Flows of Nematic Liquid Crystals

ANNUAL REVIEWS Ann. Rev. Fluid Mech. 1978. 10: 197-219

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FLOWS OF NEMATIC LIQUID CRYSTALS

:-:8121

James T. Jenkins Department of Theoretical and Applied Mechanics, Cornell University, Ithaca, New York 14853

INTRODUCTION The first liquid crystal was discovered about a century ago by Reinitzer (1888), who observed that the organic crystal cholesteryl benzoate melted at a temperature of 145°C yielding a turbid fluid which, upon the further addition of heat, became clear at a temperature of 179°C. The turbid fluid was found to exhibit optical birefringence, so the intermediate state apparently possessed the mechanical pro perties of a liquid and the optical properties of a crystal. Further research on the melts of organic crystals composed of relatively long organic molecules indicated that there was often a succession of liquid crystalline states between the crystal and the isotropic fluids. Mixtures of an isotropic liquid and either amphiphilic molecules as in soaps, or the rodlike molecular configurations of synthetic poly peptides, were also found to exhibit birefringence over ranges of concentration and/or temperature. The liquid crystalline states can be broadly classified into two types, smectic and nematic, on the basis of the molecular order that is characteristic of each. In smectic liquid crystals the molecules are arranged in parallel layers, often with their long axis oriented normal to the layers. Usually the molecules of each layer possess the mobility of a fluid within the layer. The orientation of the long axis with respect to the layer and the degree of mobility of the molecules within the layers provide the basis for further subclassifications of the smectics. Nematic liquid crystals are less ordered than smectics. In nematics the centers of mass of the molecules are unordered, as in isotropic fluids ; but the relatively long molecules exhibit an orientational ordering and, locally, seek to remain parallel to one another. In an ideal sample of nematic there is uniform parallel alignment of the molecules, and short-range molecular forces resist any departures from this homogeneous state. Chiral molecules with a liquid crystalline phase exhibit a variation of the nematic ordering. In these materials, called cholesteric liquid crystals, the molecules, locally, lie on parallel planes in which nematic order prevails ; how ever, as the normal to the planes is traversed, the direction of the molecules rotates with a pitch that is comparable to optical wavelengths.

0066-4189/78/0115-0197$01.00

1 97


198

JENKINS

Nematic and cholesteric liquid crystals are optically uniaxial, the local optic axis corresponding to the average orientation of the long axes of the molecules. The orientation of this axis usually varies smoothly from point to point; however, nematics derive their name from the existence of lines that often thread through a sample on which the orientation changes abruptly. The regular twist . of the optic axis in cholesterics is responsible for the rather spectacular optical properties of these materials. Smectic liquid crystals are' optically uniaxial or biaxial depending on whether or not the molecules in the phase orient normal to the layers. Nematics flow readily with an apparent viscosity which is about that of water. Because of their greater structure, the apparent viscosity of cholesterics and smectics may be many orders of magnitude larger than the nematics. As a result, data on flows of cholesterics and smectics is scarce, the continuum theories for smectics are still rather tentative, and the existing theory for cholesterics has been extensively tested only in static situations. Consequently, the focus here is on nematic liquid crystals. The molecular forces that act to maintain a uniform parallel orientation of the optic axis in nematics are small, typically of the order of 10-6 dynes. Because of this, the orientation of the axis can be influenced by electric fields, magnetic fields, thermal gradients, solid boundaries, fluid interfaces, and by flow. The possibility of controlling the optical properties of the material in such a variety of ways is the basis for many of the technological applications of nematic liquid crystals, particularly in display devices. Observations of the effect of flows on the orientation of the optic axis, the influence of a fixed orientation on the flow, and the generation of flows by changes in orientation providc thc motivation for dcveloping a continuum theory to describe the interaction between the local orientation of the molecules and their gross, or hydrodynamic, motion. Here one such continuum theory is presented in outline, and the applications of this theory to plane and cylindrical Poiseuille and Couette flows are reviewed in order to illustrate the variety of new phenomena that are observed in these familiar situations and to assess the success of the theory in describing the experiments. Recent reviews of the physicochemical properties of liquid crystals (Gray & Winsor 1974) and the physics of liquid crystals (de Gennes 1974, Stephen & Straley 1974) contain elaborations of this continuum theory in the context of their larger topics, while Ericksen (1976) has lately reviewed the statics of this theory and Leslie (1978) has done the same for the flow phenomena. All of these reviews contain extensive references to earlier work and a discussion of other continuum theories.

THE CONTINUUM THEORY In the continuum theory for nematic liquid crystals initiated by Oseen (1925) and developed by Ericksen (1961) and Leslie (1968), a unit vector field n(x, r), called the director, describes the local orientation of the optic axis. In these presumably in compressible materials, the director field, detailing the average molecular orienta tion, and the velocity field v(x, t) of the hydrodynamic motion of the molecules must both be determined in order to specify the dynamical state of a material


199

particle. The necessity for this more elaborate kinematics for the liquid crystals is supported by the observations, discussed by de Gennes (1974), of situations in which director motions are unaccompanied by velocity fields. The balance of energy for an arbitrary material volume V of liquid crystal surrounded by a material surface S is, in this continuum theory,

.!!.- r (j-PvoV+1ulioli+E)dV= r (F'V+Goli+q)dV+ dt Jv Jv

f

s

(t'v+s'li-h)dS.


(1) The unfamiliar terms in the energy balance are associated with the independent motion of the director. Treating the familiar terms first, p, E, F, and q are, respectively, the volume density of mass, internal energy, body force, and external heat supplied;

while t and h are, respectively, the contact force and the heat lost per unit area. The superposed dot indicates a material time derivative. The second term in the total energy corresponds to the kinetic energy of rotation of the director, and u is the associated inertial parameter. The form of this contribution is the same as that for the rotational kinetic energy of a slender rigid rod about its mass center. Finally, G is an external generalized force per unit volume that works to change the orientation of the director, and s is a similar generalized contact force. Invariance arguments applied to the energy balance (Leslie 1 978) yield the standard integral balance laws for mass and linear momentum and the expression

L

(PXAV+UIiAii)dV=

L

(XAF+nAG)dV+

I

(XAt+nAS)dS.

(2)

The form of the first term in each integral leads to the identification of this as the balance of angular momentum. Consequently, K == nAG and I == n AS are, respectively, the external couple per unit volume and the contact couple per unit area. The second term in the integral on the left has the same form as the angular momentum of a slender rigid rod about its mass center. An example of an external couple is that due to an applied magnetic field. In a liquid crystal the magnetization M induced by an applied magnetic field H depends upon the orientation of the optic axis. If the relationship between M and H is linear, (3) where X 1 and X2 are the magnetic susceptibilities perpendicular to and parallel to the optic axis, respectively. The usual expression for the couple per unit volume of a magnetic material is M A H. Adopting this, K= nA [(X2 - XI)(n'H)H],

(4)

sO that G= (x2-Xl)(n'H)H,

(5)

up to an arbitrary term in the direction of n. Nematic liquid crystals are diamagnetic, so both susceptibilities are negative and small, generally of the order of 1 0-6 to 1 0-7 in cgs units. In these materials the magnetic anisotropy Xa == X2 - Xl

200

JENKINS

is positive, so the magnetic field seeks to orient the optic axis parallel to itself. Associated with an inhomogeneous magnetic field is a body force per unit volume


F= (M·V)H.

(6)

Contact couples are exerted, for example, at a solid boundary that has been mbbed or scored in one direction. In this event the molecules at the surface lie on the surface and are aligned in the direction of rubbing (Berreman 1973). An orientation perpendicular to a solid boundary may be obtained by cleaning the surface with acid or by coating it with an oricnted monolaycr (Proust, Ter-Minassian-Saraga & Guyon 1 972). In each case contact couples applied at the boundary maintain the director orientation against the competing orienting influences of external. fields, flows, and other boundaries and against the short-range molecular forces that strive to maintain uniform orientation in the material. To obtain local forms of the balance laws, one appeals to the usual tetrahedron argument to relate t, s, and h to the unit outward normal v to S and to, respectively, the stress, director stress, and the heat flux:

h = hiVi'

(7)

The order of indices on the stress and the director stress is important because neither is, in general, a symmetric tensor. In terms of the stress and the director stress, the local expr�ssions of the balance of linear momentum and angular momentum are (8) and (9) The last equation may be written more compactly by introducing a vector g, called the intrinsic director force, which is determined, up to a vector parallel to n, by ( 1 0) With the indeterminacy of g in mind, the balance of angular momentum (9) is, then, equivalent to (1 1) which is called the director balance law. Although the main concern here is the purely mechanical theory for nematics, it is worthwhile to write down the energy equation because of the insight it provides into the nature of the d issipative mechanisms in the theory. One form of this equation, involving kinematical variables that are important to the constitutive theory, is (Leslie 1968)

(12) in which ( 1 3)


201

and


(14) The last two quantities are the familiar stretching tensor and vorticity tensor, respectively ; while the first two are, respectively, the rate of change of the director and the spatial gradients of this time rate, each calculated relative to the local rotation of the material. The stress, director stress, and intrinsic director force have equilibrium parts, which apply in isothermal situations involving at most rigid motions, and dissipative parts, which contribute to changes in the local entropy density 1/. The equilibrium parts are those of the static isothermal theory discussed in detail by de Gennes (1974) and recently reviewed by Ericksen (1976). The theory is based upon a free-energy density W , given in terms of E , 1/ , and the temperature T by W E - TI/, which is supposed to depend upon the temperature, the director, and the director gradients. The form of the free-energy function, =

W=

(15)

W(T, nhni.d,

is to be unaffected by rigid rotations of the material and, because the ends of the nematic molecules are indistinguishable, insensitive to a reversal of sign of the director. One such function, quadratic in the director gradients, has been in common use since it was obtained by Frank (1958) as an expansion about the uniform state: (16) The coefficients kJ, k2, and k3 are at most functions of the temperature and measure the resistance of the nematic to three types of distortions of the uniform parallel alignment called, respectively, splay, twist, and bend. The resistance of the nematic to local departures from the homogeneous alignment is often referred to as curvature elasticity. In order for the uniform director configuration to correspond to a local minimum of W, these coefficients must all be positive (Ericksen 1966a). Their magnitudes are typically of the order of 1 0-6 dyne (de Jeu, Claassen & Spruijt 1976). Thermodynamic arguments based upon an entropy inequality lead Leslie (1968) to constitutive relations of the form (17) aw

. = n·f3. + - +s ..

s. Jl

I

J

_

ani,j

(18)

Jh

and and

1/ =

-

aw

'

aT

'

(19)

where tildes denote the dissipative parts, p is an arbitrary scalar field, and P is an arbitrary vector field. The scalar p is associated with the constraint of incompressi-

202

JENKINS

bility; the vector field P is associated with the constraint on the magnitude of the director. As a consequence of the invariance of W under rigid rotations (Ericksen 1961) W ejik n a nk

(/

+ni,p

nk,p

np '

=

np,k

0;

(20)

so g is given by (10) as g'


l

=

f3 aw An.+n·· t,} J. - -- + g. l

ani

"

(21)

or, with y == A-V· P, (22) In the above, g is determined in terms of the dissipative parts of the stress and director stress by Equation (10), and the arbitrary scalar A or, equivalently, y is present due to the indeterminacy of g. The terms in ( 1 8) and (22) involving the vector P contribute nothing to the director balance (11) or energy equation (12); however, the scalar y, called the director tension, is essential to the director balance because of the constraint on the director. Leslie (1968) takes the dissipative parts in his constitutive theory for liquid crystals to be those introduced by Ericksen (1960a) and Leslie (1966) in a theory for anisotropic fluids that do not possess the short-range molecular forces of the nematics. This constitutive theory allows for the interaction between the orientation and the flow or heat flux and includes a viscous resistance to changes in orienta tion. Here the stress, intrinsic director force, and the heat flux are assumed to be functions of the temperature and the director and to depend linearly upon the temperature gradient, the corotational derivative of the director (13) 1, and the stretching tensor (14) 1. Because the director gradients are not included in the list of constitutive variables, Equation (10) requires that the dissipative part of the director stress vanish. If it were necessary to include the director gradients in order to describe the dissipation, a far more complicated constitutive theory would result. The constitutive assumptions could also be expanded to include dependence upon the rotationally invariant gradients (13)z of the director time rate, and in this event the dissipative part of the director stress need not vanish (Leslie 1968): However, the experiments on flows of nematic liquid crystals to be discussed below provide a strong indication that the constitutive relations of the anisotropic fluid theory are sufficient to describe the dissipative processes in nematic liquid crystals even in flows in which the director gradients are large. The forms of the dissipative stress and heat flux that are compatible with their indifference both to the sign of the director and to rigid rotations of the material are


203

and (24) where the viscosity coefficients temperature. From (10) 9, = }'1N,+}'2Aijnj,

a

and the thermal conductivities

K

depend upon the (25)

where


(26) The viscosities and the thermal conductivities are restricted by the condition that the local entropy production be non-negative, (27) and equal to zero only in rigid motions. From this, Leslie (1966) obtains the inequalities (28) (29) and (30) There is an Onsager relation noted by Parodi (1 970) that may be used to reduce the number of independent viscosity coefficients from six to five: (31) This equality has been adopted by many but not all researchers, some preferring to wait until it is supported by experiment. An experimentally determined relation ship between the viscosities, which was originally thought to confirm (31), does not in fact depend upon it (Kiry & Martinoty 1977, Leslie 1978), so this support is still lacking. Currie (1973) shows that the ability of nematics to propagate certain waves in the presence of a magnetic field hinges upon this equality and observes (Currie 1 974) that it may also be viewed as a stability condition. The viscosity coefficients for the nematic materials with the acronyms PAA, MBBA, HBAB, CBOOA, and PCB have been determined using a variety of experi mental techniques, induding spectroscopy of the light scattered by the long wavelength fluctuations of the director (Orsay Liquid Crystal Group 1971), measurement of the adsorption of ultrasonic shear waves (Kiry & Martinoty 1977), and capillary viscometry employing an orienting magnetic field (Gahwiller 1973). The Onsager relation (31) is invariably used in interpreting the experiments so that fivc viscosity cocfficients are obtained. The magnitude of the viscosities is typically of the order of 10 - 2 to 10 - 1 poise. Tn the mechanical theory generally used to describe flows of nematics, the temperature is assumed to be constant, and the energy Equation (12) is ignored.

204

JENKINS


In this event the equations of interest are the linear momentum balance (8) used with the constitutive relations (17) and (23), the director balance ( 1 1 ) used with the constitutive relations (1 8), (22), and (25), the constraint on the magnitude of the director, and the constraint of incompressibility. The boundary conditions are the familiar no-slip condition on the velocity and the specification of the director orientation, usually as either parallel to or perpendicular to a boundary. A solution of these eight equations gives the vector fields of the velocity and the director and the scalar fields of pressure and director tension associated with the constraints.

FLOWS OF NEMATIC LIQUID CRYSTALS Porter & Johnson (1 967) reviewed the viscometry of liquid crystals just prior to a dramatic increase of activity in the field. At the time of their writing it was clear that external electric and magnetic fields could be used to orient "molecular aggregates" of nematic liquids and that this orientation had a definite effect on the apparent viscosity in simple flows. It was also certain that the shear flows themselves influenced the orientation of the molecules and that the orienting influence of the flow was often in competition with that of the boundaries. Evidence of this com petition was the variation of the apparent viscosity with shear rate in a capillary viscometer. Here the apparent viscosity was highest at the lowest shear rates and decreased to a limiting value as the shear rate was increased. Presumably the limiting constant viscosity at the highest shear rates corrcsponded to the molecular orientation, thought then to be parallel to the streamlines, which was dictated by the flow. However, because the gap widths used in the early experiments were relatively large, the nematic material was turbid; so observations of the orientation of the optic axis were impossible. Also, no attempt was made to obtain a definite orientation at the boundary. In the more recent work, care is taken to control the boundary orientation, gap widths less than 500 Itm are used, and a variety of novel optical techniques are employed to monitor the orientation of the optic axis. The flows considered here illustrate how the shear, the local director orientation, the curvature elasticity, and, sometimes, an external magnetic field act together to maintain usually steady flows in relatively simple and familiar geometries. Often these steady flows exhibit instabilities that have no counterpart in Newtonian fluids. As will become apparent, the continuum theory just outlined has been reasonably successful in describing these steady flows and at least the onset of the instabilities. In some instances analysis has preceded experiment, which later verified specific predictions; more often analysis has been undertaken in order to explain experimental observations. In practical situations the inertia of the director is small (Saupe 1969); so, in what follows, the inertial parameter (J in the director balance equation (1 1 ) is set equal to zero. Also, when considering external fields, attention is confined to magnetic fields; consequently the convective electrohydrodynamic instabilities in a layer of nematic subjected to an ac electric field arc not discussed. An outline of this topic and the relevant references are given in the monograph by de Gennes (1974). Finally, because only isothermal flows are treated, an account of the convective


205

thermal instabilities induced i n thin films of nematics is not included. These instabilities, which depend upon the anisotropy of the heat-flux vector, are analyzed in a recent paper by Dubois-Violette (1974), which contains additional references to both theory and experiment.


Steady Flows

Steady flows of nematic liquid crystals are characterized by a competition between inertial, viscous, elastic, and when an orienting field is present, electromagnetic forces and couples. The relative importance of the terms of each type in the linear momentum equation and the director balance is determined by the magnitudes of the nondimensional ratios Re == pUL/rx, Er == pUL/k, and Z02 = XaL2H2/k, given in terms of a characteristic length L, velocity U, field strength H, and typical values rx and k of the viscosities and the curvature elasticities, respectively. The first ratio is the Reynolds number; the second and thi rd ratios measure the relative importance of, respectively, the viscous couple and the magnetic couple relative to the elastic couple. For gap widths in which optical measurements are possible, situations in which the viscous terms are comparable to the elastic terms (Er 1) correspond to a Reynolds number, based upon the order of magnitude of the elastic and viscous coefficients given earlier, of 10-4. In flows in which Er is much greater than unity, the viscous effects dominate the elasticity, and the orienting influence of the boundaries is expected to be confined to regions close to the boundaries. Mathematically, the highest spatial derivatives of the orientation are, in this event, multiplied by a small parameter; however, theory exploiting this has yet to be developed. Outside of these boundary layers, the orientation is determined by the flow or, if a magnetic field is present, by a competition between the flow and the magnetic field. Suppose that a liquid crystal is experiencing the simple shearing flow =

Vx

=

Ky,

(32)

where K is a positive constant, and that the components of the director are written as

nx = cos 0 sin cp,

n)' = sin 0,

nz = cos 0 cos cp,

(33)

where here the angles 0 and cp are at most functions of time. For this spatially uniform director field, the terms in the constitutive relations (1 8) and (22) involving .the free energy contribute nothing to the field equations, and the director balance (1 1 ) may be written, upon elimination o f the director tension, as (34) and (35) where the dots indicate a derivative with respect to time. These equations express the requirement that the two components of the viscous couple exerted by the

206

JENKINS

flow upon the director vanish. In this situation, the linear momentum Equation (8) is satisfied if the pressure is a function of the time only. One steady solution of tliese equations is ¢ = 0 = 0, with the director perpendi cular to the plane of shear. In the event that I ),dA21 � 1, there are others given by


¢ = n/2,

(36)

with the director in the plane of shear. Recalling that A\ must be negative and assuming A2 to be positive, Ericksen's (1960b) stability analysis shows that of these steady orientations the only one that is stable to uniform time-dependent perturba tions of the orientation alone is that with 0 � 00 � n/4 (or, because nand n are indistinguishable, 5n/4 � 00 � 3n/2) and that all time-dependent solutions of (34) and (35) tend toward this stable orientation as time passes. In liquid crystals the director often, but not invariably, adopts a steady orientation in a shear flow. When it does, it lies in the plane of shear almost parallel to the streamlines. In the context of the theory, this indicates that A2 is positive with a value greater than but close to that of A\. Gahwiller (1972) reports, for example, that in MBBA 00 varies from 5° to 17.5° as the temperature is increased from room temperature to the clearing point (43°C). If I AdAll > 1, the nature of the solutions to (34) and (35) is entirely different. With the exception of 0 = ¢ = 0, there are no steady solutions. In this regime the components of the director are periodic functions of time, and the director tumbles in the shear flow. In this case these equations are equivalent to those derived by Jeffery (1922) for the motion of an ellipsoid of revolution with major axis a and minor axis b in a shear flow of a Newtonian fluid, provided one makes the identification -

-

(37) In the nematic liquid HBAB, for example, there is a temperature range over which flow alignment is not observed (Gahwiller 1972); however it is not anticipated that such a uniform tumbling will persist in a liquid crystal when orienting boundaries are present. Ericksen (1966b) has determined the steady orientations adopted by those fluids for which I AdA21 :$ 1 when a magnetic field is applied either in the plane of shear or perpendicular to this plane. In materials with a positive magnetic anisotropy Xa, the magnetic field attempts to orient the director parallel to itself. Thus, a uniform magnetic field applied normal to the plane of shear with a sufficiently large magnitude H might be expected to stabilize the steady orientation ¢ = (} = 0, which is unstable when the shear flow alone is present. Ericksen shows that this is the case provided that (38) while, if the inequality is reversed, the stable orientation of (36) dictated by the flow is the only steady orientation that is stable to uniform perturbations of the director. For a uniform magnetic field applied in the plane of shear,

Hx = H cos 1/1,

Hy = H sin 1/1,

Hz=O,

(39)


207

the situation is more complicated. For materials with positive Xu> the only possible stable, steady orientations lie in the plane of shear, so ¢ = n/2. In this event the director equation reduces to


/(8) == K(}'1 + A2 cos 28)+ XaH2 sin 2(1/1- 8) = 0,

(40)

which expresses the balance between the viscous and magnetic couples exerted upon the director. Even for materials that align in shear, there need not exist solutions for all values of 1/1 and H2/K. For example, suppose that in a material of this type A2 is positive, then there are three possibilities (Leslie 1971) which, because nand -n are indistinguishable,may be considered by restricting attention to - n/2 � 1/1 � n/2. If 00 ;;;; 1/1 � n/2, the orientation of the magnetic field is close to that of the shear orientation and there always exists a first solution 0" smallest in absolute value, between 1/1 and 00, If 00 - n/2 < 1/1 < 00, a solution need not exist, but does so for sufficiently small or sufficiently large values of H2/K. In the first case, 01 is close to 00; in the second, 01 is close to 1/1. Finally, if 00 ;;;; 1/1 < n/2, a solution always exists with 01 > 1/1 if 1(0) is positive, and 81 < - 80 if 1(0) is negative. Ericksen (1966b) provides stability criteria that, when a steady solution exists, determine whether or not it is stable to uniform perturbations of the orientation. Miesowicz (1936) has used a magnetic field to align the optic axis in a layer of nematic liquid crystal between two parallel plates, one of which was slowly oscillating to create a shear ftow. By suitably orienting the magnetic field and measuring the decay of the oscillations, he was able to determine the apparent viscosities of the liquid crystal when the optic axis was perpendicular to the plane of shear, in the plane of shear and perpendicular to the plates, and in the plane of shear and parallel to the flow. The values of these apparent viscosities, interpreted in the context of the anisotropic fluid theory using the stress relation (18), determine the viscosity coefficient ()(4 and provide two relations between the coefficients ()(2, ()(3, ()(S, and ()(6' The reproducibility of his results by other means indicates that the experiments were performed with magnetic-field strengths that maintained a steady orientation of th e optic axis against the flow and with a combination of field, gap width, and shear rate that minimized the influence of the boundaries. In order to assay the importance of the curvature elasticity in transmitting the boundary orientation into regions of flow in these and similar experiments, Leslie (1968,197.1) used the constitutive relations (17),(18),(22), (23), (25) that he proposed for liquid crystals to analyze shear flow between parallel plates. The flow was supposed to be maintained in a gap of width h by the steady motion of the upper plate with velocity V. For the steady shear flow -

-

-

Vx

=

u(y),

vr

=

Vz

=

0,

(41)

with the directors in the plane of shear, nx

=

cos 8(y),

ny

=

sin O(y),

(4 2)

the linear momentum equation reduces to an expression for the pressure and an equation for the velocity,

[M(O) + N(O)]u' = 2b.

(43)

208

JENKINS

Here, M( B) ==ct4+( cts-ct ) sinz B, N ( B) == (2ctl sinzB+iX3+c(6) cosz B, the prime z denotes a derivative with respect to y, and b is the constant shear stress tyx• The velocity profile is clearly influenced by the director orientation and is not the linear profile of a Newtonian fluid. In the event that the uniform magnetic field (39) is applied in the plane of shear, the director balance simplifies to


2F( B) B" +F.e( B) B'Z + (AI +A2 cos 2B) u' +X"Hz sin 2( 1jJ - 8)

""

(44)

0,

where F( 8) == k, cosz B+k3 sin28. The first two terms in this equation are the elastic couple acting upon the director. Leslie supposes that the boundary orientation is parallel to the plates and restricts attention to solutions in which 8 is symmetric about the center of the gap and varies monotonically from zero at the lower plate to an extreme value 8m at the center. He considers only materials that align in shear. In the absence of a magnetic field, such a solution (Leslie 1968) has the property that the absolute value of 8m never exceeds that of 80. Also, as the ratio bhzI( kl + k3) becomes large, the angle 8 goes to the shear orientation 80 everywhere in the gap except in a boundary layer at each wall. This agrees with the interpretation by Porter & Johnson (1962) and Porter, Barrall & Johnson (1966) of the viscometry experiments in terms of thin adsorption layers at each boundary to which their orienting influence is confined at the high rates of shear. When this ratio is small, the orientation in the gap is, according to the solution, essentially that at the boundaries. When a magnetic field is present, the character of such a steady solution (Leslie 1971) depends upon whether or not there exist angles satisfying (40). When there are, Leslie shows that, as the ratio h2(XaHZ +b)/(kl +k3) becomes large, 8 approaches 81 everywhere in the gap except in the boundary layers. When this ratio is small, the boundary orientation dictates that in the gap. Also, the apparent viscosity {l, defined as the ratio of the shear stress b to the nominal shear rate K V Ih, attains different limiting values in each extreme. Numerical solutions (Finlayson 1974) of the differential Equations (43) and (44), for perpendicular wall orientations and a magnetic field normal to the plates, illustrate this limiting behavior of the apparent viscosity for PAA and MBBA, both of which align in shear. On the other hand, de Gennes (1972) shows the nature of steady solutions of the type considered by Leslie when there is no steady orientation 81 determined by the competition between the flow and the magnetic field. In this case, de Gennes indicates that the steady orientation of the director rotates rapidly in one direction as the gap is traversed from the bottom to the center. In his example the number of these rotations depends upon the nominal shear rate. This solution is also relevant to the behavior of materials which do not align in shear, for by choosing IjJ -11/4 in Equation (40) it becomes identical in form to Equation (36), and there exists a range of H for which this equation can not be satisfied. For a material that does not align in shear, de Gennes (1974) gives an illustration of the strongly deformed, steady director configuration that might be anticipated in a shear flow between parallel plates in the absence of a field. However, Pikin's (1974) analysis of moderately =

=


209

distorted director configurations in rectilinear shear flows of these materials indicates that such steady solutions are likely to be unstable. When no magnetic field is present, the velocity field associated with Leslie's solution leads to a scaling law for the apparent viscosity


(45) For a given material, the form of this function is fixed, so that experiments involving different gap widths and wall velocities may be correlated provided that the director field in each experiment is that obtained by Leslie. The function f(Kh2) approaches finite limits as Kh2 becomes small or large . The first limit is the apparent viscosity associated with the uniform orientation dictated by the wall; the second limit is that associated with the shear orientation. Consequently,the theory has the capacity for predicting both the relatively high apparent viscosities at low rates of shear and the lower apparent viscosity over a range of high shear observed by Peter & Peters (1955). Ericksen (1969) shows that the relation (45) between the apparent viscosity, nominal rate of shear,and gap width applies to more complicated parallel shearing flows as a consequence of a general scaling of the balance laws and the constitutive relations of the continuum theory. When the director inertia is neglected and atten tion is focused on a given material,the time, lengths, and the kinematic and kinetic variables appearing in the linear momentum equation and the director balance may be scaled in a way that preserves the form of these equations. To apply this scaling to the flows between parallel plates in a way that takes a solution of the boundary value problem into a solution, a restriction on the director, which may vary with position and time in the interior of the gap, is imposed on the boundaries. The orientation of the director at each boundary must be constant and must either remain so as V and h are varied or depend upon V and h only in the combination Vh. Then, if the apparent viscosity fl is defined as the ratio of a representative shear stress T to the nominal shear rate K, and a subscript zero denotes quantities associated with a unit gap width, this scaling requires that T h-2To and V = h-1 Yo, so that =

/t

==

T/K = To/Vo = flo.

(46)

Consequently, provided that some criterion is available to distinguish a unique solution for each Vo so that �10 is determined uniquely by Yo, (47)

fl = flo = f(Vo) = f(Vh) = f(Kh2).

The last proviso is important because, as Currie (1970) indicates, even the plane problems involving parallel orientation at each plate possess a variety of steady solutions other than those considered by Leslie (1968, 1 971), de Gennes ( 1 974), or Pikin ( 1 974). For example, if 8(0) 0 and 8(h) 2mr, where n is an integer, the director rotates n times in passing from the bottom plate to the top plate in a far more complicated flow. As is discussed below, steady plane configurations in these shear flows are observed to be unstable, and the resulting secondary flows and =

=


210

JENKINS

director orientations are often not plane. Thus, a failure of a scaling law like (47) need not indicate a deficiency in the continuum theory, but may result from com paring the apparent viscosities of entirely different flows. Recent experiments indicate that a scaling of this type does obtain in a somewhat different shear flow. Wahl & Fischer ( 1 973) study the shear flow of the nematic liquid crystal MBBA between parallel discs in relative rotation. The discs were treated to obtain a normal orientation of the director at their surface. Using the birefringence of the liquid crystal, they determine the average orientation of the optic axis over the gap at various gap widths and nominal shear rates. After scaling their optical data in the way suggested by Ericksen (1969), they find that the results of experiments run at different angular velocities and gap widths do indeed all fall upon a single curve. Wahl and Fischer argue that, at the relatively low rates of shear that they employ, Leslie's (1 968) analysis of rectilinear shear flow between paraliel plates may be used to interpret the results of their rotational flow experiments. Using it, they obtain values for the elastic constants, viscosity coeffi cients, and the shear orientation flo that are in relatively good agreement with those obtained by other methods. These results and the success of the scaling lend support to the continuum theory. A similar partial experimental confirmation of the scaling of the equations of the continuum theory has been obtained in cylindrical Poiseuille flow. Atkin (1970) analyzed the axial flow of a nematic liquid crystal under the influence of a pressure gradient in an annulus bounded by coaxial circular cylinders and in a capillary tube with a circular cross section. The director was assumed to have only radial and axial components which, like the velocity, varied only with the radius. The director orientation at the walls was supposed to remain constant. On the center line of the capillary the director must be parallel to the axis in order for the stresses to remain finite. Restricting attention to materials that aligned in shear, Atkin was able to obtain an exact solution for the flow in the annulus provided that the angle between the director and the streamlines never exceeded the shear orientation angle. For flow in the capillary, Atkin found it impossible to establish even the existence of a solution for arbitrary orientations at the wall ; however, numerical solutions to his differential equations were later obtained for PAA (Tseng, Silver & Finlayson 1972) and MBBA (Finlayson 1 974) for perpendicular alignment at the wall. In both cases Atkin did carry out the scaling analysis. If, in the capillary, the director orientation at the boundary is supposed to be at most a function of the azimuthal angle, Atkin found that the apparent viscosity must depend upon the ratio of the volume flux Q to the radius R of the capillary jJ. =

g(QR

1 ,

)

(48)

where, for a given material and wall orientation, the function 9 is fixed provided that jJ. can be uniquely related to Q in a capillary of unit radius. In experiments on Poiseuille flow of PAA through thin capillaries, Fisher & Fredrickson (1969) obtained data for four capillaries ranging in diameter from 78 to 516 �lm. Before one series of tests, they cleaned the capillaries with an acid in order to obtain a boundary orientation perpendicular to the wall. Before a second



211

series, they first cleaned two of the capillaries and then passed a bundle of fibers through each and drew the fibers back and forth in an effort to obtain a boundary orientation parallel to the axis of the capillary. In the first series of tests, their data relating the apparent viscosity to the volume flux and the capillary radius scaled as predicted by Atkin; in the second series, it did not. Because the agreement between the theory and the experiments was so remarkable with the perpendicular wall orientation and because there was no certainty that the axial orientation at the wall was obtained by rubbing the wall with the fiber bundle, Currie (1975) sought to explain the results of the second series of tests using the continuum theory. He approximated Atkin's system of equations and obtained solutions which, for the perpendicular wall orientation, were in good agreement with the numerical solutions of the exact equations and which gave the observed behavior of the apparent viscosity in the experiments involving this boundary condition. Then, for the two capillaries in the second series of tests, he was able to reproduce the experimental variation of the apparent viscosity by choosing different boundary orientations somewhere between axial and perpendicular for each capillary. Thus,

the failure of the scaling here could have resulted from comparing experiments with different wall orientations. This result is important, because the success of the scaling for the perpendicular wall orientation in capillaries with such small diameters is a strong argument in favor of Leslie's constitutive assumptions. If it were necessary to include the director gradients in the dissipative parts of the constitutive relations, the scaling should fail in this situation where these gradients are so large. In a sequel to Leslie's analysis of rectilinear flow between parallel plates, Atkin & Leslie (1970) considered Couette flow of a nematic liquid crystal between coaxial rotating cylinders. The director was again supposed to lie in the plane of shear with parallel orientation obtaining at each cylinder, and their attention was restricted to materials that oriented in shear. The analysis is more complicated than that for the rectilinear flow because the boundary conditions in the cylindrical geometry force the director field to be nonuniform even when both cylinders are at rest. They obtained a steady, axisymmetric solution in which the angle between the director and the circular streamlines varies monotonically from zero at the inner cylinder to an extreme value, smaller in absolute value than the shear orientation Ao, and then returns monotonically to zero at the outer cylinder. As might be expected, as the axial torque applied per unit length to each cylinder is increased, this angle approaches the stable shear orientation everywhere in the gap except in orientation boundary layers at each wall. For this solution, the apparent viscosity is a function of the torque and the ratio of the radii of the inner and outer cylinders; however, the scaling also applies to a much wider class of steady, plane solutions and director boundary conditions. Currie (1970) has investigated the other steady, plane, axisymmetric solutions corresponding to arbitrary, and possibly different, orientations of the director at the inner and outer cylinder. In an exhaustive analysis of the phase plane of the non linear first-order differential equation governing the orientation angle, he determined the qualitative behavior of the solutions to this wide class of boundary value problems. Currie treated both materials that do and do not orient in shear and

212

JENKINS

included the possible presence of circumferential and radial magnetic fields. He did not, however, address himself to the stability of these solutions.


Flow Instabilities Before considering the instabilities observed in plane Poiseuille and cylindrical Couette flows and the interpretation of these in terms of the continuum theory, it is worthwhile to discuss the instabilities in the shear flows between parallel plates. These evolve from a uniform flow and orientation, involve unambiguous determina tions of the director during their evolution, and have been rather successfully analyzed using the continuum theory. In common with the instabilities observed in Couette flow, these instabilities are unique to liquid crystals and may initiate at a Reynolds number as low as 10-2. In a shear flow of MBBA maintained between parallel plates by the constant velocity of the upper plate, Pieranski & Guyon (1973) observed that a uniform director field, parallel to the plates but perpendicular to the plane of shear, began to distort as the velocity of the upper plate exceeded a critical velocity. Above this threshold, the director in the interior of the gap began to turn into the plane of the shear and to tilt relative to the plane of the plates. They determined that the critical velocity varied inversely with the distance between the plates. When a magnetic field was applied parallel to the undistorted director, the critical shear rate increased, for moderate fields, with the square of the field, but did not vary with the gap width. The behavior of each threshold is consistent with the scaling of the continuum theory. As the strength of the magnetic field was further increased, a second instability was observed to initiate at a critical value of the field (Pieranski & Guyon 1974a). In this second regime, they observed a roll pattern of secondary flow and director distortion with the axis of the rolls parallel to the velocity of the upper plate and with a half wavelength normal to the plane of shear equal to the gap width. A similar roll instability involving either of two distinct types of rolls was obtained in the absence of a magnetic field by oscillating the upper plate. In similar circumstances, Pieranski & Guyon (1974b, 1976) found the behavior of HBAB and CBOOA to be somewhat different over the range of temperature in which these materials do not shear align. Here the first, or homogeneous, instability was not observed; however, over a part of this temperature range, the roll instability was obtained in the absence of a magnetic field above a critical shear rate. As the low temperature limit of this part of the range was approached, the wavelength of the rolls decreased and the critical shear rate increased without bound. Below this limiting temperature, a weakly damped oscillation was observed in which the frequency of oscillation was proportional to the shear rate. For materials that align in shear, a rough explanation of the homogeneous instability is possible. The elastic couples are successful in maintaining the uniform orientation consistent with the boundary conditions until the viscous couples are strong enough to begin to force the director in the interior of the gap into the plane of shear, towards the stable shear orientation. The magnetic field applied parallel to the undistorted director provides a stabilizing couple and increases the velocity threshold. In an analysis of steady perturbations of the director based on

21 3



these considerations, Pieranski & Guyon (1973) obtain values for the critical shear for both the magnetic and the nonmagnetic threshold that are in qualitative agreement with their experimental results. Manneville & Dubois-Violette (l976a) and Leslie (1976a,b) analyze the onset of the homogeneous instability in greater detail, recognizing, at the outset, that the continuum theory requires that a transverse flow must accompany such an instability. Manneville and Dubois-Violette report that such a transverse flow has been observed in the experiments. Each analysis proceeds from the linear-momentum equation and the director-balance equation. Adopting the form (33) for the components of the director, where the angles are here assumed to be functions of y and t, they suppose t hat the components of the velocity are Vx

=

u(y, t),

Vy

=

Vz

0,

=

wry, t).

(49)

In this case, if the inertia of the fluid is ignored, the linear-momentum equation reduces to an equation for the pressure and [M(I:J) + N(I:J) cos 2 4>Jw' +N(I:J)u' sin 4> cos 4> +2K(I:J)e cos 4>-2L(I:J)� sin I:J 2a, =

(50)

N(I:J)w' sin 4> cos 4> + [M(8) +N(O) sin2 4>]u' + 2K(8)e sin 4> +2L(I:J)¢ cos 4> = 2b. (51) Here, K(8) =0 C(3 cos21:J- C( sin2 0, L(O) =0 C( sin 0 cos 0, and a and b are the constant 2 2 values of the shear stresses tyz and tyx, respectively. The components of the director balance that remain after the elimination of the director tension are

2F(0)B" + F,o(8)B'2 - G,o(O)4>'z +2AI8+ (AI + A2 cos 20)(w' cos 4> +u' sin 4» sin 20 cos z 4> = 0, XaH2

(52)

and

2G(O)4>" +2G.o(I:J)I:J'4>' + 2AI cos zl:J � + (AI - Az) sin I:J cos l:J(w' sin 4>- u' cos 4» -

z XaH sin 24> cos2 0

=

0,

(53)

where G(0) (k2 cosz 0 + k3 sin2 0) cosz 0 and H is the magnitude of the stabilizing field. Considering first the possible existence of steady infinitesimal perturbations, Leslie (1 976a,b) and Manneville & Dubois-Violette (1976a) linearize these equations about the undistorted director configuration 0 =0 4> =0 ° and undisturbed flow u = Vy/h, w == 0, a = 0, b = C(4V /h. They find that such 'steady perturbations are not possible in materials that do not align in shear, and this is in accord with the experimental observations. For those materials that align in shear, Leslie obtains analytical expressions for the critical shear in terms of the field, gap width, and material parameters for one such solution that involves a transverse velocity but no net transverse flux. Manneville and Dubois-Violette obtain numerical values for the critical shear associated with this solution as a function of the magnetic field for =


214

JENKINS

MBBA. A second solution, which necessitates a net transverse flux, is shown by Leslie (1 976b) to exist at a smaller value of the critical shear than the first over a range of moderate to large magnetic fields. Thus there'is a possibility of an exchange of stability between the two solutions as the strength of the stabilizing field increases ; however, the second instability does not appear to be observed in the experiments. Pieranski & Guyon (1 974a) give a qualitative explanation for the onset of the roll instability when the motion of the upper plate is steady and a magnetic field is present. Here variations in the director orientation in the direction z normal to the plane of shear are crucial. R oughly, a fluctuation or variation of ¢ with z results in a component of velocity normal to the plates that also depends on z. The shear flow associated with this component of velocity gives rise to a couple, a portion of which is destabilizing and works to increase e. For large values of the magnetic field, the onset of a homo geneo us instability is suppressed and the roll instability takes place first at a critical value of the shear rate. The detailed consideration of the onset of this instability is far more complicated and has been undertaken by Manneville & Dubois-Violette (1 976a). They carry out a modal analysis ofthe infinitesima� steady perturbations in the full linear momentum equation and director balance. For a fixed value of the magnetic field, these equations determine the vertical component qy of the wave vector in terms of the horizontal component qz and the shear rate K. With this, the boundary conditions

can be satisfied for a nontrivial solution, provided K and qz are s uitably related. The absolute minimum of the curve K (q,) gives the shear threshold and the critical horizontal wavelength. Manneville and Dubois-Violette plot numerical values of this threshold for MBBA versus the applied magnetic field. Above a moderate value of the field, this threshold is less than that for either of the homogeneous instabilities ; again this is in qualitative agreement with the experiments. Their analysis may also be applied to the onset of roll instabiIlty seen in materials that do not align in shear. Similar two-dimensional modal analyses have been used with equal success to determine the appropriate thresholds for the convective roll instabilities in a thin layer of a nematic subjected to an ac electric field (Pikin 197 1 , Penz & Ford 1972) or a temperature gradient (Dubois-Violette 1 974). Pieranski & Guyon (1 974a) also analyze the roll instabilities observed when the upper plate is oscillated. They employ the linearized equations for the perturbations, but suppose that the spatial variations in the y and z directions are of the order of the gap width, and essentially ignore the boundary conditions. When strong stabilizing fields are present and the upper plate is supposed to oscillate as a square wave, their resulting system of ordinary differential equations governing the spatially uniform but time-dependent perturbations of ¢ and (J possess two types of solutions. For a fixed value of the average shear, below a frequency threshold of the excitation, and for moderate values of the stabilizing field, the relevant solution involvcs an oscillation in ¢ about zero with no change in sign of &. As the strength of the field is increased, this instability may disappear, to be replaced at still higher values of the field by a solution in which () oscillates about zero with no change in the sign of ¢. A second possibility is for the transition to take place gradually without the disappearance of the instability at intermediate values of the field. The results of an



215

extensive series of experiments carried out by Pieranski & Guyon (J 974a) are remarkably similar to these predictions of thcir analytical model. As Guyon & Pieranski (1 974) emphasize, their differential equations are of the same form as those that have been used to describe the onset of electrohydrodynamic convective roll instabilities i n a thin layer of a nematic (Smith et al 1 975). Guyon & Pieranski (1975) and Janossy, Pieranski & Guyon (1 976) have extended the experimental studies of shear-flow instabilities to plane Poiseuille flow main tained between fixed parallel plates by a pressure gradient. The orientation of the director at the boundaries was again parallel to the plates and perpendicular to the plane of shear. Their attention was restricted to materials that aligned in shear, and both steady and oscill ato ry pressure differences were employed. It is not surprising that some fe�tures of the instabilities observed in the steady and oscillatory simple shea ring flows persist ; however, there are interesting diffe rences due to the curvature of the velocity profile of the Poiseuille flow. Here, because the highest shear rate occurs at each plate, the instabilities initiate near each boundary. When the pressure difference is steady, these instabilities are locally similar to the homogeneous instabilities in simple shear. Above a critical value of the pressure difference, the directors in the neighborhood of each plate may begin to turn into the plane of shear and to tilt towards the shear orientation i n either of two ways. Consequently, two types of steady homogeneous instabilities are possible. In one, cf> is symmetric about the center of the gap and e is antisymmetric ; the average tilt is zero. In the other, e is s ymm etric and cf> is antisymmetric ; the average twist is ze ro. The first type of instability is observed to occur in the e xperiments above a critical pressure difference. Manneville & Dubois- Violette (1976b) present a compact analysis of the onset of these instabilities based upon the appropriate modification of Equations (50)-(53). The right-hand side of (51) must be replaced by the shear stress tyx = - dy + e, where d is the pressure gradient and e is a constant. They obtain solutions of the linear equations governing the steady perturbations about the uniform detector configuration and parabolic velocity profile that correspond to the two instabilities. The lower value of the threshold, which is close to that of the experiment, is associated with the solution having an average twist. This is the director configuration seen in th� experiment. Because ofthe curvature of the velocity profile, a secondary flow involving a net transverse flux is anticipated whenever the director has a nonzero average twist (Pieranski & Guyon 1 974c). This transverse flux favors the initiation of the first type of instability. In order to accomodate the transverse flux in the flow cell of the experiment, domains form in the direction of the flow with the fluxes in neighboring domains oppositely directed. At large values of the pressure difference, another steady homogeneous configuration is observed in the experiment. In this, there is an average twist but no net transverse flux ; the flux here may be suppressed by a transverse pressure gradient. As the pressure difference is slowly reduced, there is a transition from this nonlinear regime to the linear instability. When an oscillatory pressure difference is applied, the threshold of the homo geneous instability increases as the frequency of the oscillation is increased from zero. However, above a critical frequency, this instability gives way to a convective


216

JENKINS

roll instability with the axes of the rolls parallel to the imposed pressure gradient. The behavior of the director is similar to that at the threshold in periodic simple shear; rjJ oscillates about zero, but e does not change sign. Here, however, because the shear is of different sign in the upper and lower halves of the cell, convective rolls occur in each half. In this case, the wavelength of the rolls is observed to decrease as the pressure difference is increased. Manneville & Dubois (1976b) indicate that an analysis of these instabilities is forthcoming. Janossy, Pieranski & Guyon (1 976) also report on the influence of stabilizing fields on the behavior of the homogeneous and roll instabilities. In the steady case, the dependence of the threshold on the field is that predicted by Manneville & Dubois-Violette (1976b) in an extension of their analysis of the homogeneous instabilities. Cylindrical Couette flows of the nematics CBOOA and HBAB exhibit a cascade of instabilities over the range of temperature in which these materials do not align in shear (Cladis & Torza 1 975, 1 977). The first two of these instabilities seem to be peculiar to the cylindrical geometry, the third is similar to the convective roll instability of these materials in simple shear between parallel plates, and the fourth is the familiar Taylor roll instability seen in Newtonian fl uids. The first three instabilities take place at nominal shear rates that are several orders of magnitude smaller than that at which the Taylor rolls initiate. Cladis and Torza treat the walls of their coaxial cylinders to obtain a radial orientation there and rotate the inner cylinder. As the angular velocity of rotation is increased from zero, they observe a gradual change in orientation in the plane of shear towards the streamlincs. At a critical .value of the angular velocity, the orientation at some point in the gap changes abruptly by n, and this change of orientation propagates as rotations of n/2 to the orientation boundary layers at each wall. The resulting orientation in the gap is in the plane of shear and roughly parallel to the streamlines; however, there are indications that around the outer perimeter of each boundary layer the director has left the plane of shear and is aligned parallel to the axis of the cylinders. Throughout, the velocity profile is the same as that in a Newtonian fluid except in the boundary layers. The material in the inner boundary layer appears to rotate with the inner cylinder, while the material in the outer layer is at rest. As the angular velocity is increased further, this essentially plane configuration persists until, at a second critical value of the angular velocity, the directors suddenly leave the plane of shear, and convective rolls develop with their axes aligned along the circumference. As in the case of simple shear, the size of the rolls decreases as the temperature is lowered, and the critical angular velocity increases without bound as a limiting temperature is approached. Below this temperature, the rolls do not develop, and the orientation remains in the plane of shear roughly parallel to the streamlines. Cladis and Torza provide a qualitative explanation of these events based upon equations (52) and (53). I nitially, the viscous couple is balanced by the curvature elasticity ; however, as this couple increases with the shear rate, a point is reached at which this balance can not be maintained. The director then tumbles once to a configuration involving extreme local distortions in which the elastic and viscous couples are again balanced. Apparently, the large local distortions


217

near each wall permit the curvature elasticity of the material in rigid motion to support the shear stresses and the couples exerted by the flow. At a higher shear rate, the equatioi1'S' indicate that this plane configuration again becomes unstable, this time to out-of-plane perturbations. As in simple shear, an orientation per pendicular to the plane of shear leads to the convective rolls. In simple shear of CBOOA and HBAB between parallel plates with a boundary orientation parallel to the velocity, the initial instability seen by Pieranski & Guyon ( l 974b 1 976) and Pieranski, Guyon & Pikin (1976) differs from that seen in the cylindrical geometry. Here, as the velocity of the upper plate is increased from zero, the parallel orientation gradually distorts following Pikin's ( 1 974) plane solution to Equations (5 1 ) and (52) for small e. At a critical value of the shear rate, this distorted configuration becomes unstable as predicted by Pikin's (1 974) modal instability analysis of the system (50)·-(53) linearized about the distorted state. At this point, the directors in the experiment begin to turn out of the plane of shear and continue to do so as the shear rate is increased, eventually becoming


,

perpendicular to the plane of shear. I f the shear rate is increased further, the

convective rolls develop at a rather high critical shear rate. The difference between the initial instabilities observed in the cylindrical and plane geometries is probably due to the difference in the boundary orientations. In the plane geometry, the instability initiates after a moderate distortion of the parallel orientation. In the cylindrical geometry, a relatively greater viscous couple induces a large distortion of the initially inhomogeneous director field before the steady configuration becomes unstable. Even if the directors were oriented perpendicular to the boundaries in both the plane and cylindrical geometries, the flows would not be strictly comparable because of the initial elastic energy of the director configura tion in the annulus. Leslie (1 964) and Ericksen (1 966c) made an early start toward analyzing the Taylor instability using the anisotropic fluid theory, and Cladis & Torza (1 975) indicate that Leslie's extension of the classical analysis provides a value for the threshold of this instability that is in rough agreement with their observations. However, because of the fascinating variety of instabilities seen at extremely low Reynolds numbers in nematic liquid crystals, the formation of the Taylor rolls is not so singular an event as in the Newtonian fluids. Literature Cited

A tki n,

R . J. 1 970. Poiseuille floW of liquid crystals of nematic type. Arch. Ration.

Mecho Anal.

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Atkin, R. J., Leslie, F. M. 1970. Couette flow of nematic liquid crystals. Q. J. Mech.

. Appl. Math.

23 : S I-S24 W. 1973. A lig nme nt

Berreman, D. of liquid crystals by grooved surfaces. Mol. Cryst.

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Phys. Rev. Lett.

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Cladis, P. E., Torza, S. 1977. Flow instabilities in Couette flow in nematic liquid crystals.

Proc. 50th Colloid and Su rf. Sci. Symp., SanJuan, P.R. 1 976. New York : Academic.

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1 970. Couette flow of a nematic liquid crystal in the presence of a magnetic field. Arch. Ration. Mech. Anal. 37 : 222-42 Currie, P. K. 1973. Implications of Parodi's

relation for waves in nematics. Solid State

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Currie, P. K . 1975. Approximate solutions for Poiseuille flow ofliquid crystals. Rheol. Acta 1 4 : 688-92 de Gennes, P. G. 1 972. Mecanochromatic effect in nematics. Phys. Lett. 4 1 A : 479 de Gennes, P. G. 1 974. The Physics of Liquid Crystals. Oxford : Clarendon. 333 pp. de Jeu, W. H., Claassen, W. A. P., Spruijt, S. M. 1. 1 976. The determination of the elastic constants ofnematic liquid crystals. Mol. Cryst. Liq. Cryst. 3 7 : 269-80 Dubois-Violette, E. 1 974. Determination of thermal instabilities thresholds for homeo tropic and planar nematic liquid crystal samples. Solid St. Commun. 1 4 : 767-71 Ericksen, J. L. 1 960a. Anisotropic fluids. Arch. Ration. M eeh. Anal. 4 : 23 1-37 Ericksen, 1. L. 1960b. Transversely isotropic fluids. Kolloid-Z. 1 7 3 : 1 1 7-22 Ericksen, J. L. 1 96 1 . Conservation laws for liquid crystals. Trans. S oc. Rheol. 5 : 23-24 Ericksen, J. L. 1 966a. Inequalities in liquid crystal theory. Phys. Fluids 9 : 1 205-7 Ericksen, J. L. 1 966b. Some magnetohydro dynamic effects in liquid crystals. Arch. Ration. Mech. Anal. 23 : 266-75 Ericksen, 1. L. 1 966c. Instability in Couette flow of anisotropic fluids. Q. J. Mech. A pp l. Math. 1 9 : 455-59 Ericksen, J. L . 1969. A boundary-layer effect in viscometry of liquid crystals. Trans. Soc. R heol. 1 3 : 9-1 5 Ericksen, J . L . 1 976. Equilibrium theory of liquid crystals. Adv. Liq. Cryst. 2 : 233-98 Finlayson, B. A. 1974. Numerical computa tions for the flow of liquid crystals. In L iquid Crystals and Ordered Fluids, Vol. 2, ed. J. F. Johnson, R. S. Porter, pp. 2 1 123. New York : Plenum. 783 pp. Fisher, J., Fredrickson, A. G. 1 969. Interfacial effects on the viscosity of a nematic meso phase. Mol. Cryst. Liq. Cryst. 8 : 267-84 Frank, F . C. 1 958. On the theory of liquid crystals. Discuss. Faraday Soc. 25 : 1 9-28 Giihwiller, C. 1972. Temperature dependence of flow alignment in nematic liquid crystals. Phys. Rev. Lett. 28 : 1 5 54-56 Giihwillcr, C. 1973. Direct determination of the five independent viscosity coefficients in a nematic liquid crystal. Mol. Cryst. Liq. Cryst. 20 : 301- 1 8 Gray, G . W., Winsor, P . A., eds. 1 974. LiqUid Crystals and Plastic Crystals, Vols. 1 , 2. New York : Wiley. 383 pp., 3 1 4 pp. Guyon, E., Pieranski, P . 1 974. Convective instabilities in nematic liquid crystals. Physica 73 : 1 84-94 Guyon, E., Pieranski, P. 1975. Poiseuille flow instabilities in nematics. J . Phys. (Paris) 36-CI : 203-8

Janossy, I., Pieranski, P., Guyon, E. 1976. Poiseuille flow in nematics : experimental study of the instabilities. J. Phys. (Paris) 37 : 1 1 05-1 1 3 Jeffery, G . B . 1 922. The motion o f ellipsoidal particles immersed in a viscous fluid. Proc. R. Soc. London Ser. A 102 : 1 6 1 -79 Kiry, F., Martinoty, P. 1 977. Ultrasonic investigation of anisotropic viscosities in a nematic liquid crystal. J. Phys. (Paris) 38 : 1 5 3-57 Leslie, F. M. 1 964. The stability of Couette flow of certain anisotropic fluids. Proc. Cwilhridge Phi/os. Soc. 60: 949-55 Leslie, F. M. 1 966. Constitutive equations for anisotropic fluids. Q. J. Mech. Appl. Math. 1 9 : 357-70 Leslie, F. M. 1 968. Some constitutive relations for liquid crystals. Arch. Ration. Meeh. Anal. 28 : 265-83 Leslie, F. M. 1 9 7 1 . Continuum theory of liquid crystals. R heol. Acta 1 0 : 9 1 -95 Leslie, F. M. 1 9 76a. An analysis of a flow instability in nematic liquid crystals. J. Phys. D 9 : 925-37 Leslie, F. M . 1 976b. Magnetohydrodynamic instabilities in nematic liquid crystals. Mol. Cryst. Liq. Cryst. 37 : 335-52 Leslie, F. M . 1 978. Theory of flow pheno - inena in liquid crystals. Adv. Liq. Cryst. In press Manneville, P., Dubois-Violette, E. 1 976a. Shear flow instability in nematic liquids : theory of steady simple shear flows. J. Phys. (Paris) 3 7 : 285-96 Manneville, P., Dubois-Violette, E. 1 976b. Steady Poiseuille flow in ncmatics : theory of the uniform instability. J. Phys. (Paris) 37 : 1 1 1 5- 1 24 Miesowicz, M. 1 936. Der Einfluss des magnetischen Feldes auf die Viskositiit der Fliissigkeiten in der nematischen Phase. Bull. Int. Acad. Pol. (Cl. Sci. Math. Nat.) Ser. A, pp. 228-47 Orsay Liquid Crystal Group. 1 97 1 . Viscosity measurements by quasi elastic light scattering in p-Azoxyanisol. Mol. Cryst. Liq. Cryst. 1 3 : 1 87-9 1 Oseen, C. W. 1 925. Beitriige zur Theorie der anisotropen Fliissigkeiten. Ark. Mat. Astron. Fys. 1 9A : 1 - 1 8 Parodi, 0. 1 970. Stress tensor for a nematic liquid crystal. J. Phys. (Paris) 3 1 : 58 1-84 Penz, P. A., Ford, G. 1 972. Electromagnetic hydrodynamics of liquid crystals. Phys. Rev. A 6 : 4 14-25 Peter, S., Peters, H. 1 955. U ber die Struktur viskositiit des Kristallin-fliissigen p-p' Azoxyanisols. Z. Phys. Chem. 3 : 1 03-25 Pieranski, P., Guyon, E. 1 973. Shear-flow induced transitions in nematics. Solid State Commull. 1 3 : 435-37

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of certain shear flows in nematics. Phys.

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