R

Ma

Cb

c

d

a

ARRA

KSNNS

P6666

1

wsscmtmtfllfl

0d

J. Non-Newtonian Fluid Mech. 155 (2008) 130–145

Contents lists available at ScienceDirect

Journal of Non-Newtonian Fluid Mechanics

journa l homepage: www.e lsev ier .com/ locate / jnnfm

obustness of pulsating jet-like layers in sheared nano-rod dispersions

. Gregory Foresta, Sebastian Heidenreichb,∗, Siegfried Hessb, Xiaofeng Yangc, Ruhai Zhoud

Departments of Mathematics & Biomedical Engineering, Institute for Advanced Materials, University of North Carolina at Chapel Hill,hapel Hill, NC 27599-3250, United StatesInstitute for Theoretical Physics, Technische Universität Berlin, Hardenbergstrasse 36, D-10623 Berlin, GermanyDepartment of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3250, United StatesDepartment of Mathematics and Statistics, Old Dominion University, Norfolk, VA 23529, United States

r t i c l e i n f o

rticle history:eceived 16 February 2008eceived in revised form 2 May 2008ccepted 2 June 2008

eywords:hear flowematic polymersumerical simulationtructure formation

ACS:1.30.v1.30.Vx7.40.Fd

a b s t r a c t

Nano-rod dispersions in steady shear exhibit persistent transient responses both in experiments andsimulations. The rotational contribution from shear flow couples with orientational diffusion, excluded-volume interactions, and distortional elasticity to yield complex dynamics and gradient morphology ofthe rod ensemble. The classification of sheared responses has mostly focused on “nematodynamics” of thecollective particle response known as tumbling, wagging and kayaking; in heterogeneous simulations, onemonitors the variability in nematodynamics across the domain. In this paper, we focus on flow couplingand non-Newtonian feedback in transient heterogeneous simulations, and in particular on a remarkableeffect: the formation of localized, pulsating jet layers in the shear gap. We solve the Navier–Stokes momen-tum equations coupled through an orientation-dependent stress to three different orientational models(a kinetic Smoluchowski equation and two tensor models, one from kinetic closure and another fromirreversible thermodynamics). A similar spurt phenomenon was reported in 1D simulations of a modelfor planar nematic liquids by Kupferman et al. [R. Kupferman, M. Kawaguchi, M.M. Denn, Emergence ofstructure in models of liquid crystalline polymers with elasticity, J. Non-Newt. Fluid Mech. 91 (2000)

7.55.Fa 255–271], which we extend to full orientational configuration space. We show: the pulsating jet layerscorrelate, in space and time, with the formation of a non-topological “oblate defect phase” in which theprincipal axis of orientation spreads from a unique direction to a circle; the jet-defect layers form wherethe local nematodynamics transitions from finite oscillation (wagging) to continuous rotation (tumbling),and when neighboring directors lose phase coherence; and, a negative first normal stress difference devel-ops in the pulsating jet-defect layers. Finally, we extend one model algorithm to two space dimensions

lity o

bdpdflbnse

and show numerical stabi

. Introduction

The nano-rod dispersion is a general and simple model for aide range of anisotropic, non-Newtonian fluids that consist of

mall-to-large molecules or Brownian structures with propertiesimilar to rigid rods or platelets. Representative examples are liquidrystals, liquid crystal polymers, worm-like micelles, and tobaccoosaic virus suspensions. In each of these model systems, orien-

ational degrees of freedom and the possibility to form differentesoscopic phases (isotropic and nematic) lead to surprising orien-

ational behavior and flow feedback in shear-dominated rotationalows. The large literature on shear banding [2] in sheared worm-

ike micelles [3] is an example of the remarkable non-Newtonianow feedback that is possible in such systems.

∗ Corresponding author. Tel.: +49 30 314 25904; fax: +49 30 314 21130.E-mail address: sebastian@itp.physik.tu-berlin.de (S. Heidenreich).

bjiios

r

377-0257/$ – see front matter © 2008 Elsevier B.V. All rights reserved.oi:10.1016/j.jnnfm.2008.06.003

f the jet-defect phenomenon to 2D perturbations.© 2008 Elsevier B.V. All rights reserved.

In this article we present numerical results for a dramatic shearanding phenomenon, the occurrence of pulsating jet layers, in con-itions that mimic a parallel plate shear cell with imposed steadylate speeds. This behavior was first reported for a model two-imensional (2D) nematic liquid (the rods are 2D, whereas theow is one-dimensional (1D) with variations across the plate gap)y Kupferman et al. [1]. Here, we present numerical results forematic liquids with a three-dimensional (3D) rod configurationpace, for three completely different models, each solved by differ-nt numerical algorithms at our respective institutions. Numericalenchmarks are provided for each code to confirm these pulsating

et layers are model and not numerical phenomena, and one codes extended to two physical space dimensions to show the behavior

s stable even to 2D perturbations in physical space. We use the restf the Introduction to give some background and motivation for thepecial phenomena reported here.

In steady shear driving conditions, nano-rod dispersions (theeader may think of the particular example of liquid crystal

nian F

pshodndasp

stccorbcoa

enboaeneaa

et

dbofldstae

etad(totmtmticd

arfl

towsdDp

sdtsifstp

irrtigaoaptcnsoreogtflifj

dwetiat

sswtadok

M.G. Forest et al. / J. Non-Newto

olymers as the most studied of these fluid systems) exhibit bothtationary and transient orientational responses. This behavioras been modeled primarily in the monodomain limit, wherene posits simple shear and analyzes the resulting orientationalynamics of the rod phase. This analysis decouples the hydrody-amics and suppresses spatial dependence of the orientationalistribution. This dimensional reduction of physical space, leavingpurely time-dependent model in orientational configuration

pace, has worked remarkably well in interpreting liquid crystalolymer experiments dating back to Kiss and Porter [4].

For example, the novel observation of negative first normaltress differences in steady shear were subsequently duplicated byhe monodomain models of Doi–Hess type [16–19] that are spe-ial limits of the models presented in this paper. Most notably, signhanges in the first normal stress difference are associated in eachf these models with transitions to or among periodic orientationalesponses in steady shear. This model reduction has worked eitherecause the flow response functions are well approximated by so-alled viscometric flow in the experimental operating conditions,r because any strongly nonlinear shear flow does not qualitativelylter the measurable orientation and stress response functions.

As noted just above, nematic liquids in steady shear exhibit sev-ral distinct “modes” of transient behavior, which have picturesqueames in the liquid crystal and nematic polymer literature. Tum-ling refers to continuous rotation of the peak axis of orientationf the rod ensemble in the plane of shear (normal to the vorticityxis). Wagging refers to a finite oscillation of the major axis of ori-ntation, again in the shear plane. More limit cycles were computedumerically by Larson and Ottinger [5] in which the major directorither rotates around the vorticity axis (called kayaking) or rotatesround some axis midway between the shear plane and vorticityxis (tilted kayaking).

These long-time transient responses have been identified inxperiments [4,6–8] and confirmed through theoretical descrip-ions [9–15].

More complex, apparently chaotic behavior is also found iniverse models [12,14,20–22], arising through a period-doublingifurcation route, which Berry [6] associated with the rapid devel-pment of turbidity (strong heterogeneity) in experiments. In thisow regime, the two assumptions of a homogeneous orientationalistribution and of a simple, steady linear shear become especiallyuspect, motivating several research groups to undertake spatio-emporal numerical studies. To do so, one relaxes the above twossumptions, proceeding first to models and simulations of 1D het-rogeneity.

For spatially inhomogeneous systems, it becomes possible toxplore the impact of physical boundary conditions on the orienta-ional distribution at the moving or stationary plates. The standardpproach has been to impose quiescent equilibrium boundary con-itions, i.e., a uniform nematic phase with the principal axis fixedexperimentally, this is done by mechanical or chemical prepara-ions). The fundamental issue is whether plate anchoring on therientational distribution is felt only in a wall boundary layer orhe anchoring at solid walls penetrates into the shear gap. Diverse

odels show that limit cycles found for the monodomain assump-ions persist in the presence of heterogeneity (for example, if one

onitors the orientational distribution at a specific spatial loca-ion). Indeed, all limit cycles and complex (chaotic) behavior foundn monodomain models have been observed, and phase transitionsontinue to occur as one varies gap dimensions, plate speeds, and

istortional elasticity potentials [23–26].

In some studies, the flow is imposed as simple linear shearnd only orientational gradients are allowed, while other, moreesolved simulations perform a self-consistent computation of theow.

lp

na

luid Mech. 155 (2008) 130–145 131

A particularly interesting flow feedback phenomenon wherehere was compelling experimental evidence is the formationf steady roll cells at very low shear rates. These structuresere reported experimentally by Larson and Mead [27,28], and

uccessfully modeled by Feng et al. [29] with a liquid crystalirector theory, and more recently by Klein et al. [30] with aoi–Hess–Marrucci–Greco tensor model applicable to nematicolymers and nano-rod dispersions.

The flow feedback behavior of interest in this paper is in thetrongly nonlinear regime, where both anisotropy and focusing-efocusing of the orientational distribution are important. Yet,he phenomenon (generation of a nonlinear and non-monotonehear profile across the plate gap) is predominantly 1D in phys-cal space. Thus, a full orientation tensor, or a full kinetic modelor the orientational distribution function, is necessary, but a onepace-dimensional flow-orientation solver is sufficient to capturehe phenomenon. (We return at the end to explore stability to 2Derturbations).

The pulsating flow phenomena turn out to arise when theres a significant orientational mismatch in the local monodomainesponse functions across the gap: the interior has a tumblingesponse function, whereas near the plates the local response func-ion is wagging. These two monodomain responses are spatiallyncompatible for the following reason. The principal axis of a wag-ing orbit cannot continuously reside next to a tumbling orbit,nd thus the major director of neighboring wagging and tumblingrbits will become significantly out of phase. Either the principalxis will develop enormous local gradients (which the distortionalotential penalizes), or the orientational distribution must defocushrough a degeneracy in the principal values. This latter scenarioosts much less energy, creating what we call an order-parameter oron-topological defect. This happens locally: in space, at the tran-ition site from tumbling to wagging, and in time, during the partf the tumbling cycle when the principal axis rapidly rotates beforeesetting in phase with the neighboring wagging orbit. This picturexplains intuitively why the flow feedback effect, whatever it is,ccurs locally in space and in periodic pulses. Since this behaviorenerates gradients in the principal values of the orientational dis-ribution, the extra stress from the rod distribution will generateow feedback. The intriguing nature of this flow structure is that

t leads to local pulsating layers where the flow accelerates, andurthermore we can tune parameters to amplify the effect so thatet-layers form that speed ahead of the flow above and below.

Since our primary numerical studies are confined to one-spaceimensional heterogeneity, the issue still remains as to stabilityith respect to higher space-dimensional perturbations. We find

vidence in our comprehensive studies [32] of 1D orientational dis-ributions that are stable in two space dimensions. The flow wasmposed in that study. We proceed for this paper to perform annalogous numerical stability study of the full coupled system inwo space dimensions.

One might also surmise that these jet layer oscillations are exclu-ive to special anchoring conditions at the plates. We present oneimulation below where the flow-orientation behavior, persistshen the wall anchoring conditions are shifted out of plane. In

hese conditions, starting from the same parameter specificationss with tangential anchoring, the tumbling-wagging compositeynamics across the gap is replaced by their out-of-plane mon-domain analogs: kayaking in the center of the gap, and tiltedayaking near each plate. Again, the jet layer forms at the transition

ayer where the local monodomain responses get completely out ofhase, and order parameter defects form during the jet phase.

Given all of these studies, the hydrodynamic feedback mecha-ism leading to a pulsating jet layer in each half of the shear gapppears to be considerably more robust that previously anticipated.

1 nian F

oortttbsNcweneltlmtbta

bTtwttfebrsdtsLmtlttptfdedat

2a

mmmttda

gta

2e

yr

witmtsd

D

Tnfl

hd

v

T(˛ttbRsF

n

aod

wdvtp

V

w

32 M.G. Forest et al. / J. Non-Newto

We turn now to the modeling of this paper and the orderf presentation. We consider a dispersion of nano-rods betweenppositely translating, parallel plates at velocities −v0 and v0,espectively. We present three different models for the orien-ational distribution and the associated stress constitutive lawhat couples to the Navier–Stokes momentum equations. Wehen perform one-space dimensional simulations of the initial-oundary-value problem for flow between the translating plates,tarting with an equilibrium, nematic phase across the shear gap.umerical convergence studies are presented in an Appendix B toonfirm these results are not numerical artifacts. In each model,e impose consistent parameter values for the strength of the

xcluded-volume potential, Frank elasticity constants, Reynoldsumber, and Deborah number. The spatio-temporal attractor ofach system is periodic, and the remarkable non-monotone, jet-ike, primary shear profile emerges and disappears. Thus ourerminology of an oscillating or pulsating jet layer. When the jetayers reside sufficiently near the plates, as observed in the Kupfer-

an et al. model [1], they move faster than the translating plateshat drive the experiment! We proceed to show the jet layers cane translated toward the interior or shifted closer to the plates, andhe strength of the jet can be controlled, by varying the Ericksennd Deborah numbers.

The jet layers are then correlated with the orientational distri-ution and stress tensor, with a consistent picture for each model.he jet layers coincide with an order parameter degeneracy inhe orientational distribution, a non-topological defect structurehose existence is independent of physical space dimensions. In

hese defect phases, the principal axis of orientation (often calledhe major director) coincides with the secondary principal axis toorm a circle of maximum likelihood. This is a strong defocusingvent, with a broadening of the peak of the orientational distri-ution, which always arises when local anisotropic monodomainesponses get strongly out of phase. This order parameter defecttructure is the tensorial and kinetic theory mechanism to avoidiscontinuities in the principal axis of orientation, providing a con-inuous transition between wagging and tumbling at neighboringpatial sites. We note that this phenomenon is not captured by aeslie–Ericksen–Frank theory nor by the model 2D liquid of Kupfer-an et al. In a planar liquid, the only order parameter degeneracy is

he isotropic phase; we never observe isotropic phases in our simu-ations, which involve all three eigenvalues of the second-momentensor to collide. This is an interesting observation, suggesting thathe kinetic and second-moment models select this “oblate defecthase” degeneracy over higher order degeneracy in the distribu-ion such as complete isotropy. We develop some new graphicsor our simulation results to amplify the formation of these oblateefect phases during the pulsating jet layer events. Finally, wexplore (and confirm) robustness of these phenomena to two-spaceimensional perturbations in both flow and orientation, usingspectral-Galerkin code for the so-called Doi–Marrucci–Greco

ensor-flow model.

. The model equations for a spatially inhomogeneousnisotropic fluid

In this section, we give the descriptions of three models imple-ented for numerical simulations. The models are the kineticodel, the Doi–Marrucci–Greco model and the alignment tensor

odel. All models have the same physical ingredients to describe

he orientational dynamics: orientational diffusion, the orientingorque caused by the symmetric traceless part of the velocity gra-ient, the local molecular rotation induced by the vorticity flow,nd, elastic and diffusional terms associated with spatial inhomo-

st

M

luid Mech. 155 (2008) 130–145

eneities of the local orientation. The stress tensor of the alignmentensor model differs in some respects from that used in the kineticnd Doi–Marrucci–Greco model.

.1. The Doi–Hess kinetic model, Marrucci–Greco gradientlasticity, and flow coupling

We consider plane shear flow between two plates located at= ±h, in Cartesian coordinates x = (x, y, z), and moving with cor-

esponding velocity v = (±v0, 0, 0), respectively.There are two apparent length scales in this problem: the gap

idth 2h, an external length scale, and the finite range l of molecularnteraction, an internal length scale, set by the distortional elas-icity in the Doi–Marrucci–Greco (DMG) model. When the plates

ove relative to each other at a constant speed, it sets a bulk flowime scale (t0 = (h/v0)); the nematic average rotary diffusivity (Dr)ets another (internal) time scale (tn = (1/Dr)), and the ratio tn/t0efines the Deborah number De:

e = tn

t0= v0

h Dr. (1)

here are also scales associated with solvent viscosity and the threeematic viscosities, and with elastic distortion, which due to theow-nematic-plate interaction are not a priori known.

We nondimensionalize the DMG model using the length scale, the time scale tn and the characteristic stress �0 = �h2/t2

n . Theimensionless flow and stress variables become:

˜ = tn

hv, x = 1

hx, t = t

tn, � = �

�0, p = p

�0. (2)

he following seven dimensionless parameters arise: Re =�0tn)/(�), ˛, Er, �i and �. Here Re is the solvent Reynolds number;measures the strength of entropic relative to kinetic energy; Er is

he Ericksen number which measures short-range nematic poten-ial strength relative to distortional elasticity strength depictedy the persistence length l; �i, i = 1, 2 and 3 are three nematiceynolds numbers; and � is a fraction between 0 and 1 that corre-ponds to equal (� = 0) or distinct (� /= 0) elasticity constants [33].or other parameters, we refer to [33].

We drop the tilde on all variables; all figures correspond toormalized variables and length, time scales.

We neglect the effect of translational diffusion and employ anpproximate rotary diffusivity. The dimensionless Smoluchowskir generalized Fokker–Planck equation [18,19] for the probabilityistribution function (PDF) f (m, x, t) becomes:

Df

Dt= R · [(Rf + fRV)] − R · [m × mf ],

m = ˝ · m + a[D · m − D : mmm],(3)

here D/Dt = ∂/∂t + v · ∇ , R = m × (∂/∂m) is the rotational gra-ient operator, D and ˝ are the dimensionless rate-of-strain andorticity tensors, respectively, and a = (r2 − 1)/(r2 + 1), where r ishe rod aspect ratio. The coupled Maier–Saupe and Marrucci–Grecootential is

= −3N

2M : mm − 1

2Er[�M + � ∇∇ · M] : mm, (4)

here N is a dimensionless rod volume fraction which governs the

trength of short-range excluded volume interactions. The rank-2ensor M is the second moment of f:

= M(f ) =∫

||m||=1

mmf (m, x, t) dm. (5)

nian F

Tc

w

M

M

M

M

Iv1flp

p(

v

Wb

f

wtp

2

tsdffptflscf

wRt

F

wwot

Ttar

Q

w

2

rieDmdaA

D

Tpaerth

wp

M.G. Forest et al. / J. Non-Newto

he dimensionless forms of the balance of linear momentum, theontinuity equation, and the stress constitutive equation are [33]:

dvdt

= ∇ · (−pI + �), ∇ · v = 0,

� =(

2Re

+ �3

)D + �1(D · M + M · D) + �2D : M4

+a˛(

M − 13

I − N M · M + N M : M4

)−a

˛

6Er(�M · M + M · �M − 2 �M : M4)

− ˛

12Er[2(�M · M − M · �M) + Mc]

−a˛�

12Er[M · Md + Md · M − 4(∇∇ · M) : M4]

− ˛�

12Er[Md · M − M · Md − Me],

(6)

here

c = (∇M : ∇M − (∇∇M) : M), (7)

d = ∇∇ · M + (∇∇ · M)T, (8)

e = (∇∇ · M) · M − Mˇj,˛Mij,i, (9)

4 =∫

||m||=1

mmmm f (m, x, t) dm. (10)

n the simulations presented below, we fix the following parameteralues: Er = 500, De = 1, N = 6, � = 0.5, ˛ = 20, �1 = 0.004, �2 =.5, �3 = 0.1. We also impose tangential (director is parallel to theow direction) anchoring conditions on the equilibrium nematichase at the plates (cf. [34]).

We consider 1D physical space (the interval between the twoarallel plates). The boundary conditions on the velocity v =vx, 0, 0) are given by the Deborah number:

x(y = ±1, t) = ±De. (11)

e assume homogeneous tangential anchoring at the plates, giveny the quiescent nematic equilibrium:

(m, y = ±1, t) = fe(m), (12)

here fe(m) is an equilibrium solution of the Smoluchowski equa-ion corresponding to the tangential anchoring of particles at thelates when v = 0.

.2. Doi–Marrucci–Greco second-moment orientation model

In Landau–de Gennes models, the probability distribution func-ion f (m, x, t) of Doi–Hess kinetic theory is projected onto theecond moment tensor, M, a symmetric, trace 1, positive semi-efinite tensor of rank 2. The orientation tensor Q is the trace zeroorm of M, Q = M − (I)/(3). The distinctions among models can beramed in terms of closure rules necessary to reduce the above cou-led flow-kinetic orientation model and stress formula to a systemhat closes on the second-moment orientation tensor Q and theow variables v and p. The dimensionless forms of the stress con-titutive equation and orientation tensor equation that are oftenalled the Doi–Marrucci–Greco (DMG) model [45,46] are given asollows [35]:

� =(

2Re

+ �3

)D + a˛F(Q)

a˛( (

I)

1 1)

+3Er

�Q : Q Q +3

−2

(�QQ + Q�Q) −3

�Q

+ ˛

3Er

(12

(Q�Q − �QQ) − 14

(∇Q : ∇Q − ∇∇Q : Q))

+�1

((Q + I

3

)D + D

(Q + I

3

))+ �2D : Q

(Q + I

3

),

(13)�

TN

luid Mech. 155 (2008) 130–145 133

here Er is the Ericksen number, and �i are the three nematiceynolds numbers (normalized viscosities) all defined earlier, andhe short-range excluded volume effects are captured by

(Q) =(

1 − N

3

)Q − NQ2 + NQ : Q

(Q + I

3

), (14)

here N is a dimensionless concentration of nematic polymers,hich controls the strength of the mesoscopic approximation, F(Q),

f the gradient of the Landau–de Gennes type potential. The orien-ation tensor equation with the DMG closure rule is

DQDt

= ˝Q − Q˝ + a(DQ + QD) + 2a

3D − 2aD : Q

(Q + I

3

)−(

F(Q) + 13Er

(�Q : Q

(Q + I

3

)− 1

2(�QQ + Q�Q) − 1

3�Q

)).

(15)

he boundary condition for the scaled velocity is the same as inhe kinetic simulations (11). We assume homogeneous tangentialnchoring at the plates, given by the quiescent nematic equilib-ium:

eq = s(

exex − 13

I)

, (16)

here ex = (1, 0, 0) and s = 0.86.

.3. Alignment tensor model

In this section we introduce a different model for the secondank alignment tensor that was first derived in the framework ofrreversible thermodynamics, but also inferred from the kineticquation with a closure relation slightly different from the one ofMG. We present the model equations in the notation of the DMGodel for a more transparent comparison. This notation therefore

iffers from the original literature cited below; the correspondencend a discussion of its relationship to the DMG model are given inppendix A.

The equation for the alignment tensor Q [36,39] in theoi–Marrucci–Greco model notation reads (see Appendix 44):

DDt

Q = ˝Q − Q˝ + (

DQ + QD − 23

(D : Q)I)

+ Da

(�F(Q) − De

Er�2Q

)

− 1

DeF(Q) + 1

Er�Q −

√32

KD.

(17)

he parameters De, Er, and K are related to the DMG modelarameters (see Appendix A). The parameter Da does not have annalog in the DMG model: Da is a flow-alignment diffusion param-ter In our study we set = 0 and Da = 0. This means we presentsesults for a minimal model, which is as simple as possible, but notoo simple. The function F denotes the derivative of the spatiallyomogeneous part of the amended Landau–de Gennes potential:

∂

∂Q�AM(Q) = F(Q) = ϑQ − 3

√6(

QQ − 13

(Q : Q)I)

+ 2(Q : Q)Q

1 − ((Q : Q)2)/(Q 4max)

,

hich is a modification of the corresponding Landau–de Gennesotential:

LDG = 12

ϑQ : Q −√

6 (QQ) : Q + 12

(Q : Q)2. (18)

he parameter ϑ in (18) has been used with ϑ(N) = A0(1 −/N∗)/(1 − NK/N∗) [17]. The pseudocritical concentration N∗ and

134 M.G. Forest et al. / J. Non-Newtonian Fluid Mech. 155 (2008) 130–145

Table 1Characteristic scales

Scale DH and DMG model AT model

Time 1/Dr 1/(DeDr)Space h hVelocity h · D De · h · D

S

D(

te

ccaqa

GiufiipttMGϑtstT

s

a

toeifiDtt

b

T

ˇ

wmt

Q

Table 2Parameter values used in three models

Parameter DH model DMG model Parameter AT model

Er 500 50 Er 100De 1 0.5 De 1.0N 6 6 N 3� 0.5 – – –˛ 20 2 ˇ 1.0a 1.0 0.8 K 1.0� −3 −4

��

wn

u(−h, t) = −v0, u(h, t) = v0. (22)

In Table 2 we compare the values used in the three models.

r r

tress �(h · Dr)2 (�/m)kBT

r denotes the nematic average rotary diffusivity and De the Deborah number De =v0)/(hDr). 2h is the gap height. � is the density.

he concentration at phase coexistence NK are also model param-ters. The value of A0 depends on the proportionality coefficient

hosen between Q and 〈uu〉. The choice made above implies A0 = 1,f. [36]. The coefficients, on the one hand, are linked with measur-ble quantities and, on the other hand, can be related to molecularuantities within the framework of a mesoscopic theory based onkinetic equation for the orientational pdf [18,17,16].

In previous studies it was demonstrated that the Landau–deennes potential does not restrict the order parameter to phys-

cally admissible values, in particular the order parameter blowsp at high extensional flow rates. To cure this problem a modi-ed potential (amended potential) was introduced and discussed

n [40]). From [40], the value 2.5 is implemented for the “cut-offarameter” Qmax. This is a plausible value, e.g. liquid crystals, wherehe Maier–Saupe order parameter S = 〈P2〉 is about 0.4 at the transi-ion temperature. Thus, the maximum value of unity for S with the

aier–Saupe potential is larger by a factor 2.5. In the Landau–deennes case, one has Q = aK = 1 at the transition temperature= ϑK = 1. For the amended potential with Qmax = 2.5 one has

he transition at ϑ = ϑK ≈ 0.9883 with aK ≈ 0.9667. Because of themall difference between these values, it is convenient to main-ain the Landau–de Gennes scaling for the physical variables. In theable 1 a summary of the different scalings is given.

Note the alignment tensor Q in (17) differs from the DMG ten-orial order parameter by the constant factor aK

√15/2, i.e. QAT =

K

√15/2 QDMG.

Upon the assumption that the alignment tensor is uniaxial andhat its magnitude is constant, the spatial derivative ∇Q just actsn the director n. In this limiting case the Frank elasticity is recov-red [37,38]. For simplicity, the ‘isotropic’ case is considered whichmplies that, in the nematic phase, all three Frank elasticity coef-cients are equal. This is consistent with the choice made in theMG tensor model, whereas the kinetic model allows for two dis-

inct elasticity constants. The phenomena we report are insensitiveo one versus two elasticity constants.

The orientation-dependent stress constitutive equation is giveny

� = isoD −√

32

K

ˇF(Q) +

√32

K

ˇEr�Q

+

ˇ

(QF(Q) + F(Q)Q − 2

3(F(Q) : Q)I

)− 1

Er

ˇ

(Q(�Q) + (�Q)Q − 2

3((�Q) : Q)I

).

(19)

he parameter ˇ is inversely proportional to ˛:

= 3(v0)2

L2D0r aKAK˛

, (20)

here Dr is the rotational diffusion constant, as in the other two

odels. Again, we assume homogeneous tangential anchoring at

he plates, given by the quiescent nematic equilibrium:

(−h, t) = Q(h, t) =√

32

qeq

(exex − 1

3I)

, (21) FM

1 4 × 10 2 × 10 iso 0.12 1.5 3 × 10−3 – –3 10−1 10−3 – –

here qeq = √5aKs, here s = 0.61 (with aK = 1). For the velocity,

o-slip boundary conditions are used:

ig. 1. The space–time primary velocity profile across the gap. Top: DH model.iddle: DMG model. Bottom: AT model.

nian Fluid Mech. 155 (2008) 130–145 135

3

3Tapltc

3

Kfltmaospmgpwa

Fm

Ff

M.G. Forest et al. / J. Non-Newto

. Model robustness of pulsating jet layers

We now present results of numerical simulations of each of themodels presented above, at parameter specifications given in the

ables. The spatio-temporal attractors for each model are remark-bly consistent: We focus exclusively on the longtime attractorroperties which are independent of initial conditions and short-

ived transients. We begin with graphics of the flow structure, thenhe orientational structure, and finally the stress. We present andompare each feature from all three model results.

.1. The pulsating jet layer phenomenon

We begin with surface plots of vx(y, t) analogous to those inupferman et al. [1]. Fig. 1 (top) is from the Doi–Hess kinetic-ow (DH) model, Fig. 1(middle) is from the Doi–Marrucci–Grecoensor-flow (DMG) model, and Fig. 1(bottom) is from the align-

ent tensor-flow (AT) model. Each surface reveals the periodicppearance and disappearance of a localized layer on each sidef the shear gap within which the flow accelerates to speeds thaturpass the flow above or below it. We coin this phenomenon aulsating jet layer. Unlike the reported nonlinear shear profiles inost studies of flow-nematic coupling, these jet layers are distin-

uished by non-monotone shear profiles. The three models, at thesearameter values, differ in regard to the location of the jet layersithin the gap, the temporal period between successive jet pulses,

nd in the nonlinear flow profile across the remainder of the gap

ig. 2. Snapshots of the primary velocity profile across the shear gap from the kineticodel.

ig. 3. Snapshots of the primary flow profile, vx(y, t∗) on the interval 184 < t < 186,rom the DMG tensor model.

Fig. 4. Snapshots of the primary velocity profile across the gap in the AT model.The top snapshots are during the jet formation, while the lower plots are during the“quiet phase”.

Fig. 5. The primary velocity time series, vx(y∗, t), at the gap height y∗ where thepeak velocity of the jet occurs. Top: DH model at y∗ = 0.77; middle: DMG model aty∗ = 0.96; bottom: AT model at y∗ = 0.2.

1 nian F

dapdssn

bapnssmsDto

Fm

Tieismeemaaatt

36 M.G. Forest et al. / J. Non-Newto

uring and in between the jet pulses. These quantitative featuresre all parameter-dependent within each model, so that a com-rehensive parameter study could be undertaken to minimize theifferences among the models. Instead, we prefer to highlight theimilarities that occur without trying to force them; later we giveome results on jet pulse locations versus Deborah and Ericksenumber.

We focus next on snapshots of the primary flow profile during,efore, and after the jet pulses form. The kinetic model snapshotsre given in Fig. 2. During the relatively long “quiet phase”, the flowrofile is nearly linear. Then, prior to the jet formation, a stronglyonlinear, but still monotone shear profile develops; these snap-hots reveal a structure consisting of layers with plug-like flowandwiched between strong shear bands with local shear ratesuch greater than the bulk shear rate. Similar monotone nonlinear

hear band structures have been reported in several studies of theMG tensor-flow model [35,41]. Next, the flow gradients concen-

rate, forming the jet layer, with nearly linear flow in the remainderf the gap.

ig. 6. The space–time surface plot of the order parameter s = d1 − d2. Top: DHodel. Middle: DMG model. Bottom: AT model.

fltpbIap

so

bm

FwB

luid Mech. 155 (2008) 130–145

The corresponding DMG model snapshots are given in Fig. 3.he snapshot with the jet layers reveals jets close to each plate asn the kinetic model. The remainder of the gap has strongly nonlin-ar and non-monotone behavior with large regions of flow reversaln the top and bottom of the gap. As the jet disappears, the snap-hots reveal a composite shear band and plug flow profile, with aid-gap nearly stationary flow layer, and two strong shear bands

manating from each plate. The “quiet phase” with a nearly lin-ar profile across the gap is shown in the t = 187snapshot. The ATodel snapshots, Fig. 4, also show the formation of jet layers, which

t these parameter values are much closer to the center of the gap,nd an intermediate quiet phase where the flow is nearly linearcross the gap as in the other model simulations. To foreshadowhe correlations of these flow features with the orientational dis-ribution, we show in subsequent figures that the quiet phase in theow correlates with a phase synchrony of the major director acrosshe shear gap. The jets correlate with major director asynchrony, inarticular as one layer oscillates (wagging orbits) while the neigh-oring layer continues to monotonically rotate (tumbling orbits).n the DMG model, there is no quiet phase with nearly linear flowcross the entire gap, and the flow profile is a composite of nearlylug flow layers separated by strong shear bands.

Next, we show the time series (Fig. 5) of the primary velocitytationed at the gap height in each model where the peak velocity

f the jet occurs.

The normalized velocity at the center of the jet fluctuates:etween 0.60 and 1.25 of the plate speed at y = 0.77 in the kineticodel, and between 0.68 and 1.36 of the plate speed at y = 0.96 in

ig. 7. A snapshot of the spatial variation of the order parameter s = d1 − d2 takenhen the jet is fully formed in each model. Top: DH model. Middle: DMG model.ottom: AT model.

nian F

taTtatv

3

fdtptpaitctfltst

sr

Fta

ibsDescasca

pmtra

tDafiall figures show a “pulsating defect layer” characterized by a pre-cipitous drop of s close to zero, corresponding to an oblate defect

M.G. Forest et al. / J. Non-Newto

he DMG model. The analogous time series for the AT model showsfluctuation between 0.1 and 1.25 of the plate speed at y = 0.2.

he time series at the pulsating jet center has a consistent signa-ure between the kinetic and DMG closure models, and one can seesimilar time series from the AT model surface plot, even though

he locations of the jet layers are quite different at the parameteralues presented here.

.2. Orientational correlations with the pulsating jet layer

The hydrodynamic feedback phenomenon shown above arisesrom orientational gradients in the stress constitutive law, whoseivergence couples to the Navier–Stokes momentum balance. Thus,here must be an orientational space–time signature that accom-anies the pulsating jet layer. It is reasonable to expect, based onhe long history of liquid crystal theory in support of experimentalhenomena, that this anisotropic non-Newtonian flow behavior isssociated with defects. However, one-space dimensional behav-or cannot be associated with topological defects, which correspondo non-zero winding numbers of the unique major director around alosed curve: there are no closed loops in one space dimension. Theensor order parameter is needed to resolve the equilibrium andow phase diagrams, precisely because of behavior associated withhe degree(s) of order. These degrees of freedom are captured by the

calar order parameters defined earlier from the second momentensor of the probability distribution function.

Before giving the simulation results, recall our earlier discus-ion of the second moment tensor M, which affords a geometricepresentation of the orientational distribution in terms of triax-

ig. 8. The time series of the order parameter s = d1 − d2 at the gap height y∗ wherehe peak velocity of the jet occurs. Top: DH model at y∗ = 0.77. Middle: DMG modelt y∗ = 0.88. Bottom: AT model at y∗ = 0.22.

pt

Fstmosd

luid Mech. 155 (2008) 130–145 137

al ellipsoids. Non-degenerate nematic phases are characterizedy s > 0, for which there is a unique principal axis n1. Recall= d1 − d2, the difference in the two largest principal values of M.egenerate phases are defined by the condition s = 0. The great-st degeneracy arises in the isotropic phase, where ˇ = 0 as well,o that all directions on the sphere are equally probable, and theorresponding ellipsoid is a sphere. There remains an intermedi-te degeneracy between the nematic and isotropic phases, where= 0, yet ˇ > 0. This “oblate defect phase” corresponds geometri-ally to an oblate spheroid, in which there is a circle of principalxes.

Recall the plate boundary conditions are uniaxial equilibriumhases (s > 0, ˇ = 0) with the major director aligned with the pri-ary flow axis, corresponding to a prolate spheroid aligned with

he flow. For the parameter values chosen in Table 2, the equilib-ium value of s is 0.77 for the kinetic model, 0.86 for the DMG model,nd qeq = 1.37 for the AT model.

In Fig. 6, we show the respective surface plots of s(y, t) duringhe same evolution of the primary flow in Fig. 1, from the kinetic,MG and AT models. We do not show ˇ, which remains boundedway from zero so the phases are not isotropic; this will be con-rmed in the graphics below of the orientation ellipsoids. Clearly,

hase, which correlates precisely with the location and timing ofhe pulsating jet layer in each model! Next in Fig. 7, we present

ig. 9. The space–time plot of orientation ellipses from the kinetic-flow model, con-tructed from the shear plane (x–y) projection of the principal axes associated withhe two largest eigenvalues d1, d2 of the second moment tensor. Near the plate, the

ajor director exhibits finite, in-plane oscillations (wagging), whereas in the middlef the gap the major director continuously rotates (tumbles). A thin ellipse corre-ponds to strong focusing of the PDF, whereas a circle corresponds to the oblateefect phase.

138 M.G. Forest et al. / J. Non-Newtonian Fluid Mech. 155 (2008) 130–145

Fig. 10. The second-moment tensor ellipsoids of the DMG model simulation: asat

aslctmlweret

Mstpoaateca

vp

aoTts(sttgcitatttm

hsspfi

pace–time sequence capturing the jet-defect phase formation around t = 185, andspatial blow-up of the orientational ellipsoids from the t = 185 snapshot, capturing

he gradient morphology around the defect phase.

snapshot of the scalar order parameter s(y, t∗) from each modelimulation, coinciding with the peak jet layer snapshots shown ear-ier for each model. Since the snapshots are taken at the same time,learly there is a temporal correlation, and from the figures, clearlyhere is a spatial correlation with the minimum of s and the maxi-

um of vx. Next in Fig. 8, we show the time series of s(y∗, t) at theocation y∗ of the center of the jet layer, which is to be compared

ith the velocity time series in Fig. 5 at the same gap height forach model. One finds further confirmation of the temporal cor-elation between the oblate defect formation and the jet pulses inach model. That is, the pulsating jet layer correlates in space andime with a pulsating non-topological defect structure.

Next, we add information captured by the principal axes ofby presenting the orientation ellipsoids in Figs. 9–11, first in

pace–time for comparison with the surface plots of vx and s, andhen a snapshot of the spatial profile of the ellipsoids during theeak of the jet. The ellipsoids are presented in the coordinate framef the (x, y) plane, meaning the horizontal axis is the flow directionnd the vertical axis is y, the flow gradient axis, and the vorticityxis is normal to the paper. Notice that the ellipsoids never tilt out of

he plane. This means that the principal axis is always in-plane, andqually notable, even during the oblate phase degeneracy, the cir-le of principal orientation remains in the shear plane. The minorxis of orientation associated with d3 is always aligned with the

ftfm

Fig. 11. The ellipsoids of the AT model simulation, analogous to Fig. 10.

orticity axis. The space–time correlations between oblate defecthases and jet layers are amplified through the ellipsoid graphics.

This remarkable “in plane” symmetry of the orientation tensornd probability distribution for each model implies the dynamicsf the attractor at each gap height involves an in-plane limit cycle.he principal axis n1, when defined, is always orthogonal to the vor-icity axis, with polar coordinate angle � in the (x,y) plane. Fig. 12how the time series of �(t, yi) at several gap heights from the plateswhere � = 0 is the anchoring condition) to the center y∗ of the pul-ating jet, to the mid-gap y = 0. In each model, we find the orbit ofhe major director is wagging between the plates and the center ofhe jet layer, and tumbling in the remainder of the gap! The wag-ing orbits continuously grow in amplitude from the plates to theenter of the jet-defect layer, and then the major director tumblesn phase in the interior of the gap. This strong phase coherence ofhe major director across the middle of the gap is consistent withweak gradient morphology, since the jet-defect layers are close

o the plates and well-separated. In the AT model, for the parame-ers chosen here, the jet-defect layers are close to one another, andherefore the interior of the gap has significantly greater gradient

orphology, and less heterogeneity near the plates.We note that the formation of composite tumbling-wagging 1D

eterogeneous attractors have been observed previously, includingtudies of Tsuji and Rey [13] and the authors [31], where in-planeymmetry was imposed and pure shear was also imposed. Theresent simulations do not enforce in-plane symmetry, and solveor the fully coupled flow, yet we find the space–time attractor isn-plane; the out-of-plane degrees of freedom in the tensor and

ull kinetic distribution function simply decay to zero. The condi-ions on De and Er simply have to be tuned to amplify the floweedback jet phenomenon where the flow profile becomes non-

onotone.

M.G. Forest et al. / J. Non-Newtonian Fluid Mech. 155 (2008) 130–145 139

F gap lw 3–1 (A0

3l

DtsHiwflasc

3

sblA

wsgttDntsb

3o

ep

ig. 12. The time series of the in-plane Leslie angle of the major director at differentagging states. Gap heights: y = 0.78–1 (DH model), 0.88–1 (DMG model), y = 0.2–0.88 (DMG model), y = 0–0.22 (AT model).

.3. Shear and normal stress correlations with the pulsating jetayer

A major achievement of the Landau–de Gennes tensor andoi–Hess kinetic model lies in their prediction of sign changes in

he first normal stress difference associated with the tumbling tran-ition, in agreement with classical experiments of Kiss and Porter.ere, we of course have heterogeneous and dynamical responses

n the flow and orientational distribution, so it is of interest to seehat normal and shear stress features are associated with theseow-orientation responses. Fig. 13 gives the surface plots of N1(y, t)ssociated with the earlier plots of the primary flow vx(y, t) andcalar order parameter s(y, t). Pulsating layers of negative N1 coin-ide with the jet and oblate defect pulsating layers!

.4. Scaling behavior of the pulsating jet-oblate defect layers

So far, each model has been simulated at a special parameteret, and the features of the pulsating jet-oblate defect layer haveeen described. In particular, the locations and strengths of the jet

ayer vary from model to model, and now we give results from theT model regarding the scaling behavior of the center of the jet

doflap

ocations. Top: DH model. Middle: DMG model. Bottom: AT model. Left column: theT model). Right column: the tumbling states. Gap heights: y = 0-0.76 (DH model),

ith respect to Deborah and Ericksen numbers. In the AT modelimulation presented, the jet center is close to the middle of theap, at y∗ = 0.2. In the kinetic and DMG tensor model simulations,he jet center was near the plates. Fig. 14 and then Fig. 15 showhe location of the jet-layer depending on the parameter Er and˜ e, respectively. This indicates model robustness of the jet phe-omenon, and shows how the jet center can be moved relative tohe plates in each half of the shear cell. We omit similar parametertudies of the other two models, which also reflect a similar scalingehavior.

.5. Persistence of the pulsating jet-oblate defect layer tout-of-plane anchoring

The tangential anchoring condition has led to an in-plane ori-ntational distribution for all three models, where the jet layerulsates and oblate defect pulses coincide with the maximum

ephasing of the wagging orbits near the plates and the tumblingrbits in the center of the gap. One might question whether theseow-orientation phenomena are specific to the special in-planenchoring conditions. We choose the kinetic model to illustrate theersistence of the pulsating jet-oblate defect layer phenomenon

140 M.G. Forest et al. / J. Non-Newtonian Fluid Mech. 155 (2008) 130–145

Fig. 13. Space–time surface plots of the first normal stress difference N1 = �xx − �yy .Top: DH model. Middle: DMG model. Bottom: AT model.

Fig. 14. Scaling of the mid-point of the jet-defect layer with Ericksen number. Thescaled distance � = y/(2h) between the wall and the middle of the nearest hydro-dynamical jet in values of the gap width 2h is plotted versus Ericksen number Er forthe AT model. The points are the averages and the lines are the error bars. The curve

is the best fit of the function ˛/

√Er (˛ ≈ 3.9).

Fig. 15. Scaling of the mid-point of the jet-defect layer with Deborah number. Thescaled distance � = y/(2h) between the wall and the middle of the nearest hydro-dynamical jet in values of the gap width 2h is plotted vs. the Deborah number Defor the AT model. The points are the averages and the lines are the error bars.

Fp

wt

w�catifp

4l

awFcsaamtT1

ig. 16. The space–time surface plot of the primary velocity with an out-of-planelate anchoring condition from the kinetic-flow model.

ith out-of-plane anchoring, and thus an out-of-plane orienta-ional response across the shear gap.

Fig. 16 shows the velocity profile from out-of-plane anchoringith major director n = (sin �0 cos �0, sin �0 sin �0, cos �0), where

0 = 85◦, �0 = 5◦, which is close to the tangential anchoring. It islearly seen that the jet phenomena persists. Fig. 17 shows the polarnd the in-plane Leslie angle from the same simulation. The attrac-or exhibits tilted kayaking in layers near the plates, then kayakingn the interior of the gap. The pulsating jet arises at the transitionrom kayaking to tilted kayaking during the times when nearbyrincipal axes become strongly out of phase.

. Numerical stability of the pulsating jet-oblate defectayers to 2D spatial perturbations

Since the simulations reported here, for all three models,re confined to one space dimension, it is natural to questionhether the phenomena are stable in higher space dimensions.

or the DMG tensor-flow model, we have a spectral-Galerkinode with which we can perform a numerical stability analy-is. Following a recent study of the authors [32], we use thettractor above to populate initial data. The snapshots shown

bove are combined with noisy perturbations in full 2D spatialodes, which obey the boundary conditions on flow and orien-

ation at the plates, and which are periodic in the z direction.he results of the simulations reveal a convergence back to theD attractors, shown in Fig. 18. A snapshot of the secondary flow,

M.G. Forest et al. / J. Non-Newtonian Fluid Mech. 155 (2008) 130–145 141

Fig. 17. The orbit of the polar (�) and the in-plane Leslie angle (�) at different gap heights for the out-of-plane anchoring condition in the kinetic-flow model. The rightcolumn shows kayaking orbits (the out of plane analog of tumbling) at locations in the middle of the gap, while the left column shows tilted kayaking orbits (the out of planeanalog of wagging) near the plate.

Fig. 18. Two-space (y, z) dimensional simulations of the DMG tensor-flow model with initial data consisting of the 1D attractor above superimposed (y, z)-dependentperturbations in flow and orientation. Snapshots of (a) the primary velocity vx(y, z), (b) the y velocity component vy(y, z), (c) the velocity component vz(y, z), (d) the shearstress, and (e) first normal stress difference N1, at a fixed time after transients have passed.

142 M.G. Forest et al. / J. Non-Newtonian Fluid Mech. 155 (2008) 130–145

Fig. 19. Two-space (y, z) dimensional simulations of the DMG tensor-flow modelwith initial data consisting of the 1D attractor above superimposed (y, z)-dependentperturbations in flow and orientation. The snapshot of the secondary flow (vy ,vz)(y,z) is displayed at a fixed time after transients have passed.

(wo(

5

flpdtPflftioca

Table 3Error of the numerical solutions versus the number of mesh points, compared withthe numerical solution using 801 mesh points

Mesh points

KA

lt

ttwcmpftcott

adiarpd

djrm

A

oeFA0a

Aa

uoo

Q

w

Fig. 20. The mesh convergence rate of the velocity for the DMG model.

vy, vz)(y, z), is also presented, showing extremely weak roll cellshere the values of these flow components are O(10−11) orders

f magnitude below the primary flow and is therefore negligibleFig. 19).

. Concluding remarks

We have presented results of simulations for three differentow-nematic models for nano-rod dispersions in a steady parallel-late Couette experiment. Each model is simulated with one-spaceimensional heterogeneity and physical flow-orientation condi-ions at the plates, and with full orientational space resolution.ulsating jet layers are shown to arise in each model, where theow profile across the shear gap is non-monotone. The jet layer

orms precisely at the location and time when neighboring orbits ofhe orientational distribution transition from tumbling to wagging,

.e., monotone rotation versus oscillation in the principal axis. Therientational distribution forms an oblate defect phase that coin-ides with the jet layer “pulse”, which occurs when nearby tumblingnd wagging orbits develop director phase incoherence. When the

dp

GT

51 101 201 401

inetic model (error) 3.64 × 10−3 1.94 × 10−3 8.83 × 10−4 2.99 × 10−4

T model (error) 1.11 × 10−1 3.63 × 10−2 1.82 × 10−2 7.33 × 10−3

ocal axes of orientation regain phase coherence, the flow returnso monotone shear profiles.

The pulsating jet layer-oblate defect phase phenomenon washen shown to be robust as two constraints were released. First, theangential anchoring of the orientational distribution at the platesas shifted to an out-of-plane boundary condition. The result is a

omposite spatio-temporal attractor which exhibits kayaking in theiddle of the shear gap and tilted kayaking in layers buffering each

late. Again, the pulsating jet layers form precisely at the transitionrom monotone major director rotation (kayaking) to finite oscilla-ions (tilted kayaking), and furthermore, oblate defect phases form,orrelated in space and time with the pulsating jet, when the localrbits lose phase coherence. Finally, we use one model to establishhe numerical stability of these one-space dimensional attractorso 2D perturbations.

As pointed out by one of the referees the strong tangentialnchoring boundary condition is a first step to model nano-rodispersions or liquid crystals in Couette flow cells. A more real-

stic approach would impose an orienting field associated with annchoring energy at the wall, which could be driven out of equilib-ium by stresses propagating from the interior. While an intriguingroject, this treatment of non-equilibrium anchoring conditions iseferred to the future.

In summary, three separate models and codes predict model andimensional robustness of the non-Newtonian effect of pulsating

et layers in a sheared nano-rod dispersion, first discovered andeported by Kupferman et al. [1] in a reduced (planar) flow-nematicodel.

cknowledgement

This research has been performed under the auspicesf the Sonderforschungsbereich 448 ’Mesoskopisch strukturi-rte Verbundsysteme’ financially supported by the Deutscheorschungsgemeinschaft (DFG). This research is also sponsored byFOSR grant FA9550-06-1-0063, NSF grants DMS-0604891, DMS-604912, DMS-0605029 and DMS-0626180, NASA URETI BIMatward No. NCC-1-02037, and the Army Research Office.

ppendix A. The relationship between the AT tensor modelnd the DMG tensor model

The alignment of the effectively uniaxial particles with a molec-lar axis parallel to the unit vector u is characterized by anrientational distribution function f (u, t). In the tensor model therder parameter is the second rank alignment tensor:

=√

152

〈uu〉 ≡∫

f (u, t)

√152

uu d2u, (23)

hich is the anisotropic second moment characterizing the

istribution. The symbol x indicates the symmetric tracelessart of a tensor x, i.e. with Cartesian components denoted by

reek subscripts, one has x� = (1/2)(x� + x �) − (1/3)xı� .he alignment tensor is also often referred to the S-tensor or

nian F

a�

msfist

a

−tPt

ctdsktf

TvdaTcat

pt[

b

iaddFtowto

H2itks

Q

T

w

�

T

D

tE

E

ToFct

T

ϑ

Ctettcsid

�

i

P

H(εavfi

pltflpp

p

w

p

M.G. Forest et al. / J. Non-Newto

-tensor. The symmetric traceless part � of the dielectric tensorwhich gives rise to birefringence is proportional to the align-

ent tensor, viz. � = εaQ, with the characteristic coefficient εa

pecifying the optical anisotropy. The shear flow induced modi-cations of the alignment can be detected optically [42]. For thepecial case of uniaxial symmetry (uniaxial phase) the alignmentensor Q can be parameterized by one scalar order parameter q

nd the director n, i.e., Q = q(3/2)1/2nn, such that q2 = Q : Q, and√

5/2 ≤ q = (3/2)1/2Q : nn ≤ √5. The parameter a is proportional

o the Maier–Saupe order parameter S2 ≡ 〈P2(u · n)〉 = q/√

5, where2 denotes the second Legendre polynomial. Clearly, by definition,he order parameter q is bounded, just as S2.

The nonlinear relaxation equation for the alignment tensor Q,oupled to the velocity gradient field and an expression for the con-ribution to the stress tensor associated with the alignment wereerived in [36,39]. The influence of a shear flow on the phase tran-ition isotropic-nematic was studied in [43]. A derivation from ainetic equation was first given in [18]. The generalization to a spa-ially inhomogeneous situation was presented in [44], see also [45]or related works. The equation reads:

dQdt

− 2˝ × Q − 2 � · Q + ∇ · b + �−1a �a(Q) = −

√2

�ap

�a�. (24)

he equation involve characteristic phenomenological coefficients,iz. the relaxation time coefficient �a > 0, as well as �ap whichetermines the strength of the coupling between the alignmentnd the pressure tensor (stress tensor) or the velocity gradient.he dimensionless coefficient generalizes the corotational andodeformational derivatives and �a denotes the derivative of themended Landau–de Gennes potential that describes the isotropic-o-nematic phase transition ([40]).

The (third rank) alignment flux tensor b ∼ 〈cuu〉, where c is aeculiar molecular velocity, has to be taken into account for a spa-ially inhomogeneous system. Following the arguments given in44], the constitutive relation:

= −Da∇�a (25)

s used with the alignment diffusion coefficient Da. This ‘isotropic’pproximation is used for simplicity. It is valid when the threeiffusion coefficients linking the vector, second and third rank irre-ucible tensor parts of b with those of ∇�a are practically equal.or an experimental situation where the anisotropy of an alignmentensor diffusion matters cf. [47]. According to the general principlesf irreversible thermodynamics, a coupling of the vector part of bith the heat flux vector exists. This is akin to the coupling between

he ‘Kagan-vector’ and the heat flux considered in the kinetic theoryf molecular gases [48,49]. Here such an effect is disregarded.

For numerical studies it is convenient to scale the variables.ere the time is scaled by t0 = h/v0 (inverse shear rate), whereh denotes the distance of the plates. The alignment tensor is used

n units of the equilibrium value of the alignment at the isotropic-o nematic phase transition (aK), the stress tensor is scaled by theinetic pressure pkin = (�/m)kBT and the length is scaled by h. Thecaled variables read:

∗ = Qak

, v∗ = v

v0, x∗ = x

h, �∗ = �

pkin, t∗ = t

t0. (26)

he relaxation Eq. (24) in scaled form is given by

ddt∗ Q∗ = 2˝∗ · Q∗ + 2�∗ · Q∗ + Da�∗�a∗ − 1

De�a∗ +

√32

K�∗,

(27)

Htp

r

luid Mech. 155 (2008) 130–145 143

ith the derivative of the amended Landau–de Gennes potential:

a∗(Q)=ϑQ∗ − 3√

6Q∗ · Q∗+2(Q∗ : Q∗)Q∗

1 − ((Q∗ : Q∗)2)/(Q 4max)

− De

Er�Q∗.

(28)

he scaled parameter introduced here are the Deborah number:

˜ e = �av0

AKh= v0

6DrAKh, (29)

he scaled diffusion constant Da, and the parameter related to theriksen number:

˜r = De(

h

l

)2

AK. (30)

he quantity AK = 1 − NK/N∗ sets a scale for the relative differencef the concentrations NK and N∗. The parameter l2 is related to therank elastic persistence length. The tumbling parameter at thelearing point K is given by the ratio of two different relaxationimes:

K = − 2√3

�ap

�aaK. (31)

he parameter ϑ characterizes the reduced concentration:

= (1 − N/N∗)(1 − NK/N∗)

. (32)

omparison of the terms linear in Q in Eqs. (14), (28) and (32) showshat N = N∗ corresponds to N = 3 and ϑ = 0. The quantity aK is thequilibrium value of a at phase coexistence concentration NK. Inhe following we introduce the constitutive equations for the stressensor. In Cartesian tensor notation (Greek subscripts indicate theomponents 1, 2 and 3 pertaining to x-, y- and z-directions, theummation convention is used) the pressure tensor P � occurringn the momentum balance equation (no external field, � is the massensity):

dv�

dt+ ∇ P � = 0 (33)

s decomposed according to

� = Pı � + 12

ε �p + p �. (34)

ere P = (1/3)P is the trace part, p� = (1/2)(P� + P �) −1/3)Pı� is the symmetric traceless part of the tensor and p = �P � is the component of the pseudo vector associated with thentisymmetric part of the pressure tensor. The antisymmetric partanishes in the absence of torques caused by external orientingelds.

In the following, the trace part P is identified with the hydrostaticressure linked with the local density and temperature by the equi-

ibrium equation of state. The symmetric traceless friction pressureensor consists of an ‘isotropic’ contribution as already present inuids composed of spherical particles or in fluids of non-sphericalarticles in an perfectly ‘isotropic state’ with zero alignment, and aart explicitly depending on the alignment tensor:

= −2�iso� + pal, (35)

ith [39]:

al = �

mkBT

(√2

�ap

�a�a − 2Q · �a

). (36)

ere m and � are the mass of a particle and the mass density, �/m ishe number density, and pkin = (�/m)kBT is the equilibrium kineticressure which is used as reference value for pressures. In equilib-

ium one has �a(a) = 0 and consequently pal = 0. The occurrence

1 nian F

oOefls

P

a

Hs

ˇ

T

Iiutv

F

ˇ

D

Io(

w

F

H(E

Wst

�t

A

wp

lsfii

ul8iTmcao

fiTiei8w5smt

ssttvi

R

44 M.G. Forest et al. / J. Non-Newto

f the same coupling coefficients �ap in (36) as in (24) is due to annsager symmetry relation. For studies on the rheological prop-rties in the isotropic and in the nematic phases with stationaryow alignment, following from (24) and (36) see [36,39,44,50]. Thecaled constitutive equation is given by

∗ = p∗ı − 2 iso�∗ −√

32

K�a∗ − 2Q∗ · �a∗ (37)

nd the momentum equation yields:

ddt∗ v∗ = −∇ ·

(p∗I + 1

ˇP∗

)= ∇ · (−p∗I + �). (38)

ere � is the scaled stress tensor and the parameter ˇ measures thetrength of inertial related to viscosity forces, viz.:

= �(v0)2

pkina2KAK

. (39)

he coefficient iso is referred to the second Newtonian viscosity:

iso = �isov0

pkinha2KAK

. (40)

n the following we omit the asterisks. To rearrange the equationsnto the DMG model notation we relate the parameters to thosesed in the DMG model. The tumbling parameter K involves tohe molecules shape and can be related to the DMG parameter a,ia:

K = 2√5aK

r2 − 1r + 1

= 2√5aK

a. (41)

urther, via inspection we find:

= 3v20

�20DraKAK˛

, iso =(

1Re

+ �3

2

)ˇ, (42)

˜ e = 16AK

De, Er = 148

De Er N. (43)

n the DMG model the strain rate tensor is denoted by D insteadf � . We use the DMG notation, the relations for the parameters39–43) and rewrite the relaxation equation as

DDt

Q = ˝Q − Q˝ + (

DQ + QD − 23

(D : Q)I)

+ Da

(�F(Q) − De

Er�2Q

)

− 1

DeF(Q) + 1

Er�Q −

√32

KD,

(44)

ith the function:

˜(Q) = ϑQ − 3√

6(

QQ − 13

(Q : Q)I)

+ 2(Q : Q)Q

1 − ((Q : Q)2)/(Q 4max)

.

(45)

ere we used the expression A� = (1/2)(A� + A �) −1/3)Aı� for a Cartesian tensor A. Analogous, the constitutiveq. (38) in the DMG notation yields:

� = isoD −√

32

K

ˇF(Q) +

√32

K

ˇEr�Q

+

ˇ

(QF(Q) + F(Q)Q − 2

3(F(Q) : Q)I

)1

(2

) (46)

−Er ˇ

Q(�Q) + (�Q)Q −3

((�Q) : Q)I .

hereas the AT model equation for the orientational dynamics isimilar and have the same physical ingredients as the DMG model,he DMG stress tensor contain some additional terms, involving �1,

[

luid Mech. 155 (2008) 130–145

2. Furthermore, since we use = 0 the expression of the stressensor is simpler, but does contain the essential ingredients.

ppendix B. Numerical scheme

The partial differential equations of the three different modelsere solved by different numerical schemes and validated inde-endently by mesh convergence studies.

The convergence of the numerical solution was tested by calcu-ating the error ei := ‖vi

x − vtx‖2 of the velocity vx for different mesh

izes. Here vix is referred to the velocity on mesh i and vt

x on thenest mesh (our target solution), respectively. The symbol ||· · ·||2

ndicates the L2-Norm at a value of t specified later.For the kinetic model, 4th-order finite-difference formulas are

sed in the space discretization. Adaptive step size control is uti-ized in time integration. The numerical solution presented uses01 uniform mesh points with an adaptive time step size (approx-

mately O(10−5) to O(10−4)). This solution is taken as the target vtx.

hen we find the numerical solutions vix using 401, 201, 101 and 51

esh points, respectively. The final time is t = 5. The error ei wasomputed and shown in Table 3, which clearly shows convergence,lthough there is order reduction because of the extreme stiffnessf the underlying equations.

To solve the alignment tensor model equations a fourth-ordernite difference approximation for the space variable y was used.he nonlinear terms are discretized by nonstandard methods asntroduced in [51]. The resulting system of ordinary differentialquations were integrated by an adaptive Runge–Kutta–Fehlbergntegrator (approximately O(10−4)). The numerical solution using01 uniform mesh points gives the target solution vt

x. Further,e find the numerical solutions vi

x using 401, 201, 101 and1 mesh points, respectively. The computed error ei at t = 5 ishown in Table 3, which shows convergence. As in the kineticodel, the order reduction is due to the stiffness of the equa-

ions.For the DMG model, a Spectral-Galerkin method together with a

econd-order pressure correction method are adopted to handle thepace and time discretizations, respectively. All nonlinear terms arereated explicitly. The convergence of the numerical solution wasested by calculating the error ei at t = 10. Here the target solutiontx is given on 1024 mesh points. The convergence rate is displayedn Fig. 20.

eferences

[1] R. Kupferman, M. Kawaguchi, M.M. Denn, Emergence of structure in models ofliquid crystalline polymers with elasticity, J. Non-Newt. Fluid Mech. 91 (2000)255–271.

[2] S.M. Fielding, P.D. Olmsted, Early stage kinetics in a unified model of shear-induced demixing and mechanical shear banding, Phys. Rev. Lett. 90 (2003)224501.

[3] M.R. Lopez-Gonzalez, W.M. Holmes, P.T. Callaghan, P.J. Photinos, Shear bandingfluctuations and nematic order in wormlike micelles, Phys. Rev. Lett. 93 (2004)268302.

[4] G. Kiss, R.S. Porter, Rheology of concentrated solutions of helical polypeptides,J. Polym. Sci., Polym. Phys. Ed. 18 (1980) 361.

[5] R.G. Larson, The structure and rheology of complex fluids, Oxford UniversityPress, NY, 1999.

[6] Z. Tan, G.C. Berry, Studies on the texture of nematic solutions of rodlike poly-mers. 3. Rheo-optical and rheological behavior in shear, J. Rheol. 47 (1) (2003)73–104.

[7] M. Grosso, S. Crescitelli, E. Somma, J. Vermant, P. Moldenaers, P.L. Maffettone,Prediction and observation of sustained oscillations in a sheared liquid crys-talline polymer, Phys. Rev. Lett. 123 (2003) 145–156.

[8] M.P. Lettinga, Z. Dogic, H. Wang, J. Vermant, Flow behavior of colloidal rod-likeviruses in the nematic phase, Langmuir 21 (2005) 8048.

[9] Y.-G. Tao, W.K. den Otter, W.J. Briels, Kayaking and wagging of rods in shearflow, Phys. Rev. Lett. 95 (2005) 237802.

10] Y.-G. Tao, W.K. den Otter, W.J. Briels, Periodic orientational motions of rigidliquid-crystalline polymers in shear flow, J. Chem. Phys. 124 (2006) 204902.

nian F

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[[

[

[

[

[

[

[

[

[

[

[[

nomena in Polyatomic Gases: Cross sections, Scattering and Rarefied Gases,vol. 2, Clarendon Press, Oxford, 1991.

M.G. Forest et al. / J. Non-Newto

11] V. Faraoni, M. Grosso, S. Crescitelli, P.L. Maffettone, The rigid-rod model fornematic polymers: an analysis of the shear flow problem, J. Rheol. 43 (1999)829–843.

12] M.G. Forest, Q. Wang, Monodomain response of finite-aspect-ratio macro-molecules in shear and related linear flows, Rheol. Acta 42 (2003) 20–46.

13] T. Tsuji, A.D. Rey, Effect of long range order on sheared liquid crystalline mate-rials: flow regimes, transitions, and rheological phase diagrams, Phys. Rev. E 62(2000) 8141–8151.

14] M.G. Forest, Q. Wang, R. Zhou, The flow-phase diagram of Doi–Hess theoryfor sheared nematic polymers. II. Finite shear rates, Rheol. Acta 44 (1) (2004)80–93.

15] J. Ding, Y. Yang, Brownian dynamics simulation of rodlike polymers under shearflow, Rheol. Acta 33 (1994) 405–418.

16] M. Doi, Rheological properties of rodlike polymers in isotropic and liquid crys-talline phases, Ferroelectrics 30 (1980) 247–254.

17] S. Hess, B.R. Jennings, in: Electro-Optics and Dielectrics of Macromolecules andColloids, Plenum Publ. Corp., New York, NY, 1979;M. Kröger, H.S. Sellers, L. Garrido, Complex Fluids, Lecture Notes in Physics, vol.415, Springer, NY, 1992, pp. 295–301;G. Sgalari, G.L. Leal, J.J. Feng, The shear flow behavior of LCPs based on a gen-eralized Doi model with distortional elasticity, J. Non-Newt. Fluid Mech. 102(2002) 361–382.

18] S. Hess, Fokker–Planck-equation approach to flow alignment in liquid crystals,Z. Naturforsch. 31A (1976) 1034–1037.

19] M. Doi, S.F. Edwards, The Theory of Polymer Dynamics, Oxford University Press(Clarendon), London/New York, 1986.

20] G. Rienäcker, M. Kröger, S. Hess, Chaotic and regular shear-induced orientationaldynamics of nematic liquid crystals, Phys. Rev. E 66 (2002), 040702(R);G. Rienäcker, M. Kröger, S. Hess, Chaotic and regular shear-induced orientationaldynamics of nematic liquid crystals, Physica A 315 (2002) 537–568.

21] M. Grosso, R. Keunings, S. Crescitelli, P.L. Maffettone, Prediction of chaoticdynamics in sheared liquid crystalline polymers, Phys. Rev. Lett. 86 (2001)3184–3187.

22] M.G. Forest, R. Zhou, Q. Wang, Chaotic boundaries of nematic polymers in mixedshear and extensional flows, Phys. Rev. Lett. 93 (8) (2004) 088301.

23] B. Chakrabarti, M. Das, C. Dasgupta, S. Ramaswamy, A.K. Sood, Spatiotemporalrheochaos in nematic hydrodynamics, Phys. Rev. Lett. 92 (2004) 055501.

24] R. Ganapathy, A.K. Sood, Intermittency route to rheochaos in Wormlike Micelleswith flow-concentration coupling, Phys. Rev. Lett. 96 (2006) 108301.

25] M. Das, B. Chakrabarti, C. Dasgupta, S. Ramaswamy, A.K. Sood, Routes to spa-tiotemporal chaos in the rheology of nematogenic fluids, Phys. Rev. E 71 (2005)021707.

26] M.G. Forest, R. Zhou, Q. Wang, Microscopic–macroscopic simulations of rigid-rod polymer hydrodynamics: heterogeneity and rheochaos, SIAM MultiscaleModel. Simul. 6 (3) (2007) 858–878.

27] R.G. Larson, D.W. Mead, Development of orientation and texture during shear-ing of liquid-crystalline polymers, Liq. Cryst. 12 (751) (1992) C768.

28] R.G. Larson, D.W. Mead, The Ericksen number and Deborah number cascade insheared polymeric nematics, Liq. Cryst. 15 (151) (1993) C169.

29] J. Feng, J. Tao, L.G. Leal, Roll cells and disclinations in sheared polymer nematics,J. Fluid Mech. 449 (179) (2001) C200.

30] D.H. Klein, C. Garcia-Cervera, H.D. Ceniceros, L.G. Leal, Computational studiesof the shear flow behavior of a model for nematic liquid crystalline polymers,ANZIAM J. 46 (E) (2005) 210–244.

31] M.G. Forest, R. Zhou, Q. Wang, Kinetic structure simulations of nematic poly-mers in plane Couette cells. II. In-plane structure transitions, Multiscale Model.Simul. 4 (4) (2005) 1280–1304.

[

[

luid Mech. 155 (2008) 130–145 145

32] X. Yang, Z. Cui, M.G. Forest, Q. Wang, J. Shen, Dimensional robustness & insta-bility of sheared semi-dilute, nano-rod dispersions, Multiscale Model. Simul. 7(2) (2008) 622–654.

33] Q. Wang, A hydrodynamic theory for solutions of nonhomogeneous nematicliquid crystalline polymers of different configuration, J. Chem. Phys. 116 (20)(2002) 9120–9136.

34] R. Zhou, M.G. Forest, Q. Wang, Kinetic structure simulations of nematic poly-mers in plane Couette cells. I. The algorithm and benchmarks, Multiscale Model.Simul. 3 (4) (2005) 853–870.

35] H. Zhou, M.G. Forest, Q. Wang, Anchoring-induced texture and shear bandingof nematic polymers in shear cells, Disc. Contin. Dyn. Syst. Ser. B 8 (3) (2007)707–733.

36] S. Hess, Irreversible thermodynamics of nonequilibrium alignment phenomenain molecular liquids and liquid crystals, Z. Naturforsch. (30A) (1975) 728–738;S. Hess, Irreversible thermodynamics of nonequilibrium alignment phenom-ena in molecular liquids and liquid crystals, Z. Naturforsch. 30a (1975)1224.

37] P.G. de Gennes, J. Porst, The Physics of Liquid Crystals, Clarendon Press, NY, 1993.38] I. Pardowitz, S. Hess, Elasticity coefficients of nematic liquid crystals, J. Chem.

Phys. 76 (1982) 1485.39] C. Pereira Borgmeyer, S. Hess, Unified description of the flow alignment and

viscosity in the isotropic and nematic phases of liquid crystals, J. Non-Equilib.Thermodyn. 20 (1995) 359–384.

40] S. Heidenreich, P. Ilg, S. Hess, Robustness of the periodic and chaotic orien-tational behavior of tumbling nematic liquid crystals, Phys. Rev. E 73 (2006)061710.

41] M.G. Forest, Q. Wang, H. Zhou, R. Zhou, Structure scaling properties of confinednematic polymers in plane Couette cells: the weak flow limit, J. Rheol. 48 (1)(2004) 175–192.

42] G.G. Fuller, Optical Rheometry of Complex Fluids, Oxford University Press, NY,1995.

43] S. Hess, Pre- and post-transitional behavior of the flow alignment andflow-induced phase transition in liquid crystals, Z. Naturforsch. 31A (1976)1507–1513.

44] S. Hess, I. Pardowitz, On the unified theory for non-equilibrium phenomena inthe isotropic and nematic phases of a liquid crystal: Spatially inhomogeneousalignment, Z. Naturforsch. 36A (1981) 554–558.

45] G. Marrucci, F. Greco, The Elastic Constants of Maier–Saupe Rodlike MoleculeNematics, Mol. Cryst. Liq. Cryst. 206 (1991) 17–30.

46] G. Marrucci, F. Greco, Flow behavior of liquid crystalline polymers, Adv. Chem.Phys. (1993) 331–404.

47] B.K. Gupta, S. Hess, A.D. May, Anisotropy in the Dicke narrowing of rotationalRaman lines: A new measure of the nonsphericity of intermolecular forces, Can.J. Phys. 50 (1972) 778.

48] S. Hess, Heat-flow birefringence, Z. Naturforsch. 28a (1973) 861.49] F.R.W. McCourt, J.J.M. Beenakker, W.E. Köhler, I. Kuscer, Nonequilibrium Phe-

nomena in Polyatomic Gases: Dilute Gases, Vol. 1, Clarendon Press, Oxford,1990;F.R.W. McCourt, J.J.M. Beenakker, W.E. Köhler, I. Kuscer, Nonequilibrium Phe-

50] G. Rienäcker, S. Hess, Orientational dynamics of nematic liquid crystals undershear flow, Physica A 267 (1999) 294–321.

51] R.E. Mickens, Nonstandard Finite Difference Models of Differential Equations,World Scientific Publishing, London, 1994.