates

the quan tita tive analysis of the data . For the rem ainder of this C hapter, th e results

described are of a more qualitative character.

4.6 Secondary Bifurcations

R esults th a t illustrate some of the properties of the secondary bifurcations in this

system m e collected in th is section. T he primary bifurcation is the transition from the

conducting s ta te to the convecting sta te or vice versa while the secondary bifurcations

refer to th e instabilities of one convecting state to another. T he current-voltage data ,

w hen transform ed into epsilon-reduced Nusselt d a ta (e, n ), is such th a t th e prim ary

104


0.25

0.2oj

CO

.2 0.15<*>■c<u<Z)

PC

0.05

a = 0.47am20 0.5 2.51 1.5

Re

Figure 4.11: The size of the hysteresis 5e as a function of the Reynolds num ber. In th is plot a = 0.47 and since 0 < / 1, the size of the hysteresis 6e = g2/4 h wheng < 0. T he different symbols denote the appropriate 'P-quartiles: 13.3 < o < 15.4 < • < 17.5 < □ < 19.6 < A < 21.7.

bifurcation is located a t e = 0. The secondary bifurcations then appear for e > 0.

In this system , the secondary bifurcations are transitions between flows w ith m

vortex pairs and flows w ith m ± n vortex pairs, m and n are integers and usually n = 1.

In the absence of shear, the only control param eter is t and the secondary bifurcations

th a t appear a t e > 0 are, when e is being increased, transitions between flows w ith

m ode num bers m and m + n. W hile e is being decreased the secondary bifurcations

are transitions between m + n and m mode states. T he two foregoing s ta tem en ts

follow from observations of the patterns, where it is seen th a t increasing e leads by

secondary bifurcations to higher m modes. T he current-voltage d a ta corroborates

105


these observations.

Figure 4 .12a plots (e. n) d a ta a t a = 0.33 in which a secondary bifurcation occurs.

From experience, it can be said th a t secondary bifurcations are seldom encountered a t

small q. In the case shown in Fig. 4.12a, the P randtl-like number is ‘sm all’, V = 1.92.

The secondary bifurcation appears a t e ~ 0.025 and is subcritical. From the results

of linear theory presented in Section 3.4 and from th e m easurem ents reported in

Table 4.1. it is know n tha t the critical mode num ber a t a = 0.33 is 4. T h e secondary

bifurcation show n in Fig. 4.12a is a 4 —> 5 transition . It is certainly naive to suppose

that the am plitude of the convection is proportional to the number of vortex pairs and

the velocity of th e fluid flow where th e vortices m eet, nevertheless note th a t (5 /4)2 =

1.56 while from Fig. 4.12a (0.056 ± 0.011)/(0.034 ± 0.010) = 1.65 ± 0.58. Such a

simple argum ent assumes tha t the radial inflows and outflows where the vortices meet

have approxim ately the same velocity ju s t before and after the secondary bifurcation.

Hence the am plitude must simply change in proportion to the num ber of inflows ?.nd

outflows and hence by the ratio 5 /4 . The reduced Nusselt number is proportional to

the square of th e am plitude and therefore to (5 /4)~. The location of th e secondary

bifurcation i.e. the value of e a t which the first secondary bifurcation occurs, say

e+, is dependent, for a given a and in the absence of shear, on only th e Prandtl-like

number.

Figure 4.12b shows (e, n) d a ta for a film at a = 0.80. W hereas secondary bifur

cations are o ften encountered a t large a they are, from current-voltage data , even

more plentiful a t small Prandtl-like numbers. In th e case illustrated V = 10.13,

which is as sm all a Prandtl-like param eter as any of the da ta at a = 0.80. Unlike a t

a = 0.33, there is a sequence of secondary bifurcations which, indicative of the order

in which they occur, are denoted e+ ,c++ etc. From the m easurem ents and linear

theory results reported in Table 4.1, the onset m ode num ber a t a = 0.80 is expected

to be 20. A ssum ing th a t the prim ary bifurcation a t e = 0 results in a convecting

state w ith m = 20 and tha t the secondary bifurcations increase th is m ode number

by 1 each tim e, so th a t a t e+ ~ 0.13 where the first secondary b ifurcation occurs,

there results a 20 —> 21 mode transition. Likewise a t e++ ~ 0.21, e+++ ~ 0.27,

106


0.08

0.06

C 0.04

0.02

0

-0.05 0 0.05 0.1e

0.16

0.14

0.12

0.1

0.08c0.06

0.04

0.02

0

0 0.2 0.4 0.6£

Figure 4.12: Representative plots of secondary bifurcations in films without shear. Plots of the reduced Nusselt number n versus the reduced control parameter e for experiments in which multiple secondary bifurcations are observed in annular electroconvection without shear. In (a) a = 0.33 and V = 1.92. In (b) a = 0.80 and V = 10.13.

107

0.118 O.j 10.

• Increasing e J

(a)

r 0.056

. 0.034

• Increasing e


e++++ ~ 0.37 and e+++++ ~ 0.43 there occurs the sequence of mode or vortex tran

sitions 21 —> 22 —y 23 —► 24 —> 25. T he bifurcations are once again subcritical and

the values of e+,e++ etc. a t which they occur are likely to be dependent on V .

T he secondary bifurcations in the sheared system have, unlike the case w ithout

shear, a w ealthy phenomenology. Figures 4.13a through c illustrate th ree repre

sentative exam ples of secondary bifurcations at m oderately diverse V , H e and a.

Figure 4.13a shows (e,n) d a ta for a sheared film dem onstrating subcritical prim ary

and secondary bifurcations. T he prim ary bifurcation appears a t e = 0 for increasing

e. T he secondary bifurcation, which results in one additional traveling vortex pair,

appears a t e+ ~ 0.22 for increasing e. W hen e is decreased, the removal of a trav

eling vortex pair occurs a t e_ ~ 0.12. Convection altogether ceases a t e ~ —0.1.

Figure 4.13b shows a case where, for increasing e, the prim ary and secondary bifur

cations coalesce into a single strongly subcritical bifurcation a t e ~ 0. However, for

decreasing e, e_ is distinct from e ~ —0.11 where convection stops. Finally Fig. 4.13c

illustrates a case w ith distinct prim ary and secondary bifurcations for increasing e but

e_ where the secondary bifurcation is ‘undone' is no longer present. Instead there is

a single transition from convection to conduction.

T h a t th e cases presented in Figs. 4.13b and c simply cannot occur in the absence

of shear is easy to understand. W hen shear is absent, there is for a given film a single

control param eter H , or equivalently the voltage V, or the reduced control param eter

e. C onduction is replaced by convection a t e = 0 with a integral num ber of vortex

pairs m; th e num ber depends on the radius ratio a . Upon further increasing e, the

first secondary bifurcation occurs a t e = e+ > 0 which results in the vortex change,

m —► m + n , where the integer n is alm ost certainly unity. It m ay occur th a t n > 1 a t

large a , say a ~ 0.85. The second secondary bifurcation occurs a t e = e++ > e+ and so

on for subsequent bifurcations; the corresponding vortex transition is m + 1 -* m + 2.

It follows th a t for a given film, the values e+,e++ etc. form an increasing sequence.

These values depend only on the m aterial param eters and geom etry which, ideally,

for a given film are constant. Hence the case illustrated in Fig. 4.13b where e+ = 0

cannot occur. An argum ent along the sam e lines as given above can be applied to

108


0.1A Increasing e T Decreasing e

* Increasing e* Decreasing e

........................ i . . . . ..............................................................

- 0.2 - 0.1 0 0.1 0.2

e

a Increasing e ▼ Decreasing e

- 0.2 - 0.1 0 0.1 0.2

£

Figure 4.13: Representative plots of secondary bifurcations in films w ith shear. P lots of th e reduced Nusselt num ber n versus the reduced control param eter e for experim ents in which secondary bifurcations are observed in annular electroconvection w ith shear. In (a) a — 0.47, V — 15.30 and Tie = 0.94. In (b) a = 0.47, V = 21.68 and Tie = 0.58. In (c) a = 0.64, V = 53.43 and Tie = 0.23.

109


sequence of removal of vortex pairs. In brief e = e— > e_ > ecm v^ catld is a decreasing

sequence th a t dem arcates the vortex transitions m + 2 —> m + 1 —> m —> conduction.

Hence the case illustrated in Fig. 4.13c cannot occur. It is concluded th a t in the

absence of shear, th e relative stability of different m modes is determ ined.

T h a t shear a lters this hierarchy of relative stab ility is well established, see Fig. 3.7.

In Section 3.4 it was dem onstrated th a t as R e is increased, the m arginally unstable

m ode decreases from m ° —> — 1 —> m° — 2 . . . Hence there is a particu lar Reynolds

num ber, say 7£e0,_i a t which bo th m°c and m° — 1 are marginally unstable. T hen

for TZe < TZeQ _ x th e prim ary bifurcation occurring a t e = 0 is to a convective sta te

described by a m ode num ber m = rn°. T he first secondary bifurcation occurs a t

e = e+ > 0 to a s ta te w ith m = m Qc + 1. For 'Re > R e 0 ^ { the prim ary bifurcation

occurring a t e = 0 is to a convective s tate described by a mode num ber m = rnac — 1.

T he first secondary bifurcation occurs a t e = e+ > 0 to a s ta te w ith m = m°c. Clearly

it follows th a t as R e R e o,-! from above, e+ —► 0, resulting in the situation depicted

in Fig. 4.13b. T he case illustrated in Fig. 4.13c can likewise be explained.

T h a t shear merely alters the relative stab ility of different convective s ta tes is

insufficient to explain the behavior of the secondary instabilities. More precisely, and

it cannot be over em phasized, it is the fact th a t the shear flow selects convective

s ta tes of a lower m ode num ber while the electrical driving favors convective s ta tes

w ith a higher m ode num ber. M athem atically, increasing the electrical driving or e

results in mode transitions m —v m + l —► m + 2 . . . whereas, increasing the shear or

R e results in transitions of the onset mode from m —> m — 1 —> m — 2 . . .

N either the experim ent nor the theory has addressed the question of w hat mode

transitions occur as R e is altered for a convective s ta te w ith m ode m a t constan t e.

I t has been the protocol to always hold R e constan t and vary e, nevertheless, it seems

certain th a t as R e is increased the mode transitions are m —* m — 1 —> • • • —v l —>

conduction. A nd as R e is decreased the m ode transitions are m —> m -f-1 —> • • • —

m (R e = 0). T he phenomonology described for the secondary bifurcations in the

sheared system can, in principle, occur in system s w ith two control param eters where

the increase of one control param eter results in selecting certain convective s ta tes

110


while the increase of th e o ther leads to selecting o ther convective states.

Figures 4.14a through h present a TZe sequence of {e . n) d a ta plots a t a = 0.56

and V = 75.80 ± 0.78. These da ta , unlike the d a ta discussed before, were obta ined

a t atm ospheric pressure. This example is selected due to th e minimal d rift in the

electrical conductivity. It is expected th a t qualitatively the d a ta closely resembles

th a t obtained a t reduced am bient pressure. In Fig. 4.14a. the (e.n) d a ta a t TZe =

0.124 depicts a subcritical prim ary bifurcation from conduction to convection with

m ode num ber 7. In the range of e investigated, secondary bifurcations were not

encountered. In Fig. 4.14b. the {e . n) d a ta a t TZe = 0.142 illustrates a subcritical

prim ary bifurcation from conduction to convection with m ode num ber 6 followed by

a subcritical secondary bifurcation a t e+ = 0.26. Hence a t som e Reynolds num ber,

0.124 < TZe < 0.142, there is nascent a t e = 0 the mode change 7 —> 6; whereas

this is expected from linear theory, the quantita tive agreem ent as to the exact TZe a t

which th is occurs is currently lacking. Perhaps this was a consequence of the inherent

air drag. W hen the experim ent is repeated a t the higher TZe = 0.160. the secondary

bifurcation is suppressed to higher e+ = 0.48, as shown in Fig. 4.14c. At TZe = 0.178

and TZe = 0.196, the secondary bifurcation is beyond the e range th a t was investigated,

see Figs. 4.14d and e. T he sequence of events encountered in Figs. 4.14a through

e repeats w ith increasing TZe. In Fig. 4.14f. the {e . n) d a ta which was ob ta ined a t

TZe = 0.214 depicts a prim ary bifurcation to a ro tating wave s ta te with m ode num ber

5 and a secondary bifurcation a t e+ = 0.07 to a s ta te with m ode num ber 6. W hen

the experim ent is repeated a t higher Reynolds numbers, TZe = 0.231 and TZe = 0.249,

the secondary bifurcation is suppressed to e+ = 0.11 and e+ = 0.16 respectively, as

is shown in Figs. 4.14g and h. Note th a t the ra te a t which th e secondary bifurcation

to m ode m = 6 was suppressed as a function of TZe is greater th a n the ra te a t which

the secondary bifurcation to mode m = 5 was suppressed.

O n analyzing the d a ta obtained while e is being reduced, one finds equally in ter

esting behavior. In Figs. 4.14a, d and e, there is a single discontinuous transition

from the convecting s ta te described by m = 6 or m = 7 to th e conducting sta te .

T h is is not unusual since th e increasing e d a ta also were characterized by prim ary

111


n

Re = 0.124

Re = 0.142

Re = 0.160

Re = 0.1781 1 1 1 1

Figure 4.14: A representative plot of a sequence of bifurcations. P lo tted are the reduced Nusselt num ber n versus the reduced control param eter e a t a = 0.56 for a sequence of increasing TZe. T he open(filled) triangles denote d a ta ob ta ined while increasing(decreasing) e. In (a) through (d) a subsidiary bifurcation appears a t e = 0 as TZe increases, replacing the existing bifurcation which eventually disappears.

112


n

Re = 0.196

0.4 - (f)

Re = 0.214

Re = 0.231

Re = 0.249

Figure 4.14: A representative plot of a sequence of bifurcations. In (e) through (h) ano ther subsidiary bifurcation appears a t e = 0 as H e increases, replacing th e existing bifurcation. For plots (a) through (d) in Fig. 4.14 and for plots (e) th rough (h) above, V = 75.80 ± 0.78

113


bifurcations only. In the cases where secondary bifurcations were encountered, sev

eral scenarios occur while e is decreased. For instance in Figs. 4.14b and c, there is a

continuous transition from m = 6 —> 7 before a discontinuous transition to conduc

tion. Note however, th a t for exactly these runs the d a ta taken when e was increasing

show discontinuous secondary bifurcations yet when w hen e decreases, the events re

verse continuously. W hile the d a ta w ith increasing e in Fig. 4.14f clearly depicts a

m — 5 —» 6 bifurcation, th e da ta w ith decreasing e carries no inkling of the secondary

bifurcation. By a single discontinuous transition the convecting m = 6 s ta te directly

exits to conduction w ithout recourse to the m = 5 s ta te . However, as 72e is increased,

the m = 5 s ta te is visited while e decreases, see Figs. 4.14g and h. W hat is differ

ent here from the cases in Figs. 4.14b and c is th a t th e transition m = 6 —> 5 is

discontinuous. It is expected tha t if th e experim ents were perform ed to even higher

driving more secondary bifurcations m ay well be encountered. These phenom ena are

likely to be explained by considering th e pitchfork bifurcation diagram s for th e modes

m and m — 1. for one can quite easily imagine how two bifurcation pictures can be

m anipulated to result in the variety discussed.

A conceptually useful exercise, in the field of p a tte rn form ation and nonlinear

dynam ics, is to ‘m ap ou t param eter space'. By this, it is m eant th a t one should

determ ine and describe the solution or solutions th a t exist in regions of param eter

space. T he task becomes very difficult when there are m any param eters on hand.

In th is system , a m inim al description of the solution consists of identifying the value

of m ode num ber m . A com plete description would additionally require th e traveling

ra te and am plitude.

T he param eter space for each solution consists of th e ranges of the rad ius ratio

a , th e Prandtl-like num ber V , the control param eter 72 and the Reynolds num ber

72e in which it persists. Since a is merely a geom etrical constant one m ay m ap

param eter space a t several different a . Ideally, in the absence of drift of th e electrical

conductivity, V too is constant and therefore one may m ap param eter space a t fixed

a an d V . To the ex ten t th a t this m apping is possible is illu stra ted in Fig. 4.15. In this

figure, th e abscissa is the Reynolds num ber 72e of th e shear flow while th e o rd inate

114


7

6

5

4iOJ

3

2

1

0

Figure 4.15: An exam ple of a m apping of param eter space. In this plot a = 0.56 and V = 75. T he d a ta points consist of the location in (e, IZe) space of the transitions: conduction —>■ convection, convection —> convection, convection —> conduction. D ata gleaned from Figs. 4.14a through h axe indicated by the appropria te letters. T he conduction region is below the dashed line. Electroconvection occurs above the solid line. Convection sta tes w ith m = 7, m = 6 and m = 5 are appropriately indicated. T he ‘?’ im ply th a t the precise location where the various lines intersect is no t known.

115

y i i i | i 'T r" i ; i imir i1111!1 Mi ^ > i | i i i i | i i_ O conduction to m = 7- □ conduction to m = 6 /- A conduction to m = 5 /

m = 6 to m = 7 m = 5 to m = 6 m = 7 to conduction m = 6 to conduction m = 5 to conduction m = 7 to m = 6 m = 6 to m = 5

0 0.05 0.1 0.15 0.2 0.25Re


is the suppression variable i defined in Eqn. 4.10; note tha t i is proportional to 1Z.

T h e d a ta is obtained from experiments perform ed a t a = 0.56 and V = 75. The data

consists of the values of i a t which, for a given 1Ze. the transitions from conduction

—► convection, transitions between different convecting m states and transitions from

convection —► conduction occur. The transitions between convecting s ta te s are the

m —► m + 1 and the m + 1 —>■ m transitions which when they do occur, occur upon

increasing and decreasing e respectively. Some of the da ta can be identified in the

d a ta presented in Figs. 4.14a through h and are denoted as such on the plot. Consider

the d a ta obtained from Fig. 4.14h. By increasing e and so e, conduction persists until

6 = 0 o r f = 5 where A in Fig. 4.15 a t IZe = 0.249 denotes the transition from

conduction to electroconvection with m = 5. U pon further increasing i a secondary

bifurcation m = 5 —> 6 occurs and is denoted by V- While decreasing i mode

m = 6 persists until the transition m = 6 —> 5 which is denoted by th e solid upside

down triangle a t i = 4.2. Upon further reducing e, convection gives to conduction at

I = 3.7 and is denoted by a solid upright triangle. It is in this m anner th a t the data

in Fig. 4.15 is to be interpreted.

T he lines in Fig. 4.15 simply connect relevant d a ta points giving the figure the

appearance of a foliate. Below the dashed line, th e only observed s ta te was conduc

tion. Above the solid line the only observed s ta te consisted of electroconvection under

shear. Between these lines both conduction and electroconvection are observed.- The

fact th a t these two lines exist imply the incidence of hysteresis or subcriticality. The

onset of subcriticality occurs where the dashed and solid lines intersect. Since the

intersection is imprecisely known, it is denoted by 'T . For d a ta to th e left of this

point of intersection, th a t is for data a t lower 7Ze the bifurcation betw een conduction

and convection was supercritical and hysteresis was not observed. To th e right of the

intersection the bifurcation is subcritical and hysteresis was observed.

In th e electroconvecting regime, the description of the convecting s ta te is specified

by the m ode num ber to. In Fig. 4.15 there axe three m states. T h e prim ary and

secondary bifurcations are used to identify th e m sta te . The secondary bifurcations

m —> m + 1 and th e transitions m + 1 —> m are denoted by different symbols. A

116


do tted line dem arcates the region between the m = 6 and m = 5 while a hashed line

denotes the boundary between the m = 7 and m = 6 sta tes. Once again these lines

are approxim ate and only connect the available d a ta points.

As was discussed in Section 4.3, the solid line is m easure of the suppression of

the onset of electroconvection by the shear. More acutely the solid line represents

the suppression of the prim ary onset mode. The hashed line is a m easure of the

suppression of the mode m = 7 w hether the mode appears a t a prim ary or a secondary

bifurcation. Note th a t the suppression is greater and increases rapidly w ith "Re when

the m ode m = 7 is a secondary m ode than when it is a prim ary mode. T he dotted

line concerns the suppression of the mode m — 6 and once again when m — 6 appears

as a secondary bifurcation it is further suppressed and the ’ra te ' of suppression is

greater th an th a t of the prim ary mode. Finally note th a t the ’ra te ' of suppression of

the secondary mode m = 7 is greater than th a t for the secondary mode m = 6.

T he intersections between the convection s ta te boundaries for the secondary bifur

cations and the convection sta te boundary for the prim ary bifurcation are imprecisely

known and are denoted by *?\ A t th is point, it rem ains unknow n w hether the bound

aries between convective states are single or supercritical or double and subcritical.

It is likely th a t the la tte r applies and it is clear th a t more work on this system is

required to further elucidate the picture. So it rem ains unclear w hether o ther m

s ta tes occur in the region denoted by, for example, m = 6 ,... However, it is w ithout

question th a t convective states persist above the solid line while conduction persists

below the dashed line. Between these lines or equivalently in the hysteresis of the

prim ary bifurcation, one may have, as has presently been observed, as many as three

possible s ta tes (see for instance Figs. 4.14b, c, g, and h). T h a t is not to say coexistent

s ta tes bu t three d istinct possibilities. One is the conduction s ta te , while the others

are convective sta tes w ith mode num bers m-F 1 and m. It is an open question w hether

m ore s ta tes can be found w ithin the hysteresis of the prim ary bifurcation. Armed

w ith the predicted trends from th e linear theory and w ith experience from experi

m ental work, one may be able to guess as to how th e p o rtra it presented in Fig. 4.15

changes as th e radius ratio a and the Prandtl-like num ber V are varied. Nevertheless

117


an ac tual determ ination of these trends is a t present inadequately explored.

T he foregoing exhausts, for this thesis, the investigation into the s truc tu re of the

secondary bifurcations, the m anner in which they occur and the ir loci in param eter

space. T here is clearly plenty of scope for further investigation. In the following

Section are grouped three cases of bifurcations which have been encountered and

w arrant special com m ent.

4.7 M iscellany

In the course of perform ing current-voltage m easurem ents on a m ultitude of annular

films, there were som e d a ta th a t illustrated certain features th a t were absent in th e

d a ta th a t has already been discussed. Exam ples of these d a ta and the conditions

under which they were observed or not observed are presented here.

F irst, a t q = 0.80 and in the absence of shear, the current voltage-characteristics

illustrate w hat appear to be several secondary bifurcations, see Fig. 4.16a. T he pri

m ary and secondary transitions are shown by arrows. The prim ary and secondary

bifurcations appear to be continuous. W hen scaled into (e, n) coordinates, the d a ta

takes the form shown in Fig. 4.16b. The prim ary bifurcation appears to be super

critical bu t the secondary bifurcations appear to be only weakly subcritical. I t is

clear th a t greater resolution of the experim ental d a ta is required. It becomes d eb a t

able w hether the secondary bifurcations are always subcritical. By virtue of being in

a finite system , it is likely th a t all secondary bifurcations are subcritical w ith sub

criticality getting weaker as a gets larger, all o ther things constan t. T he secondary

bifurcations necessitate a change from m —> m 4-1 vortex pairs in the absence of shear

or change in th e m ode s truc tu re of the ro ta tin g wave when sheared. Such a change

is a discontinuous change in system of finite length since the transition m —► m 4-1

has no in term ediate states. In a system of infinite length, th e m mode can be shrunk

continuously to the m 4- 1 mode. Consequently it is postu la ted th a t all secondary

transitions are subcritical. The degree of subcriticality decreases w ith increasing a

an d d isappears in th e lim it a —> 1. Note th a t a t a = 0.80, th e lim its of experim ental

118


10

9

Q.

8

7

6Increasing voltage


0.2

0.15

c

0.05

Increasing e

- 0.2 0.20 0.60.4e

Figure 4.16: M ultiple bifurcations a t a = 0.80. In (a) is p lo tted a current-voltage characteristic which is anno tated w ith arrows th a t indicate th e positions of several bifurcations. These bifurcations appear to be continuous. In (b) is p lo tted the corresponding n versus e graph. Arrows indicate th e positions of th e bifurcations which w ith the exception of the first, appear to be only weakly subcritical.

119


6.5

Q.5.5

Increasing voltage _ Decreasing voltage

4.5

2524 26 27 28 29 30Voltage (volts)

Figure 4.17: Backward and forward bifurcations in the same experiment. A plot of the current-voltage characteristic th a t shows a subcritical bifurcation for increasing voltage but a supercritical one for decreasing voltage.

resolution of the subcriticality are already being tested.

Figure 4.17 is a representative example of a current-voltage characteristic th a t

shows a discontinuous bifurcation from conduction to convection but a continuous

transition from convection to conduction. W hen prim ary subcritical bifurcations

are encountered, the transitions to and from the electroconvective state have alm ost

always been discontinuous. This is in keeping w ith the current understanding of

the subcritical pitchfork bifurcation. However, d a ta like those shown in Fig. 4.17

question w hether a bifurcation can be subcritical for increasing a control param eter

and appear supercritical for decreasing the control param eter. More likely, there

are ‘d irection7 dependent or equivalently tim e-dependent processes that lead to these

differences. W h at these processes may be are not known bu t they probably arise from

electrical effects th a t depend on the tim e-dependence of the applied voltage i.e. is

the voltage increasing or decreasing and if so a t w hat average rate? D ata of the like

120


p lo tted in Fig. 4.17 were observed a t several a and V and often a t small H e ~ 0.1.

Interestingly, such current-voltage characteristics were never observed in the d a ta

obtained a t atmospheric pressure. However, it is not conjectured from this th a t th e

anom aly is an air-drag effect, but rather the absence of air and the accom panying

w ater vapour results in different electrical properties in the film.

In some experiments w ith strong shear, th e current-voltage characteristic took

the form shown in Fig. 4.18. Such characteristics were not observed in the absence

of shear. In this plot, a linear fit to the conduction d a ta for increasing voltages

in the range 85.0 < V < 93.0 is superim posed on the data . T he ex trapo lation of

th is line makes clear th a t th e da ta for increasing voltages disagrees w ith the line by

more th an tha t which would be expected from the drift of the electrical conductivity.

O ne then questions w hether the anticipated subcritical bifurcation is preem pted by a

supercritical bifurcation a t a lower voltage. If so. does one in terpret the discontinuous

transition as a secondary bifurcation? It is m ore likely th a t the deviation of the d a ta

from the line is a result of sudden large changes in the electrical conductivity th a t arise

for reasons not known bu t different from the electrode reactions th a t account for the

drift in the electrical conductivity. Consider the scenario, in reference to Fig. 4.18,

when the voltage is increased to V = 94.5 volts, the film is m arginally unstab le

to electroconvection and a velocity fluctuation is amplified. T he flow th a t appears

results in mixing the fluid which somehow im m ediately results in an increase in th e

electrical conductivity. Suppose tha t the conductivity is sufficiently increased th a t

the film is no longer m arginally unstable to electroconvection a t V = 94.5 volts. U pon

fu rther increase of the applied voltage, the process repeats until a t V = 98.7 volts

the increase in the electrical conductivity is insufficient to prevent electroconvection

and a transition to convection occurs. This plausible scenario leads to referring to

the bifurcation in Fig. 4.18 as a ‘delayed’ bifurcation. It is likely th a t if the electrical

conductiv ity was constant independent of the flow in the film, th a t the transition

to convection would have occurred a t V = 94.5 volts. A t th is stage, the foregoing

explanation should be trea ted as a guess and only further research on th is aspect will

supply the true reasons.

121


Increasing voltage Decreasing voltage

86 88 90 92 94 96 98 100 102Voltage (volts)

Figure 4.18: A 'delayed' subcritical bifurcation. A plot of the current-voltage characteristic th a t shows a subcritical bifurcation which appears to have been 'delayed’. The line shows a linear fit to the d a ta for 85.0 < V < 93.0 volts. The da ta in the range 94.0 < V < 98.0 disagrees w ith the line by an am ount th a t cannot be accounted for by sim ply electrochemical drift of the conductivity.

4.8 Other Similar System s

At several levels of observation annular electroconvection resem bles widely different

systems. In an effort to elucidate these sim ilarities, brief a lbeit im portant contrasts

and com parisons between the system studied in this thesis and its most similar coun

terparts have been collected in this Section.

O f the several systems th a t will be discussed below, the first th a t is considered is

th a t of the rm al convection superposed on a shear flow. T here have been some the

oretical stud ies of 3D Rayleigh-Benard convection (RBC) in th e presence of a plane

Couette shear flow[12]. The canonical theoretical geometry of a fluid layer confined

between perfectly conducting flat horizontal planes of infinite ex ten t is assumed. A

finite horizontal extent has significant im plications and is discussed below. L inear

122


stab ility analysis of the plane C ouette base s ta te to RBC reveals stability differences

between transverse roll disturbances (w ith axes perpendicular to the shear flow), and

longitudinal roll d isturbances (w ith axes parallel to the shear flow). Longitudinal-roll

d isturbances have identical stab ility properties to RBC in the absence of shear, and are

always m ore unstable th an the transverse-roll disturbances. In fact the longitudinal-

roll d isturbances have stabiUty properties th a t are independent of any uni-directional

shear flow along the axis of these rolls. Transverse-roll disturbances, conversely, ex

h ib it suppression, or added stab ility due to the shear, plane Poiseuille or plane C ouette

or any m ixture of these two flows. The onset Rayleigh num ber for transverse rolls is

a m onotonically increasing function of the shear Reynolds num ber, similar to w hat

was found for 2D annular electroconvection. Furtherm ore, the critical wavenumber of

the m ost unstab le transverse d isturbance was found to be a m onotonically decreasing

function of the shear Reynolds num ber, as was observed in annular electroconvection

w ith shear. Transverse rolls (vortices, in the co-rotating frame) appear a t onset in

th e annular electroconvection system w ith circular C ouette shear. This is perhaps,

in large p art because it is a 2D system . Like annular electroconvection the presence

of shear converts the em erging roll-state from stationary to traveling.

W hereas RBC w ith plane C ouette shear has not been, perhaps cannot be, stud ied

experim entally, RBC has been studied experim entally and theoretically w ith open

through-flows.[13, 14] T he through-flow is generally a weak Poiseuille flow w ith a

very small Reynolds num ber. Its effects on RBC are well understood. In brief, the

onset of convection is again suppressed, bu t the first instability is convective (i.e.

it grows only dow nstream of a localized pertu rba tion), ra ther th an absolute. T he

resulting convection p a tte rn drifts in the direction of the through flow. It is interesting

th a t the clear d istinction betw een convective and absolute instability is blurred in

annular electroconvection w ith shear, in which th e ‘th rough5 flow loops back on itself.

T he annular geom etry is natu ra lly closed. W hether longitudinal or transverse roll

d isturbances grow in RBC w ith weak through-flows depends on th e w idth of th e

channel and the Reynolds num ber of the flow. [14] T he finite horizontal extent of th e

channel is seen to afford, in p art, a selecting m echanism betw een longitudinal and

123


transverse disturbances.

A system w ith geom etrical sim ilarity to annular electroconvection is the ex ten

sively studied is Taylor Vortex flow (TVF).[13] The similarity is in the annular ge

ometry of bo th T V F and the electroconvection system discussed in this thesis. T his

however, is where the similarity ends. T V F depends crucially on the instability to

3D disturbances, in fact the 2D circular C ouette flow is linearly s tab le .[15, 16] It is

im portant to note th a t TV F is a result of an instability of a shear flow, while w hat

is studied here is the effect of a shear flow on the electroconvectional instability.

Some electrohydrodynam ic system s consist of an •insulating’ fluid confined be

tween metallic electrodes. Charge injection, a process by which charge carriers are

created a t the electrodes, occurs when strong electric fields are applied. It is th e

interaction of th is volume charge density w ith the applied electric field that leads to

electroconvection type instabilities; see Ref. [17] and references therein. Agrait and

Castellanos have theoretically studied the effect of a Couette shear on an electrohy

drodynamic instability in TV F geometry. [18] They considered electroconvection due

to a radial field w ith charge injection on e ither cylinder. D oth cylinders were per

mitted to ro ta te to produce a general C ouette shear. Their resu lt was tha t shearing

enhanced the instability, leading to a 3D flow th a t resembles T V F . Recall th a t the

shear suppresses the 2D annular electroconvection flow and so is in sharp con trast

with this charge-injection driven instability.

The results of RBC in rotating cylinders have some sim ilarity to the results for

electroconvection in sheared films, even though shear and ro ta tion are quite different.

A concise sum m ary of some of the work on ro tating RBC can be found in Ref. [19].

Due to the sim ilar sym m etries in these various scenarios, some results are com m on

to most ro ta ting RBC system s.[19] Since these systems are three-dim ensional, the so

lutions they supp o rt can generally be classified into axisym m etric. non-axisym m etric

(‘columnar’) or mixed (combinations of axisym m etric and non-axisym m etric) solu

tions. The principal result is tha t for 3D m ixed solutions, the onset bifurcation is no

longer steady as it is for non-rotating RBC. This leads to a flow p a tte rn th a t precesses

in the co-rotating fram e. The onset of these solutions is however suppressed, raising

124


the critical Rayleigh num ber above its non-rotating value.[19. 20, 21] In this aspect

the effects of shear in annu lar electroconvection are similar to th a t of ro ta tion in

RBC. In principle, purely non-axisym m etric or colum nar solutions which are stric tly

two-dimensional can occur in RBC under ro tation . W hen they do occur, they do not

precess (in th e co-rotating fram e) and their onset occurs at the sam e critical Rayleigh

number as in th e absence of ro ta tio n .[19, 22]

There have also been theoretical studies of the interesting b u t experim entally un

realizable s itu a tio n of 2D RBC in a ro tating annular geom etry w ith purely radial

gravity and heating[22]. T hese studies found similar colum nar solutions. In fact,

purely co lum nar solutions (‘Taylor colum ns'[23]) have yet to be observed in any ro

tating RBC experim ent since the boundary conditions a t the top and bottom of the

cylinder m ust be stress free[19, 22], a requirem ent tha t cannot be attained in te r

restrial RBC experiments. In contrast, two-dimensionality, stress free end boundary

conditions an d radial driving forces all arise natu rally in the electroconvection of an

annular suspended film th a t has been described in this thesis. As a result the vor

tices th a t occur in annular electroconvection w ithout shear which are identical to

those th a t occur when the annulus is rigidly ro ta ted take the appearance of ‘Taylor

columns'. T hey are in fact th e 2D analog of ‘Taylor columns'.

The added stability in sheared annular electroconvection is a consequence of the

shear and no t of rotation. U nder rigid ro tation , where the inner and outer elec

trodes are co-rotating, one can transform to ro ta ting co-ordinates in which the elec

trodes are stationary . This transform ation introduces a Coriolis term — 2Qz x u =

—2 in Eqn. 3.5 which m ay be absorbed into the pressure gradient term V P and

elim inated.[22] Thus, in a purely 2D system, rigid rotation and the non-rotating, un

sheared case have identical stability . It also follows, since the transform ation is general

and the unsheared bifurcation is stationary, th a t the resulting nonlinear vortex p a t

tern above onset must be s ta tio n ary in the co-rotating frame. These results are very

similar to th a t for the ‘Taylor colum ns’ m entioned earlier. T h is lack of dependence

under rigid ro ta tio n may be con trasted with a large class of 3D and quasi-2D ro ta tin g

Rayleigh-Benard systems[19, 20, 21, 24], where ro tation produces added stability bu t

125


the absence of stric tly 2D flow results in tim e-dependence (precession) o f the convec

tion pattern in th e co-rotating frame. Chandrasekhar[25] treated the classic problem

of the linear s tab ility of RBC in a laterally unbounded layer rotating ab o u t its normal.

T he case of a laterally bounded cylindrical layer has received much recent interest

theoretically[20]. and has also been the subject of a precise experim ental study[2l].

Electrically driven convection phenomena have been observed a n d extensively

studied in nem atic liquid crystals for many years. C onsult Ref. [26] and th e ex ten siv e

bibliography there in for a detailed review. The convection in these "parallel plate ca

pacitor like’’ nem atic systems occurs due to an electrohydrodynam ic instab ility whose

origin is entirely different from th a t discussed in th is thesis. The m echanism depends

crucially on the anisotropic properties of the nem atic liquid crystal, in particu lar the

electrical conductivity. For an introduction to this, the Carr-Helfrich m echanism , see

Ref [27]. W hereas the m echanism is electrical. Carr-Helfrich electroconvection has

very little in sim ilitude to the surface driven electroconvection m echanism th a t has

been presented in this thesis.

Surface driven electroconvective phenomena have been observed an d theoretically

explored in wide variety of experim ental scenarios ranging from puddles of conduct

ing fluids to partially filled capacitors.[28, 29, 30, 31] However, th e s tu d y of elec

troconvection in freely suspended fluid films began w ith S. Faetti et. al. [32] who

experim ented w ith nem atic liquid crystal films suspended between paralle l wires in

the trad itional rectangular geometry. Their observations of vortical flow are indeed a

result of the sam e surface driven electrohydrodynam ic instability discussed in this the

sis. C ircum venting the problem of nonuniformly thick and m etastable films. S.Morris

et. al. experim ented w ith electrically driven convection in smectic A liquid crys

ta l films.[9, 33] Having established the utility of th e 2D smectic A liquid crystals,

further experim ents into nonlinear electroconvective phenom ena in rec tangu lar cells

have been perform ed. [34, 35] T he phenomena th a t are observed in each of these

cases are consistent w ith each o ther though quantitatively they differ. Furtherm ore,

the phenom ena are driven by the surface driven electroconvection m echanism th a t

is discussed in th is thesis and was originally presented for the la terally unbounded

126


geom etry in Ref. [36].

Freely suspended film of o ther smectic phases have recently been experim entally

investigated in rectangular and annular geom etries.[37. 38, 39] The smectic C and C*

phases have been used. T he smectic C phase has a layered structu re much like sm ectic

A, bu t w ithin layers, the long axes of the molecules are a t a fixed tilt, not perpen

dicular, to the layer plane. As a result the m ateria l properties, and therefore optical

properties, w ithin each layer are anisotropic. T he smectic C* phase is also endowed

w ith a layered s tru c tu re and much like sm ectic C. But unlike sm ectic C. the sm ectic

C* phase has a spontaneous polarization i.e. a perm anent electric dipole m om ent in

the layer plane. From layer to layer, the long axes of the molecules show chirality and

the polarization vector ro tates in the layer plane, however the tilt w ith respect to the

layer norm al is constant. Like smectic C, the smectic C* phase has anisotropic m a

terial properties w ith th e layer plane. The electroconvection phenom ena observed in

rectangular films of sm ectic C are consistent w ith the phenom ena observed in sm ectic

A films and w ith surface driven electroconvection theory. However, the observations

on electroconvection in sm ectic C* films differ from the experim ents on sm ectic A and

C films.[38] The sm ectic C* phase is characterized by a perm anent spontaneous po

larization. which can be altered by the extent of chirality in this phase, and in teracts

w ith the electric field in these experiments. T he effects of this polarization have been

studied theoretically in. supposedly, a freely suspended geometry. [40] Surface charges

which are invariably present in a freely suspended have been neglected in Ref [40].

As it stands, the theory advocated is dependent on bulk or volume effects and the

predictions have a qualitative resemblance to Carr-Helfrich theory. W hile the exper

im ental results on sm ectic C* films disagree w ith surface driven electroconvection, it

is not clear th a t the results are consistent w ith the theoretical predictions of Ref. [40].

Experim ents w ith sm ectic C* films have been recently perform ed in an annu lar

geometry.[39] Surface driven phenomena were clearly observed. It is now believed th a t

b o th surface and bulk effects are present and a crossover between bulk and surface

effects m ay be explored by varying the film’s thickness to w idth ratio. For a film

of thickness s and w id th d. the ratio s /d determ ines, roughly, how the bulk forces

127


com pare to the surface forces. For s /d <£. 1 the surface forces dom inate. M ost films

have s /d 1 , however, it is not unusual to alter s /d by a factor of 5 to 1 0 . W hile

surface driven electroconvection is accessible in films, it is not clear w hether the bulk

effects of the natu re described in Ref. [40] have been observed.

128


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[19] E. Knobloch, "Bifurcations in R otating System s,” in Lectures on Solar and Plan

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[20] H .F. G oldstein, E. Knobloch. I. M ercader, and M. Net, “Convection in a ro ta ting

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tion: asym m etric modes and vortex states,” J. Fluid Mech.. 249, 135 (1993).

"Asym m etric modes and the transition to vortex s tru c tu res in rotating Rayleigh-

Benard convection.” Phys. Rev. L ett. 67. 2473 (1991) : R. E. Ecke. F. Zliong,

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Rayleigh-Benard convection.” Europhys. Lett. 19. 177 (1992).

[22] A. Alonso. M. Net. and E. Knobloch, "On the transition to columnar convection.”

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[23] F. H. Busse, “Therm al instabilities in rapidly ro tating systems." J. F luid Mech..

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[24] D. Fulze. “Developments in controlled experiments on larger scale geophysical

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D. N. Jackson, and R. M . Small, "A comparison of labora to ry m easurem ents and

num erical sim ulations of baroclinic wave flows in a ro ta tin g cylindrical annulus.”

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[25] S. C handrasekhar, Hydrodynamic and Hydromagnetic Stability. Dover Publica

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[26] Pattern Formation in Liquid Crystals, edited by A. Buka and L. K ram er,

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[27] P. M anneville, Dissipative Structures and Weak Turbulence, Academic Press Inc.

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[29] VV. V. R. M alkus and G. Veronis. "Surface Electroconvection." Phys. F luids 4.

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[30] J.R . Melcher and G.I. Taylor. "Electrohydrodynam ics: A review of th e role of

interfacial shear stresses." Ann. Rev. Fluid Mech. 1 . I l l (1969)

[31] D.C. Jolly and J.R . Melcher. "Electroconvective instability in a fluid layer." Proc.

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[32] S. Faetti. L. Fronzoni. and P. Rolla. "Static and dynam ic behavior of th e vortex-


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[33] S. VV. Morris. J. R. de Bruyn. and A. D. May. "Electroconvection and P attern

Form ation in a Suspended Smectic Film." Phys. Rev. Lett. 65. 2378 (1990).

"P atte rns a t the onset of electroconvection in freely suspended sm ectic films.” J.

S tat. Phys. 64. 1025 (1991).

[34] S. S. Mao. .J. R. de Bruyn. and S. VV. Morris. "Electroconvection p a tte rn s in

smectic films a t and above onset.” Physica A 239. 189 (1997).

[35] S. S. Mao. J. R. de Bruyn. Z. A. Daya. and S. VV. Morris. "Boundary-induced

wavelength selection in a one-dimensional pattern-form ing system." Phys. Rev.

E 54. R1048 (1996).

[36] Z. A. Daya. S. VV. Morris, and J. R. de Bruyn. "Electroconvection in a suspended

fluid film: A linear stability analysis." Phys. Rev. E 55. 2682 (1997).

[37] A. Becker. S. Ried. R. Stannarius. and H. Stegemeyer. "Electroconvection in

sm ectic C liquid crystal films visualized by optical anisotropy." Europhys. Lett.

39. 257 (1997).

[38] C. Langer and R. S tannarius. "Electroconvection in freely suspended sm ectic C

and sm ectic C* films,” Phys. Rev. E 58. 650. (1998).

[39] C. Langer. Z.A. Daya, S. VV. Morris, and R. Stannarius. unpublished, (1999).

132


[40] S. Ried, H. Pleiner. VV. Zim m erm ann, and H. R. B rand. "Electroconvective

Instabilities in Sm C’ Liquid C rystal Films,” Phys. Rev. E 53, 6101 (1996).

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C hapter 5

C onclusions

5.1 Introduction

This C hap ter presents a sum m ary of the conclusions of this thesis. Also described

are the possible future directions for the experim ent and theory. For convenience, the

conclusions regarding the experim ent and the theory are separately discussed. Before

describing the detailed conclusions, the m ost significant results of th is research are

listed below.

(i) An electrohydrodynam ic model has been developed to describe annular electro

convection w ith shear. It was dem onstrated experim entally th a t th is model precisely

describes the phenom ena relating to the onset of electroconvection i.e. the predictions

of th e m odel a t the level of linear stability and the relevant experim ental m easure

m ents are in quan tita tive agreement. These consisted of the m arginally unstable

m ode m° and th e shear dependent suppression e. Given this success, it is expected

th a t a t th e level of weakly nonlinear analysis, the model will prove instrum ental in

explaining and further exploring nonlinear behavior.

(ii) I t has been dem onstrated th a t the n a tu re of the prim ary bifurcation in annular

electroconvection can be tuned from a supercritical through a tricritical to a sub

critical bifurcation by varying the radius ra tio a , th e Prandtl-like param eter V and,

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the Reynolds number 'Re. For nonzero R e the bifurcation also changes from s ta

tionary pitchfork to Hopf. For some param eter ranges, the transition between th e

same two ‘phases' or states of sym m etry i.e. conduction and a convection s ta te w ith

mode num ber m . can be supercritical, tricritical or subcritical. A complex nonlinear

regime characterized by subcritical secondary bifurcations between states with m ode

num bers m and m i l was briefly explored.

5.2 Conclusions: Experiment

The experim ental work system atically explored electroconvective flows in freely sus

pended. two-dimensional annular fluid films. T he principal exploratory tool was

current-voltage da ta under a variety of situations. The experim ents were initially

perform ed a t atm ospheric pressure, but were la ter more precisely and extensively

repeated in a reduced am bient pressure environm ent to elim inate air drag. In all.

six different radii ratios a . w ith 0.33 < a < 0.80, were investigated in both pres

sure environm ents. For all the a investigated, the Reynolds num ber of the C ouette

shear varied between 0 < R e < 3. Likewise, the Prandtl-like num ber varied between

1 < V < 150. The param eter space sampled by these a . V and R e is very broad and

sufficiently dense so tha t it is highly unlikely th a t any region of param eter space th a t

has been unexplored will have different phenomena.

C urrent-voltage characteristics were acquired for a large num ber of uniform films

of different thicknesses a t each a under varying conditions of applied shear. By a

d a ta analysis procedure, quan tita tive inform ation regarding the onset and beyond

onset am plitudes were gleaned from the current-voltage characteristics. Films a t

a = 0.33 were more prone to thickness variations than those a t a > 0.47. All

films were sim ilarly more prone to thickness variations a t reduced am bient pressure

than a t atm ospheric pressure. Experim ents a t a < 0.33 and a > 0.80 were not

a ttem p ted due to the increased likelihood of nonuniform aspects due to a ‘■broad’ and

‘narrow ’ film w idth respectively. A broad film is prone to thickness variations while

a narrow film is influenced by th e three-dim ensional aspects of the electrodes which

135


are proportionately greater in a narrow film.

T he onset phenom ena concern the transition from conduction to electroconvection

i.e. the value of the critical voltage Vc. Consequently, the onset d a ta can be com pared

directly to the predictions of linear theory. Two quan tita tive com parisons were made,

the first concerning how the critical voltage varies w ith the conductance and the

second concerning by w hat am ount the critical voltage is suppressed by shear. In

both cases, the comparison between experim ent and theory is good. What, is very

encouraging is th a t this agreem ent was over a wide range in a, V and H e.

O nset phenom ena concerning the s tru c tu re of the unstable flow were com pared

to visual observations of films using the ir slight thickness nonuniform ity for flow

visualization. Q ualitative agreement is a tta ined between the s tru c tu re of the elec

troconvective flow at and above onset, w ith and w ithout shear, when com pared with

the predicted flow field from linear theory. T he unstable mode num ber a t and above

onset in the absence of shear is in excellent agreem ent with the onset mode num ber

m ° predicted by linear theory. The foregoing comparisons between the experim ent

and the theory show th a t the theoretical model is well founded.

T he above onset phenom ena concern the nature of the prim ary and secondary

bifurcations. The d a ta analysis procedure enabled the extraction of quantita tive

inform ation regarding the nature of the prim ary bifurcation. W hen shear is absent,

the prim ary bifurcation is expected to be a pitchfork bifurcation which can be either

subcritical or supercritical. In the presence of shear, the bifurcation is a pitchfork Hopf

bifurcation, which can also be either subcritical or supercritical. Since the current-

voltage d a ta concern the to ta l transpo rt of charge through the film, they could be

modelled by the real Landau equation for the am plitude of electroconvection.

T he sim plest nonlinearity in the am plitude equation is cubic and its coefficient

was denoted g. The d a ta were fitted to determ ine the functional dependence of g

as q , H e and V varied. It was found for a = 0 .47,0 .56,0 .60.0 .64,0 .80 and V ~

13 th a t g was independent of V . Furtherm ore in this range g > 0 , showing th a t

the bifurcation was supercritical. For a = 0.33, g was found to be an increasing

function of V for 2 < V < 8 . More im portantly , it was found th a t th e bifurcation is

136


subcritical (g < 0 ) for V ~ 5 and supercritical (g > 0 ) for V ~ 5 a t a = 0.33. As

an overall trend, it was found th a t g is an increasing function of a . It was argued

th a t a = 0.80 is sufficiently close to the lim iting case of q —> 1 th a t a com parison

betw een the m easured value of g a t a = 0.80 and the calculated value of g for the

■plate' geom etry is meaningful. This quan tita tive com parison gave good agreem ent.

Finally, m easurem ents of g were obtained as a function of Reynolds num ber TZe. It

was found th a t, for a = 0 .47.0.56,0.60,0.64,0.80, there was a TZe below which g > 0

and above which g < 0. Hence, it was concluded th a t the shear can alter the n a tu re

of th e bifurcation from supercritical Hopf to subcritical Hopf via a tricritical point.

T he coefficient of the quintic nonlinearity in the am plitude model was denoted h.

M easurem ents of h were presented for all the foregoing scenarios. T he ‘im perfection'

te rm denoted / in the am plitude model was also fit. For all the d a ta analyzed,

0 < / < 1 .

Secondary bifurcations were qualitatively stud ied bu t were not modelled. All

secondary bifurcations were found to be subcritical independent of a . TZe and V . It

was argued th a t, given a , the secondary bifurcations in the absence of shear, occur

a t values of the reduced control param eter e. th a t are dependent only on V . In the

presence of shear, it was established th a t th e values of e a t which the secondary

bifurcations occur are strongly dependent on TZe. Since the secondary bifurcations

involve m ode transitions m —» m + 1 and m + 1 -4 m. it was found th a t various

possible routes from conduction to electroconvection to conduction were possible. At

a = 0.56 and V = 75, th e mode structu re of th e flow was studied in more detail. From

this resulted a partia l m ap of the persistent m odes in the param eter subspace defined

by (TZe,TZ) = (TZe. e). T h e foregoing results have been reported in references [l, 2, 3].

T he sca tte r in the fitted param eters is g rea ter th an the s ta tis tica l uncertain ty

in th e d a ta . There are three main reasons for the scatter. T he first is th e drift

in th e electrical conductiv ity of the film. In order to partially com pensate for the

d rift, a system atic correction procedure was required. However, since the d rift in

one d a ta set is different in detail from ano ther obtained under th e sam e conditions,

th e uncorrected p a rt of th e drift inevitably leads to some scatter. T he second source

137


of sc a tte r is more directly related to the apparatus. W hen a liquid crystal film is

draw n across the annulus, the electrodes m ust be preferentially wet w ith the liquid

crystal. T he w etting layer is observed to change as the am bient pressure changes and

is expected to be different for different film thicknesses and from film to film. This

non-ideal experim ental feature may also give rise to some of the sca tte r in the results.

T h ird ly non-ideal three-dim ensional effects vary from film to film. T hicker films are

m ore likely to displav weak three-dim ensional flows due to preferential driving of the

surface layers th a n th inner films.

T h e experim ental system w arrants further developm ent to elim inate some of its

non-ideal features. To reduce the effects of the drift in the electrical conductivity, one

should use electrodes th a t when in contact w ith the doped liquid cry sta l are more

inert as far as electrochem ical reactions are concerned than the curren t stainless steel

electrodes. Experim ents can be conducted to study the drift as a function of the

dopan t concentration and thereby determ ine an optim al doping level. Alternatively,

it m ay be b e tte r to use different dopants or a different smectic A liquid crystal. As far

as th e w etting problem is concerned, it can be rectified by re-designing the electrodes.

Instead of using electrodes fashioned about disks, one may use the ends of concentric

hollow cylinders or pipes. These electrodes may be inserted into, say a rectangular

film th a t was draw n on some o ther assembly. In this m anner the issue concerning

w etting is alm ost wholly avoided. If after such improvements are im plem ented, the

experim ental results differ significantly from theoretical predictions, only then should

higher order corrections in the theory be entertained.

T here are m any future directions. T he annular geom etry has a continuous az

im u thal sym m etry th a t is broken when the inner electrode is moved off-center. Elec

troconvection w ith shear in the off-centered or eccentric annular geom etry perm its the

s tu d y of pattern-fo rm ation in a system th a t has no continuous sym m etries. O ne may

question in w hat m anner the phenom ena of electroconvection change as the degree of

off-centering or asym m etry is varied. T he na tu re of the shear flow too , varies signifi

can tly from th e C ouette profile in th e centered system to a non-axisym m etric profile

in th e off-centered geometry. The experim ental appara tu s needs no m odifications for

138


studying eccentric electroconvection. A prelim inary and qualita tive study is reported

in A ppendix F, Section F .l.

Having stud ied the effects of C ouette shear on electroconvection, it is quite natural

to ask w hat effects other kinds of shear have on electroconvection. It was dem on

stra ted th a t C ouette shear has a stabilizing effect on electroconvection and one may

ask w hether th a t is a generic feature or w hether a different shear flow may have a

destabilizing effect. A candidate for such a shear is an oscillatory shear th a t is im

posed by sinusoidally oscillating the inner electrode. Dy varying the frequency of the

oscillation one may drastically change the shear profile in th e film. There is added

novelty in th a t th e base s ta te flow is now tim e periodic and not stationary. Details of

how the oscillatory shear differs from the C ouette shear are provided in A ppendix F.

Section F.2. T he current experim ental appara tus is adequate for performing these

experim ents.

In A ppendix F it has been explained th a t the Couette shear flow is independent

of the viscosity of the fluid while the oscillatory shear flow is not. As a result the

oscillatory shear flow can be exploited to directly measure the fluid viscosity, a pa

ram eter th a t has yet to be measured and th a t in this work has only been indirectly

inferred. A plausible m ethod to ascertain the viscosity by use of an oscillatory shear

in the existing apparatus is discussed in A ppendix F. Section F.3.

Finally, th e research presented in th is thesis and the foregoing future directions

can be repeated in more exotic liquid crystal phases such as sm ectic C and smectic

C*. A significant advantage of using anisotropic liquid crysta l phases is tha t their

optical anisotropy can be exploited to non-invasively visualize the electroconvective

flow. Non-invasive flow visualization techniques have yet to be developed for isotropic

fluid films. P relim inary results from recent experiments on annu lar electroconvection

with sm ectic C* [4] differ significantly from those presented in this thesis.

139


5.3 Conclusions: Theory

The theoretical work presented as p a rt of this thesis consists of a linear stability anal

ysis of the annu lar electroconvection system with and w ithou t shear. T he basic model

th a t describes surface driven electroconvection was first elucidated by the au thor and

reported in Refs. [5, 6 ]. The theoretical work in this thesis consisted of generalizing

the previous theory to include a shear flow in the base s ta te . The theory was applied

to an annular geom etry with a C ouette shear. W hereas the m ethod of linear stability

is well established, the system to which it is applied is fairly com plicated by the non

trivial base s ta te for the shear flow and the surface charge density. In brief, the theory

is electrohydrodynam ic in character, with the fluid flow confined to two dimensions

and while the electrical problem is three-dimensional it is coupled to the flow via

the free surfaces of the film. The system is described by incompressible viscous fluid

dynamics w ith an electrical body force. The fluid is a Navier-Stokes fluid driven by

a body force th a t must also satisfy a nonlocal electrodynam ics. Thus a t its core,

the model is a nonlinear, nonlocal partia l differential equation system. The geom etry

for which th is system of equations is addressed is cylindrical. T he annular system

is naturally periodic about the azim uth, closing on itself. The base sta te , though

non-trivial. is always axisym metric. The linear theory questions the stab ility of the

axisymmetric base sta te to non-axisym metric flow pertu rba tions which are driven by

radial electrical forces.

The results of the linear stab ility theory predict the onset of electroconvection H c

and the non-axisym m etric s tructu re of the flow th a t appears there as a function of the

various param eters of the system: a ,V ,T Z e. The m arginally unstable flow is defined

fully by two param eters, the m ode number m c which counts the num ber of vortex

pairs tha t ap p ea r a t the onset of electroconvection, and the traveling ra te 7 * of the

onset mode around the azimuth. It was found th a t in th e absence of shear (H e = 0 )

th a t the electroconvection flow th a t emerged was sta tio n ary (7 * = 0 ) and consisted

of pairs of counter-rotating vortices. This vortex p a tte rn has a discrete azim uthal

symmetry th a t replaces the continuous azim uthal sym m etry of the base sta te . In th is

140


sense, electroconvection is a sym m etry breaking bifurcation. Furtherm ore for H e = 0,

the critical value of the control param eter, H ac was independent of V and overall

increased w ith a. The onset m ode num ber m° was a nondecreasing function of a . At

certa in values of a. bo th m° and m ° + 1 were sim ultaneously marginally unstable.

A t these values of a , H°c a tta in ed a local maximum. In th e presence of a sh ea r flow,

it was found that the onset of electroconvection was non-stationary and th e ro ta ting

wave th a t emerged traveled a t an angular speed y lc/ m r. T h e pattern was a traveling

non-axisym m etric flow. W hen viewed in a frame th a t ro ta ted at 7 ‘/m c. th e p a tte rn

consisted of mc pairs of counter-ro tating vortices. In each pair, the vortex th a t had

the sam e sense of ro tation as th e inner electrode was narrow er than the vortex th a t

ro ta ted in the opposite sense. This additional sym m etry breaking is a ttr ib u te d to the

shear flow which distinguishes between clockwise and counter clockwise ro ta tions.

It was also dem onstrated th a t for Tie ^ 0 the onset of electroconvection is sup

pressed i.e. 7Z,. was an increasing function of H e. T he onset mode num ber m c was a

nonincreasing function of H e. For nonzero 1le. it was found th a t H {. was an increasing

function of V . More detailed predictions and num erical values are given in th e text.

T h is theoretical work has been reported in Ref. [2].

W herever possible, com parisons between the predictions of linear theory and the

experim ent were made. The qualitative features of the electroconvection flow pat

tern . w ith and without C ouette shear, are seen to agree qu ite well w ith observations

m ade on films with slight thickness nonuniformities. T h e onset mode num bers are

in excellent agreement while th e relative suppressions of th e electroconvection onset

w ith shear are in good agreem ent. These results are encouraging and show th a t the

essential physics is properly accoim ted for by the model.

Now th a t the basic m echanism and linear theory have been explored, th e re are

several avenues for further theoretical work on this p ro jec t. A nonlinear theory could

be developed to quantitatively test the predictions of th e experim ental work as far as

th e prim ary bifurcation is concerned. A theoretical value for the constant g in th e am

p litude equation tha t describes the prim ary bifurcation is currently being com puted

as a function of the o ther param eters such as a, V , and He.[7\ Coupled am plitude

141


equations for electroconvection sta tes in the presence of shear defined by m odes m

and m — 1 should be explored. It is the relative s tab ility of the m — 1 and rn ro ta ting

wave s ta tes th a t determ ines the m ode that appears a t onset and thereafter th e values

of the reduced control param eter e a t which the secondary bifurcations occur. A

fundam ental goal raised by the experiment is to understand the m — 1 —> m and the

m —> m — I transitions. It would also be interesting to see how far simple am plitude

equations m ay be used to describe the nonlinear regimes. Direct numerical sim ulation

of the system m ay well be the next step towards m apping out param eter space. This

endeavour is strongly recommended.

In term s of o ther theoretical enterprises, electroconvection in an eccentric geome

try and electroconvection in the presence of an oscillatory shear are challenging and

interesting candidates. T he former, considering th e u tte r lack of sym m etry, may

be best left to numerical sim ulation or perturbatively explored for small degrees of

off-centering. Significant new phenomena may be expected when the degree of off-

centering is large. See Appendix F for further com m ents.

Electroconvection in annular geometry in the presence of an oscillatory shear is

certainly an interesting project. It is likely to have im portan t im plications on the

stab ility of the fluid film. T he attraction of an oscillatory shear is th a t th e base

s ta te flow is now tim e-periodic as opposed to s ta tionary as was the case w ith the

C ouette flow. Furtherm ore the shear flow profile for th e oscillatory shear is m arkedly

different from the C ouette shear profile. The oscillatory shear profile is dependent

on V suggesting th a t the electroconvective flows th a t ensue will be m ore strongly

dependent on nonlinear interactions. The current theoretical model can qu ite easily

be modified to include an oscillatory shear and subsequently a linear stab ility analysis

can be perform ed. See A ppendix F, Section F.2 for further details. Finally, the

curren t theory can be generalized, a step at a tim e, to take into account th e more

com plicated effects th a t are present when anisotropic liquid crystal phases are used.

T hese effects originate from th e spontaneous polarization in smectic C* liquid crystals,

th e anisotropic electrical conductivities and dielectric properties as well as orien tation

dependent elastic torques.

142


Bibliography

[1] Z.A. Daya, V. B. Deyirm enjian. and S. VV. Morris. "Bifurcations in annular elec

troconvection w ith an imposed shear.” in progress, to be subm itted to Physica

D.

[2] Z.A. Daya. V. B. Deyirm enjian, and S. VV. Morris, "Electrically driven convection

in a th in annular film undergoing circular C ouette flow.” Phys. Fluids, 11, 3613

(1999).

[3] Z.A. Daya. V. B. Deyirm enjian, S. W. Morris, and J. R. de Bruyn, "Annular

Electroconvection w ith Shear,” Phys. Rev. Lett. 80. 964 (1998).

[4] C. Langer. Z.A. Daya, S. VV. Morris, and R. S tannarius. unpublished. (1999).

[5] Z. A. Daya. S. W. Morris, and J. R. de Bruyn, "Electroconvection in a suspended

fluid film: A linear stability analysis,” Phys. Rev. E 55. 2682 (1997).

[6 ] Z. A. Daya. "Electroconvection in suspended fluid films,” MSc. Thesis, unpub

lished (1996).

[7] V. B. Deyirm enjian, Z.A. Daya, and S. W . M orris, "Weakly nonlinear theory

of annular electroconvection w ith shear,” in progress, to be subm itted to Phys.

Rev.E.

143


Chapter 6

Afterword

‘ ‘In going where you have to go, and doing what you have to do,

and seein g what you have to see , you dull and blunt the instrument

you w rite with. But I would rather have i t bent and du ll and

know I had to put i t on the grindstone again and hammer i t into

shape and put a whetstone to i t , and know that I had something

to w rite about, than to have bright and shining and nothing to

say, or smooth and w e ll-o iled in the c lo se t but unused.

Now i t i s necessary to get to the grindstone a g a in .' ’

Ernest Hemingway

144


A ppendix A

Colourim etric D eterm ination of

the F ilm Thickness

In C hapter 2, it was indicated th a t the film thickness was m easured by observing

its colour under reflection. This Appendix, by recounting a s tan d ard calculation,

describes how th e film thickness can be inferred from its colour under reflection in

white light. Since there are several different colour system s[l] currently in use. it is

best to in troduce some definitions from G unter Wyszecki's chap ter on colouriinetry

[2]-

Colour is th e characteristic of a visual stim ulus by which an observer can d istin

guish differences between two fields of view resulting from differences in the spectra l

composition of th e stimulus. Prim ary Colours are the colours of three reference lights,

often red, green and blue, by which nearly all other colours can be produced by addi

tive mixing. T h e Commission Internationale de I’Eclairage (CIE) 1931 colourim etric

system, which is often the s tandard in calculations of th is sort, uses the non-real

primary colours X , Y and Z. Tristim ulus Values are the am ounts of each of the th ree

primary colours th a t when mixed additively give the desired colour. Colour-matching

Functions are th e tristim ulus values a t each wavelength of the stim ulus for a fixed ra

diant flux. T he C IE 1931 S tandard Colourim etric Observer uses th e colour m atching

functions (spec tra l tristim ulus values) x(A), y (A) and 2 (A). C hromaticity Coordi

nates are the ra tio of each tristim ulus value to the sum of the th ree tristim uli values.

145


Hence, only two of the three chrom aticity coordinates are independent. Specifying

the chrom aticity coordinates of a stim ulus then specifies th e colour of the stim ulus.

T he objective is to determine the chrom aticity coordinates of the reflected light from

the film. Given a colour-stimulus function 0 (A), it follows th a t the tristim ulus values

are

X = k J 0 (A) x(A) d X .

Y = k J cp(X)y(X) dX, (A .l)

Z = k f 0(A) 3(A) d X .

T he constant k is a normalizing constant. In practice th e integrals in Eqn. A .2 are

replaced by the discrete sum

X = k^2<t>(A) T(A) A A. etc. (A.2)A

T he chrom aticity coordinates (x. y , z) are then

X Y 7x = ------ . y = ----------- . ~ = . (A.3)X + Y + Z' y X + Y + Z X + Y + Z K ’

T he colour-stim ulus function 0(A) for the experim ental situation is the spectral

intensity of light reflected from the film. The spectral in tensity of the reflected light

can be calculated as a function of th e film thickness given th e spectral intensity of

the incident light, the index of refraction and the angle of incidence. For optically

anisotropic m aterials, there are added complications. However, for sm ectic A where

the optic axis is norm al to the film and for normal incidence the well known expres

sions for reflectivity can be used. Following Sirota et al. [3] closely, let /o(A) be the

incident light intensity (incident colour-stimulus function), th en the reflected light in

tensity (reflected colour-stimulus function) 0 (A, N ) off a film w ith N layers (let each

146


layer have thickness /), a t norm al incidence and index of refraction n is given by

F s i n~ rv0 ( A , J V ) = / o(A) - ■ . , (A .4)

1 + Fsin-c*

where 9

T he light source for illum ination purposes was a Foster EKE 8375 Tungsten-halogen

lam p th a t operates a t a colour tem perature th a t can be varied between 2 0 0 0 - 3200K.

It was operated a t roughly 3000K. The lamp is assumed to be a black-body radiator.

T hus the incident light intensity is given by

Iq{ X T ) = \5 (ec2/\T _ I) (A-6 )

where

ci = 3.7415 x 1(T 16 VV n r . c2 = 1.4388 x 1 0 ~ 2 m I\. T = 3000 K. (A.7)

For 8 CB. the layer thickness I = 3 .1 6 nm [4] and the index of refraction is taken to

be n = 1.5375. [5] which is appropriate for the range of wavelengths in w hite light.

Using Eqn. A .6 the reflected intensity (colour-stimulus function) given in Eqn. A.4 is

calculated. Using this colour-stimulus function for each N , the corresponding tristim

ulus value from Eqn. A.2 was calculated. T he CIE 1931 colour-m atching functions

are tabu la ted in several references [2]. The chrom aticity coordinates for each N can

then be obtained from these tristim ulus values. Since only two chrom aticity coor

dinates are required the pair { x { N) , y ( N) ) are calculated. These are p lo tted in the

chrom aticity diagram Fig. A .l W hereas the chrom aticity diagram is a s tan d ard tool

in colourimetry, it is convenient to have a direct m ap between the observed colour

and the num ber of layers. T he tristim ulus values X , Y and Z are transform ed to

th e tristim ulus values R , G and D. T he transform ation equations betw een the XYZ

prim ary colours and RGB prim ary colours are well known [l] viz:

147


Figure A .l: A chrom aticity diagram for smectic A 8 C B films. C hrom aticity diagram for a film illum inated with w hite light a t a colour tem peratu re o f 3000K. for 2 < 'V < 100 layers.

148


X = +2.36460R - 0.51515G + 0.00520B

Y = —0.89653R + 1.42640G - 0.01441B

Z = —0.46807R + 0.08875G + 1.00921B

(A.8)

M athem atica is used to create a plot of colour versus film thickness based on

these R. G and B values. In Fig. A.2 are given colour p lo ts of observed colour

versus film thickness. M ost o f the experim ents were perform ed with films between

25 and 85 layers thick. O ver most of the m iddle of th is range, the film thickness

can be determ ined to w ithin ± 2 layers, while close to the ends of the range a more

conservative determ ination of w ithin ± 5 layers was used.

■ i0 20 40 60 80 100

Number o f L ayers

..........................40 50 60 70

Number o f L ayers

Figure A.2: Colour charts for smectic A 8 CB. These two p lo ts are colour charts for sm ectic A films of 8 CB under reflection in w hite light. T h e abscissa is the film thickness measured in num bers of smectic layers.

149


B ibliography

[1] A. Nemcsics, Colour Dynamics, Ellis Horwood (1993).

[2] G. Wyszecki. "Colorimetry.” in Handbook o f Optics, M cGraw-Hill (1978).

[3] E.B. Sirota, P.S. Pershan, L.B. Sorenson and J. Collett, "X-ray and optical

studies of the thickness dependence of th e phase diagram of liquid-crystal films,”

Phys. Rev. A, 36 , 2890-2901 (1987).

[4] A .J. L eadbetter, J.C . Frost, J.P . G aughan, G.W . Gray, and A. Mosly, "The

struc tu re of sm ectic A phases of com pounds w ith cyano end groups,” J . Phys.

(Paris) 40. 375 (1979).

[5] D. A. Dunm ur, M. R. Manterfield, W . H. Miller and J. K. Dunleavy. "The Dielec

tric and O ptical Properties of the Homologous Series of Cyano-Alkyl-Biphenyl

Liquid C rystals,” Mol. Cryst. Liq. C ryst. 46, 127 (1978).

150


A p p en d ix B

C ylinder Functions

B .l Expansion functions for the stream function

T he standard m ethods of Sturm-Liouville theory are exploited to determ ine a set

o f expansion functions for the stream function. Since the stream function <pm(r) is

constrained by rigid boundary conditions a t r = r, and r = ru and obeys Eqn. 3.36.

it can be expanded in the eigenfunctions of the square of the Laplacian operato r.[lj

Hence, the eigenfunctions sought are defined by the eigenvalue relation

( d . D - Cm,n = i34mnCm;n . (B .l)

T he boundary conditions are Cm;n(rj) = Cm;n(r0) = 0 and DCm;n(r,) = DCm;n{r0) = 0 .

T h e desired solutions of Eq. B .l are

Cm;„ (r) = J m{/3mnr) + B mnYm(/3mnr) + CmnI m{/3mnr) + DmnK m{(3mnr ) , (B.2)

w here J m and Ym are the Bessel functions of order m , Im and K rn are the m odified

Bessel functions of order m[2, 3]. The param eters (3mn a re successive solutions of th e

151


secular equation det M(/3) = 0, where

( JmiPr,) Ym<J3ri) Imd3n)

Jm{/3r0) Ym {f3r0) Im ((3ra)

Jm-dPri) Ym-dPri) / m_ 1(/?ri)

V Jm-l{0 ro) Ym. dPTo) I,n-dPra)

K n W r d N

K m(l3r0)

- K m-d&Ti)

- K m-dfira) )

(B.3)

and (I, B mn,C mn, D mn) is the eigenvector corresponding to eigenvalue zero for each

(3mn. A t each radius ratio a . the 3mn and the eigenvector ( 1 . Drnn. C mn, D mn) were

calculated by using M athem atica routines.

The functions defined by Eq. B.2 form a complete, orthogonal set w ith orthogo

nality condition

r dr rCm;nCm;p = Afcui:J n P ■ (B.4)Jrx

It is convenient to define functions[l] Um-n and Vm;„ by

(B.5)

and

It follows th a t Cm,n {r) = Um;n(r) + Vm;„ (r) and

m*D ,D — j C m n — — 3~nrliMm\n ~ Vrrt;„)

(B .6 )

(B.7)

Using the definitions Eqns. B.5 and B .6 and the relation B.7. the norm alization Afcm n

in Eqn. B.4 can be expressed quite sim ply and is given by

= r02U;n.n{r0) - ri2U%l.n (r t ) . (B .8 )

152


B.2 Expansion functions for the potential

T he po ten tia l pertu rba tion ^ m(r) satisfies homogeneous boundary conditions at the

inner and ou ter edges of the film and obeys Eqn. 3.37 so that an expansion in term s of

the eigenfunctions of the Laplacian operato r may be sought. T he relevant eigenvalue

relation is

( d . D - = -x L tP m j r (B.9)

which has solutions

n ( 0 ‘ r n ( X m n ^ ) ^ m n ^ 'm ( X m n ^ ) • (D.1 0 )

T he param eters Xmn solve the secular equation (let N (x) = 0 . where

N (x ) = (B .1 1 )

and ( 1 . bmn) is the eigenvector corresponding to eigenvalue zero for each \ mn. At

each radius ratio a , the \ „ m and the eigenvector (1. bmn) were calculated by using

M athem atica routines. The functions defined by Eq. B .10 satisfy the orthogonality

conditionf r“I d r ripm;ny rn.p = A/0m nSrtp. (B.12)

Jri

where the norm alization A/^r(i;„ is given by [2 ]

A/"*/;,,,.,, —

1

(B.13)

8

n- t v

:r“2( —1 (Xmn^o) Jm+l(X mn I'o) + fan (^m—l(XmnT’o) ^rn+l(X mu rG)

dm—liXmn^i) dm+l(X mn + fan ^ m - l ( X m n n ) K n + lfX m n ^ i)

153


Bibliography

[1] S. C handrasekhar. Hydrodynamic and Hydromagnetic Stability, Dover Publica

tions Inc. (1961).

[2] I. S. G radshetyn and I. M. Ryzhik. Integrals, Series and Products Academic

Press. New York (1980).

[3] .VI. Abram owitz and I. Stegun. Handbook o f M athematical Functions. Dover Pub

lications Inc. (1965).

154


A ppendix C

Exact N onlocal Solution

This A ppendix presents a solution tha t solves the nonlocal problem for the charge

density pertu rba tion and is referred to as the exact nonlocal solution in con trast to

the local approxim ation introduced in Eqn. 3.53. The electrostatic Eqns. 3.38 - 3.39.

in which the charge density and electrostatic potential are related nonlocally. are

solved numerically. The solution tha t is presented here is due in large p art to the

efforts of Vatche D eyirm enjian and Stephen Morris.

T he first step in the present m ethod is to find the appropriate expansion functions

of the field variables. Substitu tion of Eqns. 3.40 - 3.43 into Eqns. 3.37 - 3.39 yields

equations which can be solved for A n, 0 m;„, 'I',,,;,,. 'i-3,n,n- and Q m.n. The stream

function (f>m-,n{r) = C m;n(r) as in Eqn. 3.48. T he potentials tym;n and '$3m;n and the

charge density Q m-n are further expanded as

^m;n(f) = m;/(r) . (C. l )(

'i'3m;n(r) = Vm;nlP3mj(r ) , (C.2)I

Qm;n{r) = ^ i (^-'•3)I

where i / w is given by Eqn. 3.49 and vm.nl are complex coefficients. T he functions

and qni-i are com puted as follows. A fter substitu ting Eqns. 3.42 and C.2 into

155


Eqn. 3.38. the resulting equation

(C.4)

is solved num erically on a finite 2D grid by an over-relaxation algorithm fl] for the

functions ip3m;i(r. z), subject to the boundary conditions as in Eqns. 3.46 - 3.47 with

where the differentiation is perform ed numerically. F igure 3.4 shows a plot of the

charge density ^6;1 a t a = 0.5. and com pares the num erical result to the approxim ate

one given by Eqns. 3.53 and 3.56 in C hapter 3. T he approxim ate solu tion does not

results in an equation which is projected against 4>m;k t ° ob ta in a m a trix expression

th a t can be solved numerically for th e complex coefficients vm.n[.

Finally, su b stitu tin g th e various expansions in into Eqn. 3.36 and taking

the inner p roduct w ith C m;p, yields a set of linear homogeneous equations for the

constants A n . T his set is w ritten as the m atrix equation A n T ^ = 0 . For a

nontrivial solution, the com patibility condition is

0) - C’m;i(r) for r, < r < r 0 and 0 otherwise. T hen Eqns. 3.39, 3.42. 3.43.

and C .2 - C.4 give

q,n-,i(r) = —2dz ip3m-i{r, c ) |._ 0+ , (C.5)

contain the divergences which occur near the film's edges, due to the sharp changes

in the derivative of the potential. These are a feature of the 2D model, which treats

the film and electrodes as having zero thickness.

By substitu tion of Eqns. C .l and C.3 into Eqn. 3.37 and using

(C.6 )

R eal(det [T]) = Imag(def [T]) = 0 , (C.7)

156


w ith the elem ents of the m atrix T given by

Tp„ = (cm„(p.D -

7‘ - ™ (^)) ( D*D - % ) c j ) - imRFm;pn .

(C .8 )

T he first two inner products of Eqn. C .8 can be simplified using Eqns. D .l and D.7

and the m atrix elements Fm;pn are

F m;fm = - (Dqw )ipm^ y (C.9)

T he real values of 7Z and 7 ' which satisfy Eqns. C.7 and C .8 a t each m define the

neutra l stab ility boundary R = R (a . V, R e . m . 7 '). The critical param eters m c, R c

and 7 * are obtained when R is minimized.

T he num erical over-relaxation calculation used to solve Eqn. C.4 involved a grid

spacing such th a t there was a minim um of 160 points across the w idth of the film. For

the purposes of integration, the discrete values of qm-i found num erically from Eqn. C .5

were Chebyshev interpolated. T he series in Eqns. C .l - C.3 and C .6 were calculated up

to I = 20. Three modes (n = p = 3) were employed in the com patibility conditions

Eqn. C.7 when the shear was zero. This was reduced to one m ode (n = p = 1 )

when th e shear was non-zero. All r-in tegrations were performed by the Romberg

m ethod.[l] R eal(det [T]) = 0 (Eqn. C.7) was solved for R for a given trial 7 * and

associated coefficients vm.nl of the field variables. The R and trial v m.nl were employed

in the search for the 7 * which satisfied Im ag(def [T]) = 0 . The new value of 7 ' was

then used to find new coefficients T he iterative cycle was continued until the

param eters and field variables had converged.

157


Bibliography

[l] VV. H. Press. S. A. Teukolsky, VV. T. Vetterling, and B. P. Flannery, Numerical

Recipes in C, Cam bridge University Press, Cam bridge (1992).

158


A ppendix D

D ata at A tm ospheric Pressure

In this Appendix are collected some results from experim ents th a t were conducted

a t atm ospheric pressure. The m ajority of these experim ents were perform ed prior

to the experim ents a t reduced am bient pressure. T he experim ental protocol, while

not identical, was sim ilar to tha t described in Section 2.3. However, the analysis

of the d a ta was quite different. In fact the current-voltage characteristics were not

modelled by the m ethods described in Sections 4.2 and A ppendix E. In analysing the

d a ta discussed here, the interest was in th e prim itive param eters such as the critical

voltage Vc and the conductance c a t onset. These quantities are easily determ ined

from the current-voltage characteristics.

For the d a ta obtained in the absence of shear, the dependence of the critical

voltage on the conductance given by Eqn. 4.9 is rew ritten , a little differently, below:

voltage, is com puted as follows. From th e current-voltage characteristics are deduced

the critical voltage V® of electroconvection and the onset conductance c. T he value of

159


(D .l)

N ote th a t in the above expression, there is a single unknow n param eter ^/rj3/ er3. The

quan tity on the left hand side of Eqn. D .l, here referred to as the scaled critical

72° (a ) was the highest order numerical result of the nonlocal theory: see Table 3.1. In

Fig. D .l is p lo tted the left hand side of Eqn. D .l versus the onset conductance. The

£=U sio

0.330.510.560.640.770.80o o > 6

C 2

0 9 4 6 8Conductance (10'13 £2'1)

Figure D .l: A plot of the scaled Vc° versus the onset conductance a t several a. T he line is a one param eter least squares fit to the data .

scaled critical voltages were obtained from 108 current-voltage characteristics a t six

different a. T he d a ta encom passed a broad range of conductivities and consequently

th e range of Prandtl-like num ber V is equally broad. Despite th e diversity in the

experim ental param eters i.e. the different a , s. cr3 and V , Fig. D .l confirms the

linear relationship th a t is predicted by Eqn. D .l. A nd like Fig. 4.3, there is some

sca tte r in th e d a ta in Fig. D .l. Nevertheless, one is confident th a t the linear tren d

is indicative th a t th e geom etric scaling with respect to the film thickness s and th e

rad ius ratio a is properly accounted for. A Linear fit to the scaled critical voltage

as a function of the conductance, for the entire range of d a ta illustrated in the plot,

provides a m easure of the viscosity of the film; th e single unknown param eter. A

weighted least squares m inim ization leads to rj3 = 0.19 ± 0.05 k g /m s.[l] This value

160


for the viscosity is w ith in uncertainty identical to th a t determ ined in Section 4.3 and

is also of order 0 .1 k g /m s as is expected from other stud ies.[2] Since th is viscosity is

determ ined from onset d a ta i.e. the film has no flow, it is unlikely th a t significant air

drag effects are incorporated into it. Nonetheless it is probable th a t w hen the film

has some flow, either C o u ette shear or electroconvection, the air drag will have more

significant effects.

D ata from sheared films were treated as described in Section 4.3. T he suppression

of the onset of electroconvection by the C ouette shear is measured by Eqn. 4.10 which

is repeated below:

I (a . He. V ) =H c(a , H e . V )

712(a)/ V J a . c . H e ) \ 2 _ 4

- * ■ (D-2)

The first equality in Eqn. D .2 is used to calculate the suppression theoretically, while

the second, experim entally. The theoretical calculation is described in Section 3.4.

The experimental determ ination of i is as follows. At each radius ratio or. the critical

voltage for a sheared film Vc{c,He) is determ ined by exam ining the current-voltage

characteristic. Due to the drift in the electrical conductivity and hence of the con

ductance. it is unlikely th a t V'.°(c) can be determ ined from a single current-voltage

characteristic. Instead a current-voltage characteristic for a film in the absence of

shear and ‘nearby! conductance is used and adjusted to determ ine V^lc) for the same

conductance as the sheared film. It follows th a t the experim ental value of i can be

com puted. The uncertain ty in the suppression is due to the uncertainties in Vc(c, Tie)

and l/°(c).

Each experim ental m easurem ent of the suppression is a t a given rad ius ra tio a , a

measured conductance and therefore Prandtl-like num ber V and a m easure Reynolds

num ber He. As discussed in Section 4.3, the experim entally accessible variables are

the radius ratio a an d th e Reynolds num ber 1Ze. T he Prandtl-like num ber V , due

to the slight drift in th e electrical conductivity, varies slightly during the experim ent.

As a result, the suppression e is m easured a t several a as a function of H e while

the V is simply m easured and noted. Figures. D .2 a through d plot the experim ental

161


m easurem ents of the suppression a t a = 0.33,0.56,0.64 and 0.80 respectively. Since

the m easurem ents a t a given a span a range in "P, a reasonable comparison to theory

can be m ade by com puting the theoretical suppressions a t th e extremes of the range.

The te st is then to see how well the two theoretical suppression curves contain the

data . In Fig. D.2a is plo tted suppression d a ta a t a = 0.33. T he da ta had P rand tl-

like num bers: 9.8 < V < 12.4. T he dashed line is a local theory calculation of

the suppression a t V — 9.8 while the d o tted line is the local theory suppression at

V = 12.4. In Figs. D.2 b through d are three o ther d a ta sets a t different a. T hey too

are accom panied by the appropriate theoretical curves, w ith th e details given in the

caption to the plot.

It is clear tha t the agreement is good for a = 0.56 an d 0.64 but is poor for

a = 0.33 and 0.80. More im portant, perhaps, is th a t the disagreem ent is system atic.

At a = 0.33 the measured suppression is lower than the predicted values while a t a =

0.80 it is higher. This system atic discrepancy is a ttribu ted to the effect of air d rag on

the film, since the suppressions a t reduced am bient pressure, while agreeing no be tte r

than those in Figs. D.2 b and c, did not show any system atic trends in agreem ent.

In Figs. D .2 a through d, the range of th e Reynolds num ber is the same m aking it

quite easy to see the effect of the radius ratio on the suppression by observing the

scale of suppression axes. The (lack of) agreem ent shown in Figs. D.2a th rough d

is, as indicated in Section 4.3 fundam entally independent of th e value of q3 th a t was

determ ined by fitting d a ta to Eqn. D .l . Since the q3 dependence in the 7£e scaling

of b o th the theory (via V in Eqn. 3.20) and the experim ent (according to Eqn. 4.3)

are proportional to 1 / q 3, any change in r?3 multipies both by the same factor. This

simply results in a rescaling of the TZe axis in Figs. D.2 a th rough d. w ith no change

in th e quality of (dis)agreement.

162


0.6

0.5

tw 04

0.3

a.

a = 0.33-o0.20.05 0.15 0.250 0.1

5

4

c.2 3

a.§•2tn

a = 0.56o cto 0.05 o.i 0.15 0.25

Reynolds number. Re Reynolds number, Re

a = 0.640.05 0.1 0.15 0.2 0.25

Reynolds number. Re

10

8

6

2

*j> a = 0.80o -2 :o 0.05 0.150.1 0.2 0.25

Reynolds number. Re

Figure D.2: Comparison betw een experim ental m easurem ents of suppression and theoretical predictions. Shown are plots of the experim ental m easurem ents of the suppression i versus the Reynolds num ber 72.e. In (a) a = 0.33 and the d a ta spans 9.8 < V < 12.4, (b) a = 0.56, 72.5 < V < 82.0, (c) a = 0.64, 49.6 < V < 85.5 and (d) a = 0.80, 62.8 < V < 92.4. T he predictions of the local theo ry are calculated a t the respective minim um (dashed line) and m axim im um (dotted line) values of V.

163


Bibliography

[1] W. H. Press, S. A. Teukolsky, W. T . Vetterling, and D. P. Flannery, Numerical

Recipes in C, Cam bridge University Press, Cam bridge (1992).

[2] R.G. Horn and M. Kleman, “O bservations on shear-induced tex tures and rheol-

ogy of a smectic-A phase.’’ Ann. Phys., 3, 229 (1978).

164


A ppendix E

D ata M odelling

T his A ppendix presents the detailed procedure used to ex tract the best fit param eters

of th e am plitude equation s ta rtin g from the current-voltage characteristic. An exper

im ental run consists of current-voltage d a ta th a t is, as described in Section 2.3. ob

ta ined by first incrementing the applied voltage across the film followed by decrem ent

ing the voltage. Hence the current-voltage characteristic, see for example Fig. 2.7,

consists of four regions which are acquired in the following order: a region where the

film is does not convect while th e voltage is being incremented, a region where the

film electroconvects while the voltage is being increm ented, a region where the film

electroconvects while the voltage is being decrem ented, and a region where the film

does not convect while the voltage is being decrem ented.

In the regimes where the film does not convect, the current is due to conduction. In

these regimes the film behaves ohmically. As a result, it is easy to calculate the film 's

conductance: c = I / V , where ( / , V) are the current-voltage data . However, in th e

regimes th a t the film electroconvects, it is not possible from the current-voltage d a ta

to determ ine the conductance. In the absence of drift of the electrical conductiv ity

there would be a constant conductiv ity and thus for each film a constant conductance.

T he drift in the electrical conductivity results in a likewise variation of the film

conductance. Figure E .l plots th e conductance of a film during an experim ental run.

O n the abscissa is plotted the d a ta acquisition index which is sim ply the order in w hich

th e d a ta was acquired. It is roughly proportional to tim e, bu t not exactly since th e

165


1.47

1.46

convectionincreasing decreasing voltage voltage

1.45

o 1.44uCJ conduction

decreasing voltageconduction

increasing voltagec3o3■ocoU

1.43

1.42

1.41

0 50 100 150 2 0 0 250 3 0 0Data acquisition index

Figure E .l: T he conductance of the film during a single current-voltage run. The conductance can be m easured while the film is conducting. W hile convecting, the conductance is assumed to be given by linear interpolation.

duration spen t a t each m easurem ent is different, see Section 2.3. O n the ordinate,

in the conduction regimes, is the measured value of the conductance. The three

vertical broken lines dem arcate the four regions in each current-voltage characteristic.

Between d a ta acquisition indices 1 and 129, th e film is not convecting and the voltage

is being increm ented. Between da ta acquisition indices 130 and 168, the film is

convecting and the voltage is being increm ented. Between d a ta acquisition indices

169 and 224, the film is convecting and the voltage is being decrem ented. Between

d a ta acquisition indices 225 an d 338, the film is n o t convecting and th e voltage is being

decrem ented. Note th a t the lengths of the four intervals are unequal and therefore

the du ration in each regime is likewise different. T h e plot Fig. E .l is no t typical of the

166


conductance during an experim ental run. In fact, there is a great diversity in how the

conductance changes during the course of an experim ental run and seems to depend

in some com plex m anner on how much time and therefore how much current was

transpo rted through the film as well as on the s ta te of flow of the film, the am bient

pressure and even the perim eter of the annulus, see Sections 2.3 and 4.2.

As the objective is to approxim ate the conductance of the film while it convects,

the sim plest assum ption is th a t the conductance varies linearly from th e its value

before convection begins to the value when convection stops. In Fig. E .l the solid

line, extending between d a ta acquisition indices 122 and 233, shows the values tha t

the conductance is assumed to take during convection. Note th a t a few d a ta points to

the left of index 129 are ignored and a few are averaged so th a t the conductance does

not in terpolate between the boundary of the conduction —> convection regim e where

fluctuations are expected to be large. The foregoing also applies to the convection —>

conduction boundary. T he residual error after the linear interpolation is no t known

bu t it is certainly less than the intrinsic error in assum ing a constant conductivity.

By th is procedure, for every current-voltage m easurem ent, a corresponding conduc

tance. c can be determ ined. T he augm ented d a ta a t each acquisition index is then

(V, I , A / , c).

Assume th a t a critical voltage Vc has been determ ined; say it takes a value of V

somewhere between the values of V a t acquisition indices 128 and 129. T hen this

critical voltage corresponds to a conductance somewhere between the values th a t the

conductance assumes a t indices 128 and 129. In Section 4.2 it was explained th a t the

d rift in the electrical conductivity implies a drift in the critical voltage, see Eqn. 4.7.

Com bining Eqns. 4.1 and 4.7, the critical voltage for a uniform film of thickness s

and radius ratio a can be expressed as

<*!>

It is crucial to note th a t the critical voltage is not constan t during an experim ental

run! Since the conductance drifts, so does the critical voltage. T hen w hat precisely

167


does one m ean by the critical voltage? W hether a film is convecting or not. the critical

voltage for the film at th a t m om ent is none o ther than th a t voltage which would have

to be applied so as to make that film marginally unstable. However, th e critical voltage

th a t is deduced from the current-voltage characteristics is in fact unique and is the

voltage a t which the conduction s ta te lost stab ility to convection. T he distinction

should not be lost, the critical voltage is by definition a potentiality. however in the

experim ent it is an actuality. Henceforth, the critical voltage determ ined from the

experim ental current-voltage m easurem ents will be denoted Vj and the corresponding

conductance cK If (c*. V'J) have been deduced from the current-voltage data , then

the critical voltages Vc.(c) can be determ ined from Eqn. E .l and the m easured and

in terpolated values of the conductance. As a result the augm ented d a ta a t each

acquisition index is then (V. I, A I . c ,Vc).

T here are three ingredients th a t are required to augm ent th e raw d a ta from

(V, I . A /) to (V , / . A / , c, Vc); a critical voltage and conductance pair from the current-

voltage delta and the conductance of the film during the experim ent. In

practice, the following algorithm was employed. It is easy to ascertain bounds on

the critical voltage by observing the current-voltage characteristics, see for exam ple

Figs. 2.7a and b. Each current-voltage characteristic was observed by-eye and two

voltage intervals were chosen. The first interval S i contained th e critical voltage

a t which the conducting s ta te becomes m arginally unstable to the electroconvecting

sta te . The second interval So contained th e voltage a t which th e convecting sta te

becom es m arginally unstable to the conduction sta te . A guess a t the critical voltage

V* € S i and a voltage V* € So are chosen a t random by use of a suitable uniform

deviate random num ber generator[l]. The conductances are then determ ined for the

conduction regimes V < V* and V < . A linear interpolation between the two

regim es is effected as described earlier. T h e conductance cA corresponding to the

random ly chosen critical voltage can then be determ ined. W ith these ingredients;

(cA, V*) and the conductances, the raw d a ta can be augm ented.

For each of th e 1 0 0 random ly chosen (V .A, V*) there will be a corresponding aug

m ented d a ta set (V, / , A / , c, Vc). This d a ta set is then transform ed according to

168


Eqns. 4.4 and 4.5 for the reduced control param eter e and the convection am plitude

A, conveniently repeated below.

V \ 2 I!• A = ^ = i c V ~ l - (E-2)

Following s tandard procedures for handling measurement errors, the augm ented d a ta

set is transform ed to the set (e, A, A A). T he relevant am plitude model equation given

by Eqn. 4.6 is rew ritten more conveniently as

eA = g A 3 + h A 5 — f . (E.3)

T he transform ed d a ta is also expressed after another transform ation as {A, eA. A(e.A)).

T he dependent variable eA is fit as a function of .4 given by th e am plitude model

Eqn. E.3. In this model h > 0 and f > 0 . The fit procedure is a Levenberg-

M arquard t nonlinear routine.[1], W hereas A(e.4) = cA.4, it is found more useful to

use A(e.A) = \ f e A A for greater sensitivity of the fit to the region e ~ 0. The re

sult. consisting of the identifiers (VCA, V?) and the fitted param eters (g . h . f ) for each

augm ented d a ta set is collected as (V'A. I /v . g. h. f , x„2), where \ u2 is the goodness

of fit s ta tis tic referred to as the chi-square per degree of freedom .[l, 2] The degree

of freedom u = N — 3. where jV is the num ber of da ta points and there are three

fit param eters. From the 100 such results, the lowest \ i ? Is selected. The random

voltages { V A, V ' ) for this set are denoted {Vcn .Vcu). New intervals 3?/ and 3 / of

a ten th of the length of the original intervals and ^ 2 respectively are now cho

sen centered abou t (V .0 , Vcu). Another 100 selections are m ade a t random in these

intervals and the fitting process is repeated . The best fit param eters are selected

by m inimizing the x J 2 statistic. This random ly chosen voltages of this set are de

noted (V^.V^) and enable the transform ation: ( V , I , A I ) —> (A e A A (e A )) by the

procedures outlined above. Also corresponding to this set are the best fit param eters

from th e Levenberg-M arquardt nonlinear fit to Eqn. E.3 which are simply denoted as

{9; / ) • T he m ethod employed in the above fitting procedure is to a priori constrain

169


the two fitting param eters (V,cA,Vrcv) and fit the three param eters (g .h . f ). Dy the

M onte-Carlo m ethod of using many instances of (V * ,V ^ ) , a 3-param eter nonlinear

fit is used to determ ine the five best fit param eters for w hat would be an otherwise

unwieldy 5-param eter nonlinear fit.[l, 3].

O ften it was deem ed necessary to restrict the fit to a neighborhood of e ~ 0 . There

are two compelling reasons for this. F irstly the am plitude equation models are valid

only for f <£. 1 and secondly the shorter the range of e the less th e overall im pact the

uncorrected part of the drift in the d a ta would have on the results. As a consequence,

more often than not, only the d a ta acquired w ith increasing voltages or equivalently

increasing e was fit. Thus there are four param eters of interest ( V j . g . h . f ) , however

all five param eters are needed to transform the raw data.

Finally, it rem ains to determ ine the uncertainties on these best fit param eters.

T he uncertainty A V j is taken to be the s tandard deviation of a uniform deviate or

uniform probability d istribu tion on the interval 3 f/. The uncertainties (At/. Ah. A /)

are determ ined by a M onte-Carlo bootstrap m ethod.[l] The d a ta set (.4. e.4. A(eA))

corresponding to the best fit param eters consists of say N m easurem ents. The Monte-

Carlo boo tstrap m ethod entails choosing N d a ta points a t random and with replace

m ent from the d a ta set (.4, eA, A(eA)). Each of these instantiations, a subset of the

best fit d a ta , is fit by the Levenberg-M arquardt m ethod and the results { g . h . f . x v 2)

are collected. A to tal of 500 instantiations are modelled. A probability measure based1 1

on the chi-square s ta tis tic proportional to e~*Xu is com puted for each d a ta subset.

T he probability m easure is normalized over the 500 d a ta subsets and a weighted

s tandard deviation of the param eters (g, h , / ) is calculated. These weighted standard

deviations are m easures of the uncertainties (Ag, A h . A /) .

170


Bibliography

[1] W .H. Press. S.A. Teukolsky, W .T . Vetterling and D.P. Flannery. Numerical

Recipes in C. Cam bridge (1989).

[2] P.R. Bevington and D. K eith Robinson. Data Reduction and Error Analysis for

the Physical Sciences, McGrawHill (1992).

[3] L.E. Scales. Introduction to Nonlinear Optimization. M acmillan Publishing Ltd.

(1985).

171


A ppendix F

Som e Future Investigations

F .l Electroconvection in an Eccentric Annulus

T his Section reports results of some of the preliminary investigations into electrocon

vection in an eccentric annulus. A schem atic is shown in Fig. F .l . The outer electrode

is supported on a translation stage allowing the investigation of electroconvection in

annuli w ith differing degrees of off-centering. In a centered annulus. the centers of

the circles th a t define the edges of the inner and outer electrodes are coincident. A

m easure of the off-centering is given by the distance of th e separation between these

two centers. In dimensionless units, where lengths are m easured in units of th e film

w idth in a centered annulus. ra — r,, the dimensionless separation I varies from zero

in a centered configuration to 1 in the extrem e case where the electrodes touch.

T he m ethods are sim ilar to those in C hapter 2. T he film can be draw n with

the electrodes in a centered or eccentric configuration and it is quite easy to change

the degree of off-centering w ithout changing the film thickness. The prelim inary

results discussed here were obtained a t atm ospheric pressure and a t reduced am bient

pressure. In Fig. F .2 is shown a pair of current-voltage characteristics a t rad ius ratio

a = 0.80 bu t w ith different extents of off-centering. T he d a ta displayed here were

obtained a t atm ospheric pressure. The d a ta , which are quite sim ilar to th a t of a

centered annulus. displays a conduction region which is ohmic. A threshold voltage

or a region where there is a sharp increase in th e current can be easily identified and is

172


Figure F .l : A schem atic of the eccentric annular geometry. The centers of the circles th a t define the inner and outer electrodes are a nondimensional d istance I apart. The polar coordinate system has its origin at the center of the inner electrode.

related to the rapid increase of electroconvection. The imperfect bifurcations implicit

in the current-voltage d a ta shown appear to be continuous and there is no hysterisis

in the d a ta . T he greater the off-centering, the greater the current carried, the lower

the threshold voltage and the more im perfect the bifurcation appears.

T he resistance of the film can be obtained from th e ohmic p a rt of th e current-

voltage d a ta . A t I = 0.26, the film's resistance is approxim ately 22% larger than

a t I — 0.55 for the d a ta shown in Fig. F.2. This difference is not wholly related to

the d rift in the electrical conductivity of th e liquid crystal. As can be seen from the

slightly different resistances of the film on th e forward and reverse runs, drift in the

electrical conductivity can account for abou t a change of 3% per current-voltage run.

T he drift thus accounts for a modest 6 % increase in the resistance, yet the d a ta shows

a varia tion of 22%. This apparen t disparity can be explained when one realizes th a t a

uniform film in an off-centered annulus has a different resistance th a n a uniform film

173


^

<Q*

c 4<D

3 3U

2

1

0

^ Increasing voltage w ith 1 = 0.26* D ecreasing voltage w ith 1 = 0 .26 $° Increasing voltage w ith 1 = 0.55+ D ecreasing voltage w ith 1 = 0.55 . $

a.0 8

j , »* s

r

/ > ■u** ***

o

* * * * * * * * x*

* * > *

5 10 15

V oltage (volts)20

Figure F.2: A representative plot of current-voltage d a ta from an eccentric annular film. P lo tted are current-voltage characteristics for a film w ith radius ratio a = 0.80 b u t w ith off-centering given by I = 0.26 and I = 0.55.

of the sam e thickness in an annulus of dissim ilar off-centering. A fter all, resistance

is a property of geom etry as well as conductivity. Before sparingly presenting an

elem entary bu t tricky calculation of the resistance of an eccentric annular film, it is

noted th a t the resistance of a film a t a = 0.80 and I = 0.26 is 15.4% larger th an the

resistance of a film of th e sam e thickness and conductivity a t a = 0.80 and I = 0.55.

T his geom etrical difference and the drift in com bination account for the changes in

the film resistances in Fig. F.2.

Since any further analysis will require calculating the resistance of a film in an

a rb itra ry eccentric configuration, the necessary form ulation is presented here. Con-

174


sider a film of uniform thickness s, th a t spans the eccentric region between two circles

of radii r, and r Q shown in Fig. F .l. The centers of the two circles are a distance I

ap a rt, with distances nondim ensionalized by d = ra — r,. W hen I = 0 , th e annulus

is centered. In a polar coordinate system w ith origin a t the center of th e circle with

rad ii r*, and with the rad ius ratio a = r , / r 0, it is easy to show th a t

r.„ = (F .l)1 - a

rmt = Icosd + — l2s\n2d , (F.2)

where r in and rout param etrize the edges of the inner and outer electrodes respectively.

T h e space between th e two circles can be sm oothly filled by defining a general curve

whose radial position is given by

r(0 , tf) = rin + c - ri„), (F.3)

w here S is a variable which satisfies 0 < 6 < 1. A differential element dr(Q) is defined

as

dr{0) = r(0 , <5 + dS) - r{9 , 8)

= ( W - rin)d8. (F.4)

W ith the bulk conductiv ity denoted by <7 , an elem ent of resistance d2R is defined by

d2R = — • (F.5)a sr{8 ) d9

Integration over the azim uth 6 and the radial coordinate (in actuality over 5) gives

th e resistance R of the film:

1 + 1F ( a , l )(F .6 )

175


0.2a = 0.33

• a = 0.800.18

0.16

0.14

0.12

0.08

0.06

0.04

0.02

0.40 0.2 0.6 0.81

Figure F.3: T he resistance of an off-centered film. The dim ensionless resistance versus the off-centering param eter for annular films with a = 0.33 and a = 0.80.

where F ( a . l ) is the numerical integral

l [ / ( l - a )a

- i

0 - sia2{2^ }'1(1 - O ) dip .

(F.7)

By use of Eqns. F .6 and F.7 and from the experim ental determ ination of the film

resistance, it is possible to apply much of the analysis developed in Section 4.2. In

the sim pler case of a centered annulus i.e. I = 0 Eqn. F .7 simplifies considerably

resulting in a closed form expression for the resistance:

fi(a'°) = 2 ^ lo60 (F.8)

It is a t a first glance, perhaps strik ing th a t the resistance of the annular films

of th e sam e thickness w hether centered o r otherwise, are independent of th e rad ia l

176


dim ensions and vary only w ith the geometric proportions a and I. A little thought

makes clear why th is is so. A plot of the dimensionless resistance versus the off-

centering variable I is given in Fig. F.3. N ote th e resistance is a m axim um when

the annulus is centered and vanishes for com plete off-centering. Also no te th a t the

resistance varies inversely with the radius ratio.

A prelim inary investigation was also conducted into eccentric electroconvection

experim ents in the presence of shear. The simple C ouette shear profile is increasingly

pertu rbed as I increases from zero. The shear profile is significantly different from the

C ouette profile for large I and large ro tation rates ui of the inner electrode. A sys

tem atic study of the correlation between the shear profile and electroconvection may

prove inform ative. Figure F .4a illustrates the effect of shear on the current-voltage

d a ta a t q = 0.8 and I = 0.26. It is unlikely th a t a t this ra ther large radius ratio

and weak off-centering th a t the shear profile is significantly different from C ouette

flow. T he d a ta displayed were obtained a t atm ospheric pressure. It is evident th a t the

shear lengthens the conduction regime w ith rapid electroconvective flow suppressed to

higher voltages. T he stronger the shear the greater this suppression, however the rela

tive degree of suppression is likely to be a function of off-centering. There is currently

no system atic s tu d y of this suppression, experim ental or theoretical. F igure F .4 b

plots the current-voltage characteristics of an eccentric annular film a t a = 0.33,

I = 0.55 and a t a reduced am bient pressure ~ 1 torr. In spite of the d rift in conduc

tivity, it is evident th a t higher shears lead to increased suppression even though a t

the sm aller radius ratio and greater off-centering, the shear profile is hardly expected

to resemble the C ouette profile. It is however, th e relative suppressions th a t are likely

to be very different. In the few experim ents conducted in an eccentric geom etry, a t V

and H e num bers com parable to the study of the centered annuli, there is no evidence

of strongly subcritical behavior.

From this brief study of electroconvection in freely suspended fluid films in off-

centered annuli, it is clear th a t some further experim ents should be dedicated to

m aking a simple s tudy of the relative rates of suppression a t a few a a n d I. More

in teresting however, is the s tudy of the n a tu re of secondary vortex changing transi-

177


5

4<a'w'wc63

u

3

co = 0 .0 rad/s co = 0.2571 rad/s co = re rad/s

. 001050 2015

Voltage (volts)

6.5

< 5.5

4.5

co = 0.957t rad/s co = 1.0571 rad/s co = 1.15ti rad/s

3.5

42 44 46 48 50 52 54 56 58 60

Voltage (volts)

Figure F.4: Representative plots of the current-voltage characteristics of sheared films in eccentric annuli. Shown are current-voltage characteristics of electroconvection in eccentric annuli a t a few different rates of shear. In (a) a = 0.80 and I = 0.26. In (b) a = 0.33 and I = 0.55. W hile the plot in (a) was obtained a t atm ospheric pressure, the plot in (b) was a t a reduced am bient pressure of ~ 1 to rr.

178


tions. Do they always occur in the region of the film which is m ost narrow? Are these

transitions ever subcritical? O ther interesting questions concern the effect of the ap

pearance of the stagnation point in th e shear flow of a sm all a and large I annulus on

the flow p a tte rn of electroconvective flow. Theoretical work m ay also be w arranted,

however com putational simulation m ay be the fastest rou te to an exploration of the

param eter space.

F.2 Electroconvection w ith Oscillatory Shear

Having studied the effect of a circular C ouette shear flow on electroconvection, it

is na tu ra l to ask how other shear flows alter electroconvection. C ircular C ouette

flow is very special in certain respects. Specifically, it is a steady sta te flow which

is independent of the fluid viscosity, see for exam ple Eqn. 3.19. The viscosity and

hence the P randtl-like number V do not appear in Eqn. 3.19 which describes th e

azim uthal velocity of the fluid under C ouette flow. This absence of the viscosity can

be seen to orig inate from the fact th a t the to ta l frictional to rque on any ring of fluid

is zero, see for instance Ref. [2]. Since it has been established in C hapter 3 th a t

a rb itra ry independent rotations of the inner and outer edges of the annulus can, by

changing one's fram e of reference, be transform ed away, one m ay question as to how

other mechanically imposed shears can be studied. The one th a t is discussed here is

a tim e-dependent shear that is imposed by sinusoidally oscillating the inner edge of

the annulus in the theory or the inner electrode in the experim ent. Such a shear will

hereafter be referred to as an oscillatory shear.

T he experim ental apparatus, as described in Section 2 .2 , is adequate for im ple

m entation of th is oscillatory shear. T h e theoretical trea tm en t would require som e

generalization to time-dependent shears. It should a t once be clear th a t while th e

s teady s ta te C ouette shear is dependent only upon the ra te of ro ta tion of the inner

electrode, the time-dependent oscillatory shear depends on b o th the am plitude and

frequency of sinusoidal oscillation of th e electrode. As shall be estabhshed below, th e

tim e-dependence necessarily implies a dependence on the viscosity, unlike the Cou-

179


e tte sheax flow. Even though the tim e-dependence of the oscillatory shear is simple

it has significant consequences on the shear profile and therefore one expects im por

ta n t effects on electroconvection. Like the C ouette flow, the oscillatory shear is an

axisym m etric flow.

Determining the shear profile of the oscillatory shear is simply an exercise in

fluid mechanics or p a rtia l differential equations. Using the relevant electroconvection

length and time scales, the film w idth d and tim e scale f Qd / n (see Section 3 .2 ),

the oscillatory shear is described by a dimensionless frequency u and dim ensionless

azim uthal velocity am plitude of the inner electrode T 0. T he Navier-Stokes equation

for th e azimuthal dim ensionless velocity Y (r, t) in polar coordinates is

d ( d 2 I d T \(F.9)

T he boundary conditions a t the inner and outer edges of the annulus are

T ( r i .«) = T 0e,w‘. T ( ro. £ ) = 0 . (F.10)

Leaving out the details, the solution for the radial derivative of the stream function

0 L b

^ £ = -T ( r ,£ ) = A J x{Xr) + B N x{Xr) e , (F .l l )

where

. J ( u j \ 2

l \ v ) !M X * ) n - 1

B = - T o

M X r 0)

Ni(Xrt) - (F.12)

A = - BJ i { X r 0) •

and J i (N i ) is the first order Bessel function of th e first(second) kind.

T he oscillatory sh ear is dependent on V and as a consequence, it is expected

th a t the V dependence o f electroconvection under oscillatory shear will be m arkedly

180


— Oscillatory shear Couette shear

oO

>3

3s*** ~ViVi

JOCo

"33 1 c<u£5

o

4.24 4.4 4.6 4.8 5Dimensionless radius

Figure F.5: A com parison of the C ouette and Oscillatory shear profiles. A 'snapsho t’ of the shear profiles for C ouette and Oscillatory shears in an annulus a t a = 0.80.

different from th a t under C ouette shear. Recall th a t in the theory, the expressions

regarding stab ility were simplified due to C ouette shear satisfying D(D,D4>ti°'1) = 0-

For the oscillatory shear this is not true and leads to different s tab ility properties. It

is enlightening to com pare the C ouette and oscillatory shear profiles. A com parison

a t a = 0.80 is provided in Fig. F.5. P lo tted are the steady s ta te C ouette profile

and a ‘snapsho t’of the oscillatory' shear profile a t t = 0 . T he param eters are chosen

such tha t the velocity am plitude of the inner electrode for th e oscillatory shear is

identical to th a t for the C ouette flow. T he oscillatory profile is then dependent on

th e X , or u> and V . For the param eters chosen, the oscillatory profile is very different

from the C ouette profile, in fact, there is in th e oscillatory case a retrograde flow.

Keep in m ind th a t w hat is illustrated in Fig. F .5 is only a 'sn ap sh o t’ and the profile

oscillates in tim e. A prelim inary and approxim ate theoretical work on the stab ility

properties ind icate markedly different results from the C ouette shear case, in fact there

181


is some expectation tha t a t high oscillation frequency the oscillatory shear flow would

be destabilizing to electroconvection. A more accurate analysis will be undertaken

by the au thor. Experim ental work on the oscillatory system, which is quite easily

performable in th e current appara tu s, should be undertaken.

F.3 M easurement of V iscosity

One of the problem s tha t was encountered in interpreting the experim ents th a t were

performed was th a t the m aterial properties of the liquid crystal w ith dopant were

poorly characterized. W hereas the electrical conductivity could be m easured directly,

the viscosity was only indirectly determ ined, see Section 4.3. A direct m easurem ent

of the viscosity is thus desirable. Since the oscillatory shear, unlike the Couette shear,

is dependent on the viscosity(see Eqns. F . l l and F.13), it can be exploited to m easure

the viscosity.

The experim ent recommended is to suspend particles on a film while it executes

an oscillatory shear. It is expected th a t the particles will execute an oscillation

much like th a t depicted in Fig. F .6 . In the experim ent to determ ine the viscosity,

the protocol would be to determ ine the ‘angular spread’ of th e paxticles a t several

radial positions a t a given am plitude and frequency of oscillation. Since it is likely

tha t experim ent would be difficult to perform in a reduced pressure environm ent, the

effects of air d rag can be sub tracted away by considering the lim it —> 0 . Hence th e

experim ent would have to be perform ed a t several a /s . The ‘angu lar spread’ or the

angular displacem ent can also be calculated theoretically. For instance if the chosen

experim ental measurem ent is twice the am plitude 2 Z( r ) of oscillation as a function

of r , then one need only determ ine th is from the expressions for th e velocity field of

the oscillatory shear:

where Y (r, t) is given by Eqns. F . l l and F.13. W ithout doubt, th e expressions will get

complicated, b u t they are com prised essentially of well behaved cylinder functions.

(F.13)

The details are left to the interested experim enter. A nonlinear fitting routine would

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Figure F.6 : A schematic for m easuring the in-plane viscosity. M easurem ents of the angular deviation of particles suspended in the film a t several different radii and for various am plitude and frequency of oscillation will provide the necessary d a ta to o b ta in the in-plane viscosity.

be used to determine X and thereby with a knowledge of V through the knowledge

of the film conductivity and thickness, one would determ ine the viscosity. This is

a m easurem ent th a t is highly recom mended for it determ ines a heretofore unknown

m aterial parameter.

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Bibliography

[1] H.K. Moffat, Six Lectures on G eneral Fluid Dynamics and Two on H ydrom ag-

netic D ynam o Theory, Les Houches, 1973, in Fluid Dynamics edited by R. Dalian

and J.L . Peube. G ordon and Breach Science Publishers (1977).

[2] G .K . Batchelor, A n Introduction to Fluid Dynamics , Cam bridge (1967).

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