Orientational instabilities in nematic liquid crystals with weak anchoring under combined action of steady flow and external fields I. Sh. Nasibullayev,1,2 O. S. Tarasov,3 A. P. Krekhov,1,3 and L. Kramer1,* 1

Physikalisches Institut, Universität Bayreuth, D-95440 Bayreuth, Germany Institute of Mechanics, Russian Academy of Sciences, 450054 Ufa, Russia 3 Institute of Molecule and Crystal Physics, Russian Academy of Sciences, 450075 Ufa, Russia 共Received 10 March 2005; revised manuscript received 9 September 2005; published 9 November 2005兲 2

We study the homogeneous and the spatially periodic instabilities in a nematic liquid crystal layer subjected to steady plane Couette or Poiseuille flow. The initial director orientation is perpendicular to the flow plane. Weak anchoring at the confining plates and the influence of the external electric and/or magnetic field are taken into account. Approximate expressions for the critical shear rate are presented and compared with semianalytical solutions in case of Couette flow and numerical solutions of the full set of nematodynamic equations for Poiseuille flow. In particular the dependence of the type of instability and the threshold on the azimuthal and the polar anchoring strength and external fields is analyzed. DOI: 10.1103/PhysRevE.72.051706

PACS number共s兲: 61.30.Hn, 64.70.Md, 61.30.Cz

I. INTRODUCTION

Nematic liquid crystals 共nematics兲 represent the simplest anisotropic fluid. The description of the dynamic behavior of the nematics is based on well established equations. The description is valid for low molecular weight materials as well as nematic polymers. The coupling between the preferred molecular orientation 共director n兲 and the velocity field leads to interesting flow phenomena in nematics. The orientational dynamics of nematics in flow strongly depends on the sign of the ratio of the Leslie viscosity coefficients = ␣3 / ␣2. In typical low molecular weight nematics is positive 共flow-aligning materials兲. The case of the initial director orientation perpendicular to the flow plane 共spanned by the primary flow velocity and its gradient兲 has been clarified in classical experiments by Pieranski and Guyon 关1,2兴 and theoretical works of Dubois-Violette and Manneville 共for an overview see Ref. 关3兴兲. An additional external magnetic field could be applied along the initial director orientation. In Couette flow and low magnetic field there is a homogeneous instability 关1兴. For high magnetic field the homogeneous instability is replaced by a spatially periodic one leading to rolls 关2兴. In Poiseuille flow, as in Couette flow, the homogeneous instability is replaced by a spatially periodic one with increasing magnetic field 关4兴. All these instabilities are stationary. Some nematics 共in particular near a nematic-smectic transition兲 have negative 共non-flow-aligning materials兲. For these materials in steady flow and in the geometry where the initial director orientation is perpendicular to the flow plane only spatially periodic instabilities are expected 关5兴. These materials demonstrate also tumbling motion 关6兴 in the geometry where the initial director orientation is perpendicular to the confined plates that make the orientational behavior quite complicated. Most previous theoretical investigations of the orientational dynamics of nematics in shear flow were carried out

*Deceased. 1539-3755/2005/72共5兲/051706共10兲/$23.00

under the assumption of strong anchoring of the nematic molecules at the confining plates. However, it is known that there is substantial influence of the boundary conditions on the dynamical properties of nematics in hydrodynamic flow 关7–11兴. Indeed, the anchoring strength strongly influences the orientational behavior and dynamic response of nematics under external electric and magnetic fields. This changes, for example, the switching times in bistable nematic cells 关7兴, which play an important role in applications 关12兴. Recently the influence of the surface anchoring on the homogeneous instabilities in steady flow was investigated theoretically 关9,11兴. In this paper we study the combined action of steady flow 共Couette and Poiseuille兲 and external fields 共electric and magnetic兲 on the orientational instabilities of the nematics with initial orientation perpendicular to the flow plane. We focus on flow-aligning nematics. The external electric field is applied across the nematic layer and the external magnetic field is applied perpendicular to the flow plane. We analyze the influence of weak azimuthal and polar anchoring and of external fields on both homogeneous and spatially periodic instabilities. In Sec. II the formulation of the problem based on the standard set of Ericksen-Leslie hydrodynamic equations 关13兴 is presented. Boundary conditions and the critical Fréedericksz field in case of weak anchoring are discussed. In Sec. III equations for the homogeneous instabilities are presented. Rigorous semianalytical expressions for the critical shear rate for Couette flow 共Sec. III A兲, the numerical scheme for finding threshold for Poiseuille flow 共Sec. III B兲, and approximate analytical expressions for both types of flows 共Sec. III C兲 are presented. In Sec. IV the analysis of the spatially periodic instabilities is given and in Sec. V we discuss the results. In particular we will be interested in the boundaries in parameter space 共anchoring strengths, external fields兲 for the occurrence of the different types of instabilities.

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©2005 The American Physical Society

PHYSICAL REVIEW E 72, 051706 共2005兲

NASIBULLAYEV et al.

The Navier-Stokes equation 共momentum balance兲 has the form

共t + v · 兲vi = − p,i + 关Tvji + Teji兴,j ,

共5兲

where is the mass density of the NLC, p is the pressure, and the viscous and elastic parts of the stress tensor are Tvij = ␣1nin jAkmnknm + ␣2niN j + ␣3n jNi + ␣4Aij + ␣5ninkAkj + ␣6n jnkAki , Teij = −

F nk,j , nk,i

共6兲

where N = 共t + v · 兲n − ⍀ ⫻ n. In addition we have the incompressibility condition FIG. 1. Geometry of NLC cell 共a兲. Couette 共b兲 and Poiseuille 共c兲 flows.

The basic state is given by the stationary homogeneous solution of Eqs. 共1兲, 共5兲, and 共7兲:

II. BASIC EQUATIONS

Consider a nematic layer of thickness d sandwiched between two infinite parallel plates that provide weak anchoring 关Fig. 1共a兲兴. The origin of the Cartesian coordinates is placed in the middle of the layer with the z axis perpendicular to the confining plates 共z = ± d / 2 for the upper or lower plate兲. The flow is applied along x. Steady Couette flow is induced by moving the upper plate with constant velocity V0 关Fig. 1共b兲兴. Steady Poiseuille flow is induced by applying constant pressure difference ⌬P = P2 − P1 along x 关Fig. 1共c兲兴. An external electric field E0 is applied along z and a magnetic field H0 along y. The standard set of the nematodynamic equations 关13兴 consists of the director equation

␥1共t + v · ⵜ兲n = ␥1⍀ ⫻ n + ␦⬜共− ␥2An + h兲, =

=

n0 = 共0,1,0兲, p0 =

再

0,

␥2 = ␣6 − ␣5 = ␣3 + ␣2 ,

共2兲

冎

共8兲

v0x = V0共1/2 + z/d兲,

共9兲

and for Poiseuille flow v0x = − 共⌬P/⌬x兲共d2/␣4兲共1/4 − z2/d2兲.

共10兲

In order to investigate the stability of the basic state 共8兲 with respect to small perturbations we write n = n0 + n1共z兲eteiqy,

v = v0 + v1共z兲eteiqy ,

p = p0 + p1共z兲eteiqy .

共11兲

Guided by the experimental observations we assume that the wave vector of the destabilizing modes, if not zero, is perpendicular to the flow plane. The case q = 0 corresponds to a homogeneous instability. Here we analyze stationary bifurcations, thus the threshold condition is = 0. It follows from the director normalization n2 = 1 that n1y ⬅ 0 in linear approximation. The linearized equations 共1兲, 共5兲, and 共7兲 are 共K22z2 − K33q2 − 0aH20兲n1x − ␣2v0x,zn1z − iq␣2v1x = 0,

共3兲

共, j denotes the partial derivative with respect to the spatial coordinate x j兲 where F is

共12兲 共K11z2 − K33q2 + 0aE20 − 0aH20兲n1z − ␣3v0x,zn1x − ␣3zv1y

1 F = 兵K11共 · n兲2 + K22关n · 共 ⫻ n兲兴2 + K33关n ⫻ 共 ⫻ n兲兴2 2 − 0a共n · E0兲2 − 0a共n · H0兲2其.

for Couette flow,

where for Couette flow

共1兲

and Eqs. 共5兲 and 共7兲 below. Here ␣k denote the Leslie viscosity coefficients and we have used the Parodi relation. In Eq. 共1兲 ⍀ = 共 ⫻ v兲 / 2 is the local fluid rotation, ␦⬜ ij = ␦ij − nin j is the projection tensor which imposes the normalization of n 共n2 = 1兲, Aij = 共vi,j + v j,i兲 / 2 is the hydrodynamic strain, and h is the force on the director derived from the orientational free energy density

F F − hi = x j ni,j ni

v0 = 关v0x共z兲,0,0兴,

共⌬P/⌬x兲x, for Poiseuille flow,

where ␥1 , ␥2 are rotational viscosities,

␥ 1 = ␣ 3 − ␣ 2,

共7兲

· v = 0.

− iq␣2v1z = 0,

− v0x,zv1z + 共3z2 − 1q2兲v1x + iq共1 − 3兲v0x,zn1z = 0, 共14兲

共4兲

Here Kii are the elastic constants, a is the anisotropy of the dielectric permittivity, and a is the anisotropy of the magnetic susceptibility. 051706-2

共13兲

− iqp1 + 关2z2 − 共4 − 3 − 5兲q2兴v1y − 共3 − 2兲z共v0x,zn1x兲 = 0,

共15兲

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PHYSICAL REVIEW E 72, 051706 共2005兲

− z p1 + 关共3 + 5兲z2 − 1q2兴v1z − iq5v0x,zn1x = 0, 共16兲 iqv1y + zv1z = 0,

TABLE I. Symmetry properties of the solutions of Eqs. 共12兲–共17兲 under 兵z → −z其. Couette flow

共17兲

where 1 = 共␣4 + ␣5 − ␣2兲 / 2, 2 = 共␣3 + ␣4 + ␣6兲 / 2, 3 = ␣4 / 2, 4 = ␣1 + 1 + 2, 5 = −共␣2 + ␣5兲 / 2. The anchoring properties of the director are characterized by a surface energy per unit area Fs, which has a minimum when the director at the surface is oriented along the easy axis 共parallel to the y axis in our case兲. A phenomenological expression for the surface energy Fs can be written in terms of an expansion with respect to 共n − n0兲. For small director deviations from the easy axis one obtains 1 1 2 2 + W pn1z , Fs = Wan1x 2 2

Wa ⬎ 0,

W p ⬎ 0,

共18兲

where Wa and W p are the “azimuthal” and “polar” anchoring strengths, respectively. Wa characterizes the surface energy increase due to distortions within the substrate plane and W p relates to distortions out of the substrate plane. The boundary conditions for the director perturbations can be obtained from the surface torques balance equation ±

F Fs + = 0, n1i,z n1i

共19兲

with “±” for z = ± d / 2 and i = x , z. Taking into account Eq. 共18兲 the boundary conditions for the director perturbations 共19兲 can be written as ±K22n1x,z + Wan1x = 0,

± K11n1z,z + W pn1z = 0, 共20兲

with “±” for z = ± d / 2. It is convenient to introduce dimensionless parameters as ratios of the characteristic anchoring length 共Kii / Wi兲 over the layer thickness d,

a = K22/共Wad兲,

 p = K11/共W pd兲.

共21兲

In the limit of strong anchoring, 共a ,  p兲 → 0, one has n1x = n1z = 0 at z = ± d / 2. For torque-free boundary conditions, 共a ,  p兲 → ⬁, one has n1x,z = n1z,z = 0 at the boundaries. From Eq. 共21兲 one can see that by changing the thickness d, the dimensionless parameters a and  p can be varied with the ratio a /  p remaining constant. The boundary conditions for the velocity perturbations 共no-slip兲 are

Poiseuille flow

Perturbation

“odd”

“even”

“odd”

“even”

n1x n1z v1x v1y v1z p1

odd odd odd even odd even

even even even odd even odd

odd even odd odd even odd

even odd even even odd even

absence of flow 共Fréedericksz transition兲. Clearly the Fréedericksz transition field depends on the polar anchoring strength W p. There is competition of the elastic torque 共K11z2n1z兲 and the field-induced torque 共0aE20n1z兲. The solution of Eq. 共13兲 with n1x = 0, v1y = 0, q = 0, and H0 = 0 has the form n1z = C cos共␦z/d兲,

␦ = EFweak/EF ,

共25兲

EFweak

is the actual Fréedericksz transition field and where EF = 共 / d兲冑K11 / 共0a兲 is the critical Fréedericksz field for strong anchoring. After substituting n1z into the boundary conditions 共20兲 we obtain the expression for ␦, tan共␦/2兲 = 1/共␦ p兲.

共26兲

One easily sees that ␦ → 1 for  p → 0 and ␦ → 冑2 /  p / for  p → ⬁. For  p = 1 one gets EFweak = 0.42EF. III. HOMOGENEOUS INSTABILITY

In the case of homogeneous perturbations 共q = 0兲 from Eqs. 共14兲, 共16兲, and 共17兲 and boundary conditions 共22兲–共24兲 we deduce v1x = 0, v1z = 0, and p1 = 0. In order to simplify equations for n1x , n1z, and v1y we use dimensionless variables as in Ref. 关9兴 ˜z = z/d, N1x = n1x,

˜S =  v , d 0x,z

N1z = n1z,

˜q = qd, V1y = 223

2 = ␣32k2132,

d =

d v1y , d

共− ␣2兲d2 , K22

共27兲

v1x共z = ± d/2兲 = 0,

共22兲

v1y共z = ± d/2兲 = 0,

共23兲

where ij = i / j, ␣ij = ␣i / ␣ j, kij = Kii / K jj. This leads to the following equations 共tildes are omitted兲

v1z共z = ± d/2兲 = v1z,z共z = ± d/2兲 = 0.

共24兲

共z2 − h兲N1x + SN1z = 0,

共28兲

共z2 + e − k21h兲N1z + 23SN1x + zV1y = 0,

共29兲

z2V1y − 共1 − 23兲z共SN1x兲 = 0,

共30兲

The symmetry properties of the solutions of Eqs. 共12兲–共17兲 under the reflection z → −z is shown in Table I. We will always classify the solutions by the z symmetry of the x component of the director perturbation n1x 共first row in Table I兲. In the case of positive a, for some critical value of the electric field the basic state loses its stability already in the

and HF where h= = 共 / d兲冑K22 / 共0a兲, EF = 共 / d兲冑K11 / 共0兩a兩兲 are the critical

051706-3

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H20 / HF2 ,

e = sgn共a兲2E20 / EF2 ,

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NASIBULLAYEV et al.

V1y = C5cosh共1z兲 + C6cos共2z兲 + C7 .

Fréedericksz transition fields for strong anchoring. For the shear rate S one has, for Couette flow, V 0 d a2 =  d

S = a 2,

Taking into account the boundary conditions 共33兲 and 共34兲 the solvability condition for the Ci 共“boundary determinant” equal to zero兲 gives an expression for the critical shear rate a2c , at which the basic state 共8兲 loses its stability,

共31兲

冋 冋

and for Poiseuille flow, S = − a2z,

a2 = −

共32兲

⫻ a2 + tan

The boundary conditions 共20兲 and 共23兲 reduced to ± aN1x,z + N1x = 0,

1 2

2 2

冉 冊册 冉 冊册

 p2 + tan

2 2

− 共h − 21兲

 p1 + tanh

1 2

= 0,

共36兲

where

±  pN1z,z + N1z = 0, for z = ± 1/2,

21 =

共33兲 V1y共z = ± 1/2兲 = 0.

冉 冊册冋 冉 冊册冋

共h + 22兲 a1 + tanh

⌬P dd . ⌬x 3

共35兲

共34兲

b + 冑c2 + 4a4c , 2

22 =

b = 共1 + k21兲h − e,

− b + 冑c2 + 4a4c , 2

c = 共1 − k21兲h + e.

共37兲

For the “even” solution one obtains A. Couette flow

N1x = C1cosh共1z兲 + C2cos共2z兲 + C3 ,

For Couette flow the solution of Eqs. 共28兲–共30兲 can be obtained semianalytically. For the “odd” solution one gets

N1z = C4cosh共1z兲 + C5cos共2z兲 + C6 ,

N1x = C1sinh共1z兲 + C2sin共2z兲,

V1y = C7sinh共1z兲 + C8sin共2z兲 + C9z.

N1z = C3sinh共1z兲 + C4sin共2z兲,

The boundary conditions 共33兲 and 共34兲 now lead to the following condition 共“boundary determinant”兲:

冨

1

h共k21h−e兲−23ac4

h

− a2tan共2/2兲 + 1 共h + 22兲关−  p2tan共2/2兲 + 1兴

a1tanh共1/2兲 + 1

共h − 21兲关 p1tanh共1/2兲 + 1兴

2ac4共1−23兲 tan共2/2兲 2 tanh共1/2兲 1

where 1 , 2 are defined in Eq. 共37兲. Expressions 共36兲 and 共39兲 allow us to determine the influence of anchoring conditions 共a ,  p兲 and external fields on the critical shear rate a2c .

冨

共38兲

共39兲

= 0,

⬁

V1y = 兺 C3,nun共z兲,

共40兲

n=1

where the trial functions f n , gn, and un satisfy the boundary conditions 共33兲 and 共34兲. For the “odd” solution we write

B. Poiseuille flow

In the case of Poiseuille flow the system 共28兲–共30兲 with S = −a2z admits an analytical solution only in the absence of external fields 共in terms of Airy functions兲 关9兴. In the presence of fields we solve the problem numerically. In the framework of the Galerkin method we expand N1x , N1z, and V1y in a series,

f n共z兲 = on共z; a兲,

gn共z兲 = en共z;  p兲,

un共z兲 = on共z兲, 共41兲

and for the “even” solution f n共z兲 = en共z; a兲,

gn共z兲 = on共z;  p兲,

un共z兲 = en共z兲. 共42兲

⬁

N1x = 兺 C1,n f n共z兲, n=1

⬁

N1z = 兺 C2,ngn共z兲, n=1

The functions on共z ; 兲 , en共z ; 兲 , on共z兲, and en共z兲 are given in Appendix I. In our calculations we have to truncate the expansions 共40兲 to a finite number of modes. 051706-4

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TABLE II. Trial functions for the homogeneous solutions. Couette flow

具¯典 =

“odd”

“even”

“odd”

“even”

f共z兲 g共z兲

o1共z ; a兲 o1共z ;  p兲

e1共z ; a兲 e1共z ;  p兲

o1共z ; a兲 e1共z ;  p兲

e1共z ; a兲 o1共z ;  p兲

After substituting Eq. 共40兲 into the system 共28兲–共30兲 and projecting the equations on the trial functions f n共z兲 , gn共z兲, and un共z兲 one gets a system of linear homogeneous algebraic equations for X = 兵Ci,n其 in the form 共A − a2B兲X = 0. We have solved this eigenvalue problem for a2. The lowest 共real兲 eigenvalue corresponds to the critical shear rate a2c . According to the two types of z symmetry of the solutions 共and of the set of trial functions兲 one obtains the threshold values of a2c for the “odd” and “even” instability modes. The number of Galerkin modes was chosen such that the accuracy of the calculated eigenvalues was better than 1% 共we took ten modes in case of “odd” solution and five modes for “even” solution兲. C. Approximate analytical expression for the critical shear rate

In order to obtain an easy-to-use analytical expression for the critical shear rate as a function of the surface anchoring strengths and the external fields we use the lowest-mode approximation in the framework of the Galerkin method. By integrating Eq. 共30兲 over z one can eliminate zV1y from Eq. 共29兲 which gives + e − k21h兲N1z + SN1x = K,

冕

IV. SPATIALLY PERIODIC INSTABILITY

For spatially periodic perturbations 共q ⫽ 0兲 eliminating v1y from Eq. 共13兲 by use of the incompressibility condition 共17兲 and the pressure p1 by taking z derivative of Eq. 共15兲 one obtains the equations for n1x , n1z , v1x, and v1z. We used again the renormalized variables 共27兲 with V1x = 

共43兲

N1z = C2g共z兲,

− 共z2 + ␣23q2兲V1z = 0,

a2c = 冑c1c2/c3 ,

共46兲

with c1 = 具f f ⬙典 − h具f 2典, c2 = 具gg⬙典 + 共e − k21h兲具g2典, c3 = 具fsg典关具fsg典 − 共1 − 23兲具g典具sf典兴,

−

共45兲

where f共z兲 and g共z兲 are given in Table II and Appendix I. Substituting Eq. 共45兲 into Eqs. 共28兲 and 共43兲 and projecting the first equation on f共z兲 and the second one on g共z兲 we get algebraic equations for Ci. The solvability condition gives the expression for the critical shear rate

d v1z , d

共49兲

共50兲

iq共z2 − k31q2 + e − k21h兲N1z + iq23SN1x

−1/2

N1x = C1 f共z兲,

V1z = 223

共z2 − k32q2 − h兲N1x + SN1z + iqV1x = 0,

共44兲

We choose for the director components N1x , N1z the onemode approximation

d v1x, d

which gives

1/2

共SN1x兲dz.

共48兲

The values for the integrals 具¯典 are given in Appendix II. Expression 共46兲 can be used for both Couette and Poiseuille flow by choosing the function s共z兲 关for Couette flow s共z兲 = 1 and for Poiseuille flow s共z兲 = −z兴 and the trial functions f共z兲 and g共z兲 with appropriate symmetry. In comparison with the rigorous calculations for the material parameters of MBBA at 25 °C 关14兴 in the case of Couette flow the one-mode approximation 共46兲 for the “odd” solution has an accuracy that varies from 2.5% to 16% when H0 / HF varies from 0 to 4. The “even” solution has the accuracy of 1–8 % for 0 艋 H0 / HF 艋 3 and of 1–12 % for 0 艋 E0 / EF 艋 0.6. For Poiseuille flow for odd solution the accuracy is 30% in the absence of fields. For the even solution the accuracy is 12–15 % for magnetic fields 0 艋 H0 / HF 艋 0.5. For both Couette and Poiseuille flow the accuracy of the formula 共46兲 decreases with increasing field strengths.

where K is an integration constant. Taking into account the boundary conditions for V1y one has K = 共1 − 23兲

共¯兲dz.

−1/2

Poiseuille flow

Function

共z2

冕

1/2

共51兲

v 2 共 23兲−1SV1z + 共z2 − 13q2兲V1x + iq共13 − 1兲SN1z = 0, d 共52兲

共z4 − 42q2z2 + 12q4兲V1z + iq关共1 − 23兲z2 + 53q2兴共SN1x兲 = 0, 共53兲 and dimensionless shear rate S is defined by Eqs. 共31兲 and 共32兲. The convective term in Eq. 共52兲 is proportional to the ratio of the viscous relaxation time v = d2 / 3 to the director relaxation time d = 共−␣2兲d2 / K22 and can therefore safely be neglected since for the typical NLC material parameters 关 ⬃ 103 kg/ m3, 3 ⬃ 共−␣2兲 ⬃ 10−1 Pa· s, K22 ⬃ 10−11 N兴 one has v / d ⬃ 10−6. We have the boundary conditions for the director 共33兲 and for the velocity

共47兲

where 具¯典 denotes a spatial average 共projection integral兲

V1x共z = ± 1/2兲 = 0, V1z共z = ± 1/2兲 = V1z,z共z = ± 1/2兲 = 0.

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共54兲

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r2 = 具ww共4兲典 − 42q2具ww⬙典 + 12q4具w2典.

Trial functions for the spatially periodic

Couette flow

Poiseuille flow

Function

“odd”

“even”

“odd”

“even”

f n共z兲 gn共z兲 un共z兲 wn共z兲

on共z ; a兲 on共z ;  p兲 on共z兲 on共z兲

en共z ; a兲 en共z ;  p兲 en共z兲 en共z兲

on共z ; a兲 en共z ;  p兲 on共z兲 en共z兲

en共z ; a兲 on共z ;  p兲 en共z兲 on共z兲

The system 共50兲–共53兲 with boundary conditions 共33兲 and 共54兲 has no analytical solution. Thus we solved the problem numerically in the framework of the Galerkin method ⬁

N1x = 兺 C1,n f n共z兲, n=1 ⬁

V1x = 兺 C3,nun共z兲, n=1

⬁

N1z = 兺 C2,ngn共z兲, n=1 ⬁

V1z = 兺 C4,nwn共z兲,

共55兲

n=1

where the trial functions f n , gn , un, and wn satisfy the boundary conditions 共33兲 and 共54兲 共see Table III and Appendix I兲. After substituting Eq. 共55兲 into the system 共50兲–共53兲 and projecting onto corresponding trial functions f n共z兲 , gn共z兲 , un共z兲 , wn共z兲 we get a system of linear homogeneous algebraic equations for X = 兵Ci,n其. This system has the form 关A共q兲 − a2共q兲B共q兲兴X = 0. Truncating the expansion 共55兲 we have solved the eigenvalue problem numerically to find the neutral curve a20共q兲. The minimum of a20共q兲 yields the critical wave number q = qc and the critical shear rate a2c = a20共qc兲. The number of Galerkin modes was chosen such that the accuracy of the calculated a2c and qc was better than 1% 共ten modes for odd solution and five modes for even solution兲. In order to get an approximate expression for the threshold we use the lowest-mode approximation in the framework of the Galerkin method. We used the same scheme described above for the single mode 关n = 1 in expansion 共55兲兴 and get the following formula for the critical shear rate a2c = 冑m1m2/共m3m4兲, with m1 = 具f f ⬙典 − 共k32q2 + h兲具f 2典, m2 = 具gg⬙典 − 共k31q2 − e + k21h兲具g2典, m3 = 具fsg典 + 共13 − 1兲q2具usg典具fu典/r1 , m4 = 具fsg典 + 关具gw⬙典 + ␣23q2具gw典兴 ⫻关共1 − 23兲具w关sf兴⬙典 + 53q2具wsf典兴/r2 , r1 = 具uu⬙典 − 13q2具u2典,

共56兲

共57兲

The values of the projection integrals 具¯典 are given in Appendix II. Expression 共56兲 can be used for both Couette and Poiseuille flow by choosing the function s共z兲 关for Couette flow s共z兲 = 1 and for Poiseuille flow s共z兲 = −z兴 and the trial functions f , g , u, and w with appropriate symmetry. Minimization of Eq. 共56兲 with respect to q gives the critical wave number qc. In the case of Couette flow and strong anchoring an approximate analytical expression for a2c was obtained in Ref. 关15兴 using trial functions that satisfy free-slip boundary conditions for the velocity. The formula 共56兲 is more accurate because we chose for V1z Chandrasekhar functions that satisfy the boundary conditions 共54兲. For the calculations we used material parameters of MBBA at 25 °C 关14兴. Compared with the rigorous calculations the accuracy of Eq. 共56兲 is better than 1% for Couette flow and better than 3% for Poiseuille flow. Note that Eq. 共46兲 for the homogeneous instability is more accurate than Eq. 共56兲 for q = 0 because Eq. 共56兲 was obtained by solving four equations 共50兲–共53兲 by approximating all variables, whereas Eq. 共46兲 was obtained by solving the reduced equations 共28兲 and 共43兲 by approximating only two variables. V. RESULTS AND DISCUSSION

For the calculations we used material parameters of MBBA at 25 °C 关14兴. Calculations were made for the range of anchoring lengths a = 0–1 and  p = 0–1. For strong anchoring a =  p = 0, whereas a =  p = 1 correspond to very weak anchoring when the characteristic anchoring lengths are equal to the NLC layer thickness. A. Couette flow

We found that in the case of Couette flow without and with an additional electric field the critical shear rate a2c for the even type homogeneous instability 共EH兲 is systematically lower than the threshold for other types of instability. Note that in the presence of the field the symmetry with respect to the exchange a ↔  p is broken. In Fig. 2 contour plots for the critical shear rate a2c vs anchoring lengths a and  p for different values of the electric field are shown. The difference between a2c obtained from the exact, semianalytical solution 共39兲 and from the one-mode approximation 共46兲 is indistinguishable in the figure. In Fig. 3 contour plots of a2c 共thin dashed lines兲 and the boundaries where the type of instability changes 关the thick solid lines are obtained numerically, the thick dashed lines are from Eqs. 共46兲 and 共56兲兴 are shown for different values of magnetic field. For not too strong magnetic field in the region of weak anchoring the odd type homogeneous instability 共OH兲 takes place 关Fig. 3共a兲兴. In the region of strong anchoring, 共a ,  p兲 → 0, one has homogeneous instability of opposite z symmetry 共EH兲. Note that the threshold for the EH instability becomes less sensitive to the surface anchoring 关Fig. 3共a兲兴. Increasing the magnetic field the OH region

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FIG. 2. Contour plot of the critical shear rate a2c vs a and  p for weak Couette flow. 共a兲 E0 = 0; 共b兲 E0 = Eweak F , a ⬍ 0; 共c兲 E0 = EF , a weak ⬎ 0. EF = 0.42EF.

FIG. 3. Contour plot of the critical shear rate a2c vs a and  p for Couette flow with additional magnetic field. 共a兲 H0 / HF = 3; 共b兲 H0 / HF = 3.5; 共c兲 H0 / HF = 4. Boundaries between different type of instabilities are given by thick solid lines 共full numerical兲 and thick dashed lines 共one-mode approximation兲.

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failure of the strong anchoring limit and deviations of the initial director orientation from the direction perpendicular to the flow plane. In addition, the difference in the material parameters of the substance used in the experiment and that of “standard” MBBA can also lead to the discrepancy. B. Poiseuille flow

FIG. 4. Neutral curves a20 vs q for Couette flow with additional magnetic field, a = 0.1,  p = 0.1. 共a兲 H0 / HF = 3; 共b兲 H0 / HF = 3.4; 共c兲 H0 / HF = 4.

expands toward stronger anchoring strengths. Above H0 ⬇ 3.2 a region with lowest threshold corresponding to the even roll mode 共ER兲 appears. This region has borders with both types of the homogeneous instability 关Fig. 3共b兲兴. With further increasing magnetic field the region of spatially periodic instability ER expands 关Fig. 3共c兲兴 and above H0 / HF = 4 the ER instability has invaded the whole investigated range of 共a ,  p兲. For strong anchoring and H0 / HF = 3.5 the critical wave number of ER instability is qc = 5.5. It increases with increasing magnetic field and decreases with decreasing anchoring strengths. Leslie has pointed out 共using an approximate analytical approach兲 that for strong anchoring a transition from a homogeneous instability without transverse flow 共EH兲 to one with such flow 共OH兲 as the magnetic field is increased is not possible in MBBA because of the appearance of the ER type instability 关16兴. This is consistent with our results. We find that the EH-OH transition in MBBA is possible only in the region of weak anchoring 关Figs. 3共a兲–3共c兲兴. In Fig. 4 neutral curves for different values of the magnetic field and fixed anchoring lengths are shown 共solid line for ER and dashed lines for OR兲. There are always two minima for the even mode; one of them at q = 0 that corresponds to the homogeneous instability EH. For small magnetic field the absolute minimum is at q = 0 共line a兲. The neutral curve for the odd mode OR is systematically higher than for ER. With increasing magnetic field the critical amplitude for the EH instability 共q = 0兲 increases more rapidly than the one for the ER instability 共q ⫽ 0兲 so that for H0 / HF ⬎ 3.4 the ER solution is realized 共lines b and c兲. In a small range of q 共dashed lines兲 a stationary ER solution does not exist but we have OR instead. For the EH instability under Couette flow and strong anchoring in the absence of fields we find a2c = 12.15 关from the semianalytical expression 共39兲 as well as from the one-mode approximation 共46兲 and also Eq. 共56兲 with q = 0兴. The available experimental value for the critical shear velocity in MBBA at 23 °C is V0c = 11.5 m / s for a sample of thickness 200 m 关1兴, that gives a2c = 6.3 for the material parameters 关14兴. We suspect that the lower experimental value is due to

In Fig. 5 the contour plot for a2c 关thin dashed lines are from the full numerical calculation, dotted lines are from the one-mode approximations 共46兲 and 共56兲兴 and the boundaries for different types of instabilities 关thick solid line: numerical; thick dashed line: Eqs. 共46兲 and 共56兲兴 are shown. In Poiseuille flow the phase diagram is already very rich in the absence of external fields. In the region of large a 共weak azimuthal anchoring兲 one has the EH instability. For intermediate anchoring strengths rolls of type OR occur 关Fig. 5共a兲兴. Note that even in the absence of the fields there is no symmetry under exchange a ↔  p, contrary to Couette flow. The one-mode approximations 共46兲 and 共56兲 do not give the transition to EH for strong anchoring. From the full numerical calculations follows that in the region of strong anchoring, 共a ,  p兲 → 0, the difference between the EH and the OR instability thresholds is only about 5%. By varying material parameters 共increase ␣2 by 10% or decrease ␣3 by 20% or ␣5 by 25% or K33 by 35%兲 it is possible to change the type of instability in this region. Application of an electric field leads for a ⬍ 0 共a ⬎ 0兲 to expansion 共contraction兲 of the EH region 关Figs. 5共b兲 and 5共c兲兴. At E0 / EF = 1 and a ⬍ 0 rolls vanish completely and the EH instability occurs in the whole investigated area of 共a ,  p兲. For a ⬎ 0 the instability of OH type appears in the region of large  p 共weak polar anchoring兲. In this case, increasing the electric field from EFweak to EF causes an expansion of the OH region. Note that for  p ⬎ 1, which is in the OH region, the Fréedericksz transition occurs first. An additional magnetic field suppresses the homogeneous instability 共Fig. 6兲. Above H0 / HF ⬇ 0.5 the spatially periodic instability OR occurs for all anchoring strengths investigated. The critical wave number qc in the absence of fields is 1.4. Application of an electric field decreases qc whereas the magnetic field increases qc. The wave number decreases with decreasing anchoring strengths. In the absence of fields and strong anchoring we find for the EH instability a2c = 102 关Eq. 共46兲 gives 110 and Eq. 共56兲 with q = 0 gives 130兴. The experimental value for the critical pressure gradient in MBBA is ⌬Pc / ⌬x = 245 Pa/ m for a sample of thickness 200 m 关17,18兴, that gives a2c = 130 for the material parameters 关14兴. Thus theoretical calculations and experimental results are in good agreement. Note that in the experiments 关17,18兴 actually not steady but oscillatory flow with very low frequency was used 共f = 5 ⫻ 10−3 Hz兲. In summary, the orientational instabilities for both steady Couette 共semianalytical for homogeneous instability and numerical for rolls兲 and Poiseuille flow 共numerical兲 were analyzed rigorously taking into account weak anchoring conditions at the confining plates and the influence of external fields. Easy-to-use expressions for the threshold of all pos-

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FIG. 6. Contour plot of the critical shear rate a2c vs a and  p for Poiseuille flow with additional magnetic field H0 / HF = 0.4.

sible types of instabilities were obtained and compared with the rigorous calculations. In particular the regions in parameter space 共anchoring strengths, external fields兲 where the different types of instabilities occurred were determined. The results can be used for the experimental measurements of the polar and azimuthal anchoring strengths in one single experiment. ACKNOWLEDGMENTS

We wish to thank W. Pesch for helpful discussions and critical reading of the manuscript. Financial support by DFG 共EGK “Non-equilibrium phenomena and phase transition in complex systems” and Grant Nos. Kr690/14-1 and Kr690/ 22-1兲 and RFBR 共Grant Nos. 05-02-16716, 05-02-16548, 0502-97907兲 are gratefully acknowledged. APPENDIX A: TRIAL FUNCTIONS

In the calculations we used the following set of trial functions for the director perturbations:

on共z; 兲 = sin共2nz兲 + 2n sin共关2n − 1兴z兲, en共z; 兲 = cos共关2n − 1兴z兲 + 共2n − 1兲 cos共2关n − 1兴z兲, and for the velocity perturbations

on共z兲 = sin共2nz兲, en共z兲 = cos共关2n − 1兴z兲, on共z兲 =

FIG. 5. Contour plot of the critical shear rate a2c vs a and  p for Poiseuille flow. 共a兲 E0 = 0; 共b兲 E0 = Eweak , a ⬍ 0; 共c兲 E0 = Eweak , a 0 0 ⬎ 0. Eweak = 0.42E . Boundaries between different types of instabiliF F ties are given by thick solid lines 共full numerical兲 and thick dashed lines 共one-mode approximation兲.

en共z兲 =

sin共2nz兲 sinh共2nz兲 − , sinh共2n/2兲 sin共2n/2兲

cosh共2n−1z兲 cos共2n−1z兲 − , cosh共2n−1/2兲 cos共2n−1/2兲

where on共z兲 and en共z兲 are the Chandrasekhar functions and n are the roots of the appropriate characteristic equations results from n共±1 / 2兲 = zn共±1 / 2兲 关19兴.

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Poiseuille flow

Couette flow

具gw⬙典 = −27.257− 32.441 p, 具gw典 = 0.690 43 + 16a兲 / 6, + 3.2870 p, 具w关sf兴⬙典 = −27.257− 32.441a, 具wsf典 = 0.690 43 + 3.2870a, 具uu⬙典 = −22, 具u2典 = 1 / 2, 具ww共4兲典 = 3803.5, 具ww⬙典 = −46.050, 具w2典 = 1. Even solution: 具f f ⬙典 = −2共1 + 4a兲 / 2, 具f 2典 = 共1 + 8a + 222a兲 / 2, 具gg⬙典 = −2共1 + 4 p兲 / 2, 具g2典 = 共1 + 8 p 2 2 2 + 2  p兲 / 2, 具fsg典 = 关1 + 4共a +  p兲 + 2 a p兴 / 2, 具sf典 = 共2 + 2a兲 / , 具g典 = 共2 + 2 p兲 / , 具usg典 = 共1 + 4 p兲 / 2, 具fu典 = 共1 具gw⬙典 = −6.8828, 具gw典 = 0.697 38+ 2.6102 p, + 4a兲 / 2, 具w关sf兴⬙典 = −6.8828, 具wsf典 = 0.697 38+ 2.6102a, 具uu⬙典 = −2 / 2, 具u2典 = 1 / 2, 具ww共4兲典 = 500.56, 具ww⬙典 = −12.303, 具w2典 = 1.

Odd solution: 具f f ⬙典 = −22共3 + 20a + 322a兲 / 3, 具f 2典 = 共3 + 32a + 1222a兲 / 6, 具gg⬙典 = −2共1 + 4 p兲 / 2, 具g2典 = 共1 + 8 p + 222p兲 / 2, 具fsg典 = −关16+ 92共a +  p兲 + 722a p兴 / 共182兲, 具g典 = 共2 + 2 p兲 / , 具usg典 = −共16 具sf典 = −共1 + 8a兲 / 共2兲, 2 2 + 9  p兲 / 共18 兲, 具fu典 = 共3 + 16a兲 / 6, 具gw⬙典 = −6.8828, 具gw典 具w关sf兴⬙典 = −0.876 73− 22.615a, = 0.697 38+ 2.6102 p, 具wsf典 = −0.102 92− 0.498 16a, 具uu⬙典 = −22, 具u2典 = 1 / 2, 具ww共4兲典 = 500.56, 具ww⬙典 = −12.303, 具w2典 = 1. Even solution: 具f f ⬙典 = −2共1 + 4a兲 / 2, 具f 2典 = 共1 + 8a + 222a兲 / 2, 具gg⬙典 = −22共3 + 20 p + 322p兲 / 3, 具g2典 = 共3 + 32 p + 1222p兲 / 6, 具fsg典 = −关16+ 92共a +  p兲 2 2 具sf典 = 具g典 = 0, 具usg典 = −共16 + 72 a p兴 / 共18 兲, 具fu典 = 共1 + 4a兲 / 2, 具gw⬙典 = −27.257 + 92 p兲 / 共182兲, − 32.441 p, 具gw典 = 0.690 43+ 3.2870 p, 具w关sf兴⬙典 = 4.4917, 具wsf典 = −0.122 06− 0.596 94a, 具uu⬙典 = −2 / 2, 具u2典 = 1 / 2, 具ww共4兲典 = 3803.5, 具ww⬙典 = −46.050, 具w2典 = 1.

关1兴 P. Pieranski and E. Guyon, Solid State Commun. 13, 435 共1973兲. 关2兴 P. Pieranski and E. Guyon, Phys. Rev. A 9, 404 共1974兲. 关3兴 E. Dubois-Violette and P. Manneville, in Pattern Formation in Liquid Crystals, edited by A. Buka and L. Kramer 共SpringerVerlag, New York, 1996兲, Chap. 4, p. 91. 关4兴 P. Manneville, J. Phys. 共Paris兲 40, 713 共1979兲. 关5兴 P. Pieranski and E. Guyon, Commun. Phys. 共London兲 1, 45 共1976兲. 关6兴 P. E. Cladis and S. Torza, Phys. Rev. Lett. 35, 1283 共1975兲. 关7兴 P. J. Kedney and F. M. Leslie, Liq. Cryst. 24, 613 共1998兲. 关8兴 I. Sh. Nasibullayev, A. P. Krekhov, and M. V. Khazimullin, Mol. Cryst. Liq. Cryst. Sci. Technol., Sect. A 351, 395 共2000兲. 关9兴 O. S. Tarasov, A. P. Krekhov, and L. Kramer, Liq. Cryst. 28, 833 共2001兲. 关10兴 I. Sh. Nasibullayev and A. P. Krekhov, Crystallogr. Rep. 46, 488 共2001兲.

关11兴 O. S. Tarasov, Liq. Cryst. 31, 1235 共2004兲. 关12兴 V. G. Chigrinov, Liquid Crystal Devices: Physics and Applications 共Artech House, New York, 1999兲. 关13兴 P. G. de Gennes and J. Prost, The Physics of Liquid Crystals 共Clarendon Press, Oxford, 1993兲. 关14兴 Elastic constants in units 10−12 N: K11 = 6.66, K22 = 4.2, K33 = 8.61; viscosity coefficients in units 10−3 Pa s: ␣1 = −18.1, ␣2 = −110.4, ␣3 = −1.1, ␣4 = 82.6, ␣5 = 77.9, ␣6 = −33.6. 关15兴 P. Manneville and E. Dubois-Violette, J. Phys. 共Paris兲 37, 285 共1976兲. 关16兴 F. M. Leslie, Mol. Cryst. Liq. Cryst. 37, 335 共1976兲. 关17兴 E. Guyon and P. Pieranski, J. Phys. 共Paris兲, Colloq. 36, C1203 共1975兲. 关18兴 I. Janossy, P. Pieranski, and E. Guyon, J. Phys. 共Paris兲 37, 1105 共1976兲. 关19兴 S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability 共Oxford University Press, New York, 1961兲.

Odd solution: 具f f ⬙典 = −2 共3 + 20a + 322a兲 / 3, 具f 2典 = 共3 + 32a + 1222a兲 / 6, 具gg⬙典 = −22共3 + 20 p + 322p兲 / 3, 具g2典 = 共3 + 32 p + 1222p兲 / 6, 具fsg典 = 关3 + 16共a +  p兲 + 122a p兴 / 6, 具sf典 = 具g典 = 0, 具usg典 = 共3 + 16 p兲 / 6, 具fu典 = 共3 2

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