2-d lid-driven cavity ﬂow of nematic polymers: an unsteady

PAPER www.rsc.org/softmatter | Soft Matter

Dow

nloa

ded

by U

nive

rsity

of

Nor

th C

arol

ina

at M

oreh

ead

City

on

23 A

ugus

t 201

0Pu

blis

hed

on 2

7 Ja

nuar

y 20

10 o

n ht

tp://

pubs

.rsc

.org

| do

i:10.

1039

/B90

8502

EView Online

2-D lid-driven cavity flow of nematic polymers: an unsteady sea of defects

Xiaofeng Yang,*ac M. Gregory Forest,ab William Mullinsa and Qi Wangc

Received 29th April 2009, Accepted 20th October 2009

First published as an Advance Article on the web 27th January 2010

DOI: 10.1039/b908502e

A classical benchmark for viscous and viscoelastic fluid codes is the lid-driven cavity, a model problem

which highlights effects arising from a recirculating, geometrically confined flow. Here we simulate

a square 2-D cavity for nematic liquid crystalline polymers (LCPs), which adds the physics of ordered

phases and distortional elasticity to earlier cavity simulations. The range of local flow types within the

cavity create ideal conditions for strong orientational conflicts: nematic polymers are prone to steady

alignment in extension-dominated flow, yet sustained dynamic responses (tumbling, wagging,

kayaking) in shear-dominated flow. Orientational conflicts are mediated by defects, or disordered

phases, whose genesis and evolution in fully confined 2-D flows are explored here for the first time. Our

algorithm extends previous LCP-flow solvers by implementation of physical boundary conditions on

velocity and the orientational distribution at all four walls. We impose uniform orientational anchoring

at the cavity walls (parallel on top and bottom, normal on sides) to suppress corner defect anomalies,

and apply the method of Shen (J. Shen, J. Comp. Phys., 1991, 95, 228–245) for smoothing of the corner

flow singularities due to the standard no-slip flow condition on solid walls. The same code simulates

a viscous fluid by decoupling the orientational stress from the Navier–Stokes equations. We choose

a Reynolds number for which the viscous simulation develops a stationary flow structure with a central

rotating eddy and three corner eddies. We then simulate the Doi–Hess–Marrucci–Greco orientation

tensor model for nematic polymers coupled to the flow equations, and apply level-set detection,

tracking and graphics of defect domains (X. Yang, M. G. Forest, W. Mullins and Q. Wang, J.

Rheology, 2009, 53 (3), 589–615). The viscous flow structure is qualitatively preserved, albeit with

weaker corner vortices; the orientation field, however, consists of a highly transient, defect-laden

texture. Our diagnostics continuously detect and track defect domains through strong fluctuations in

topology as defects spawn, propagate, merge, collide and annihilate ad infinitum.

1. Introduction

Lid-driven cavity flow is one of the standard benchmarks for

computational fluid dynamics, serving as a test bed, experimen-

tally and numerically, for several compelling reasons. Bench-

marks play an important role in the computational science

community as a mechanism for vetting algorithms to deal with

specific challenges, such as abrupt contractions and sharp

corners, which can be compared with results of other codes and

experimental data. Flows are typically probed using particle

imaging velocimetry, whereas viscoelastic fluids are often probed

with various light scattering instruments to gain both strain and

stress measures from light intensity and birefringence patterns.

Improved experimental techniques continue to reveal more

details of flow and polymer conformations, which provide more

targets for numerical modeling.

Complex fluids require another type of benchmark, on the

model equations for the dynamics and heterogeneity of the

microstructure and on the expression for the extra stress stored

aDepartment of Mathematics, University of North Carolina at Chapel Hill,Chapel Hill, NC 27599-3250, USA. E-mail: [email protected] for Advanced Materials, Nanoscience & Technology, Universityof North Carolina at Chapel Hill, Chapel Hill, NC 27599-3250, USAcDepartment of Mathematics, University of South Carolina, Columbia, SC29208, USA

1138 | Soft Matter, 2010, 6, 1138–1156

by the microstructure which enters the flow equations. These two

elements of constitutive modeling comprise the weakest link in

our understanding and interpretation of complex fluid behavior.

The reader is referred to two monographs by Larson,26,27 the two

volume monograph of Bird et al.,3 and the polymer physics

approaches of de Gennes and Prost,5 Doi,9 Edwards et al.,11 and

Rubinstein and Colby30 for detailed treatments of constitutive

modeling of complex fluid microstructure. Progress has been

made on theory and modeling at the macromolecular scale, e.g.

the kinetic theory of bead-spring chains (cf. Bird et al.3 Vol. II).

Progress has also been made on ‘‘upscaling’’ to the equations and

variables of continuum mechanics (cf. Bird et al.3 Vol. I).

However, for most complex fluid systems, a predictive quanti-

tative theory and simulation remains an active area of research.

The most successful models with respect to experimental

behavior and data are for dilute and semi-dilute polymer chains,

either melts or solutions, and for rigid particle microstructures,

including colloids, liquid crystals, and liquid crystalline poly-

mers. Significant current activity surrounds entangled polymeric

liquids and the concepts of tube dynamics and reptation (cf. Doi

and Edwards10).

In the present paper, we adopt the highly vetted and bench-

marked models of Doi,9 Hess,22 Marrucci and Greco31 for

nematic (liquid crystalline) polymers. The microstructure

consists of rigid, high aspect ratio, Brownian rods, uniformly

This journal is ª The Royal Society of Chemistry 2010

http://dx.doi.org/10.1039/B908502E

Dow

nloa

ded

by U

nive

rsity

of

Nor

th C

arol

ina

at M

oreh

ead

City

on

23 A

ugus

t 201

0Pu

blis

hed

on 2

7 Ja

nuar

y 20

10 o

n ht

tp://

pubs

.rsc

.org

| do

i:10.

1039

/B90

8502

EView Online

dispersed in a viscous solvent, which have equilibrium-ordered

(nematic) phases across a range of sufficiently high rod volume

fractions. We apply the models to a numerical study of the lid-

driven cavity, focusing on rod volume fractions in the nematic

equilibrium phase. This study combines the numerical challenges

of corner flow singularities with viscoelastic stress and micro-

structural orientational phenomena due to geometric confine-

ment and recirculating flow. Nematic phases experience steady

alignment in extension-dominated flow and limit cycle behavior

(tumbling, wagging and kayaking) in steady shear-dominated

flow; with a diverse range of local flow types and strengths within

the cavity, a complex texture is inevitable.

Recirculating, spatially-confined flows generated within a lid-

driven cavity are representative of realistic engineering and

industrial applications where physical confinement by solid walls

is a dominant feature. Any particular cavity flow contains a full

range of local flow types from pure rotation near the center of the

recirculation region to strong extension near the edges of the lid

and strong shear layers between counter-rotating vortices. The

driven cavity therefore captures essential flow features present in

mold- and void-filling applications while filtering out other

effects, such as arbitrary geometry, in favor of a simple Cartesian

wall geometry (the easiest to handle from a grid generation point

of view and to implement in the laboratory). This geometric

simplicity is exploited to great advantage: the geometry is self-

consistent with a reduction to two space dimensions, which can

be experimentally designed and implemented in modeling and

simulations without sacrificing key hydrodynamic features. The

ability to compute and measure 2-D features makes validation

feasible and the data sets are significantly reduced relative to full

3-D flows. In this paper, we likewise restrict the microstructure

variable, the second-moment tensor of the rod orientational

distribution function, to two space dimensions. In the next

section, we will summarize the orientation tensor and its

expanded degrees of freedom beyond the single director of liquid

crystal theory.

From another modeling and computational viewpoint, the lid-

driven cavity is a model problem for handling corner singularities.

The moving lid intersects the stationary no-slip condition at the

sidewalls, resulting in a discontinuous velocity along with sharp

changes in pressure and stresses near the corner singular points.

Thus, methods have been developed and tested to circumvent this

singularity. For numerical simulations at low Reynolds number,

Burggraf4 employed a second-order central finite difference

method and the corner nodes were assumed to be part of the

stationary wall. At higher Reynolds number, the presence of the

singularity generates oscillations, which are then convected

throughout the cavity. For viscous fluids,39 the singularities are

removed by subtracting an analytical solution that contains the

same leading order behavior of the corner singularities. Floryan

et al. likewise coupled the numerical solution with a local

analytical solution near corners.14 The singularity subtraction

method is developed in Refs 1, 2 and 35. For viscoelastic fluids,

Grillet et al.20 treated the corner singularities by incorporating

a controlled amount of leakage. Shen37 introduced a ‘‘regularized’’

velocity boundary condition. In this study of nematic polymers,

we adopt the methods introduced by Shen for regularizing the

corner flow singularities, and we apply homogenous Dirichlet

boundary conditions on the orientation tensor at all four corners.


The cavity has another appealing feature associated with

transition phenomena. For viscous fluids, lid-driven cavity flow

undergoes a classical Reynolds number transition from steady to

periodic. This bifurcation phenomenon is an ideal benchmark for

numerical analysts to reproduce, which Shen accurately captured

by 2-D modeling and simulation.37 For viscoelastic fluids, the lid-

driven cavity has been utilized for the discovery and in-depth

elucidation of elastic instabilities.20,33 Here, we highlight behavior

unique to anisotropic viscoelastic fluids, where the microstructure

has order-disorder degrees of freedom (resolved by eigenvalues

of the orientation tensor) that couple to flow time scales and

length scales. Such features are quite challenging to identify and

quantify experimentally. For nematic polymers, stresses stored in

these disordered microstructural phases can be released when

a defect domain melts, thereby generating hydrodynamic

events.15,25 Those studies were limited to 1-D heterogeneity and

confinement, so it is natural to inquire as to the flow-micro-

structure behavior in 2-D confined flows.

The jump from one to two space dimensions is significant from

the point of view of nematic defects. We devote attention to these

issues throughout the paper. The nomenclature of defects has

a rich history; we refer the reader to an article by Denn,7 and to

our recent article46, which discusses tensorial defect metrics, both

local and non-local, and illustrates defect domains with and

without non-trivial topology. These local and topological tools

for defect detection and taxonomy are motivated and summa-

rized below for purposes of self-containment. The primary

historical classification of defects is based on their topology.

Defect phases can only possess non-trivial topology in dimen-

sions two or three, measured by a non-trivial winding number of

the principal axis of orientation (the nematic director) around

a closed space curve. Defects also possess local metrics, called the

scalar order parameters (cf.5), which assess whether the nematic

director is uniquely identifiable. If not, then locally the phase is

disordered, independent of physical space or space dimension.

There are two degrees of disorder: the isotropic phase is totally

disordered, whereas the so-called oblate phase is partially disor-

dered. To distinguish them, one must monitor both scalar order

parameters, as explained below and depicted in Fig. 1. These local

defect phases have been implicated in 1-D heterogeneous flow

phenomena,15 below the minimum dimension for defect topology.

Topological and local defect metrics co-exist in dimensions two

and three. Indeed, local defect phases (oblate and isotropic) are

always present in the cores of half-integer topological defects.

These dual descriptions of defects in two and three space dimen-

sions underscored the necessity of orientation tensors or full

distribution functions to replace theories based on the nematic

director. In the present paper, we employ an orientation tensor

model and the defect diagnostics developed inRef. 46 to describe

the complex defect dynamics that unfolds in the driven cavity.

In the simulations below, we report a nematic polymer variant

of the steady-unsteady Reynolds number transition in the 2-D

driven cavity, which will take on a unique hybrid character: the

flow will saturate into a quasi-steady pattern reminiscent of

viscous cavity flows, whereas the rod orientational pattern (the

texture) is slaved to the complex 2-D flow structure and responds

in a dynamic combination of stationary and recirculating features,

including a highly transient population of defect domains. At

lower lid speeds, a stationary flow and texture emerge, whereas at

Soft Matter, 2010, 6, 1138–1156 | 1139


Fig. 1 Sketches of the ellipsoidal geometry defined by the second

moment orientation tensor: the triaxial prolate ellipsoid of a nematic

phase, with eigenvalues d1 > d2 $ d3, with principal axis n1 of orientation;

the sphere of the isotropic phase with d1 ¼ d2 ¼ d3 ¼ 1/3; and the oblate

spheroid of the disordered phase with d1 ¼ d2 > d3.

Dow

nloa

ded

by U

nive

rsity

of

Nor

th C

arol

ina

at M

oreh

ead

City

on

23 A

ugus

t 201

0Pu

blis

hed

on 2

7 Ja

nuar

y 20

10 o

n ht

tp://

pubs

.rsc

.org

| do

i:10.

1039

/B90

8502

EView Online

higher lid speeds, a partial transition emerges in what we call

a hybrid response: the flow saturates into a stationary pattern,

whereas the texture is highly non-stationary. Furthermore, the

orientation tensor field is rife with complexity, and punctuated by

many defect domains and transition events, which we detect and

characterize below. This behavior of our model for flowing

nematic polymers arises when viscous stress dominates elastic

distortional stress, which is typical for experimental nematic

polymer shear cell experiments (cf. Larson and Mead28).

Cavity numerical experiments correspond to a 2-D generali-

zation of the ‘‘zero’’ (monodomain) and 1-D experiments that have

been studied in great detail for nematic polymers in a steady shear

cell. We refer the reader to the survey article by Rey and Denn.34

The 0-D experiment involves a parallel plate shear cell in which

either the top or both plates are translated at a steady speed, setting

up a stationary, nearly linear shear flow in the plate gap. If one

posits a simple linear shear flow, then the flow equations are

automatically satisfied and the coupled system reduces to a dynamic

equation for a spatially uniform (therefore ‘‘0-D’’) orientational

distribution. Remarkably, the nematic polymer microstructure

responds in a variety of transient limit cycle responses known as

tumbling, wagging and kayaking. These dynamic responses to

steady shear were first observed experimentally by Kiss and

Porter,23 and subsequently reproduced with the Doi–Hess kinetic

theory12,17,29 and its second moment closure models.16,40

The ‘‘0-D’’ bulk monodomain phase diagrams were general-

ized to 1-D heterogeneity in the shear gap, both for imposed

linear flows,40 as well as full coupled flow-orientation simul-

ations.15,18,19,45 The parallel plate shear cell simulations of

nematic polymers were then allowed to develop 2-D structure24,46

to model the Larson–Mead experiments on roll cells and roll cell

instabilities.28 These experiments and simulations avoid effects

due to full spatial confinement by applying periodic boundary

conditions in the vorticity direction.

1140 | Soft Matter, 2010, 6, 1138–1156

The lid-driven cavity benchmark has not been previously

investigated for nematic polymers to the best of our knowledge

and search capabilities. This lag in nematic polymer simulations

and experiments for a cavity flow is due to the fact that we have

only very recently achieved a satisfactory understanding of the

simpler benchmark of a parallel plate shear cell, which filters out

recirculation as well as confinement with reasonable device

design (large aspect ratios in the directions transverse to the plate

gap). This simpler geometry only requires physical boundary

conditions at the top and bottom plates, and one can impose

periodicity (therefore no walls) in the second dimension.24,46

The lid-driven cavity experiment for nematic polymers pres-

ents the next logical challenge and ideal additional level of

complexity beyond the parallel-plate shear cell. The lid-driven

cavity code is ideal first because it naturally limits to previous 2-

D shear cell model simulations by taking the sidewalls to infinity,

reducing to one-dimensional confinement. Through this

comparison at identical model parameters, numerical experi-

ments can explore consequences of the added physics and

geometric challenges introduced by a second or third dimension

of physical confinement. These comparisons are also accessible

experimentally; one of our goals is to identify salient features,

which can be validated by wet experiments.

For high-performance materials applications, one cannot

simply worry about the power consumption of a process and

estimates through accurate modeling of viscous dissipation.

Rather, the spatial morphology of the rod orientational

distribution, or texture, and the stored elastic stresses convey

the unique performance properties of rod nano-composites. If

the texture is riddled with defects, then it is valuable to know

how the unique conditions of a driven cavity contribute to

their formation, transport around the cavity or stationary

location. Most of this information is typically beyond

experimental resolution, so high-fidelity numerical modeling is

essential. Our group has developed direct connections between

flow-induced orientational bulk phases and 1-D structures and

conductive and mechanical properties of nematic polymers.48

These cavity simulations will be subsequently analyzed for

2-D property tensors. We also emphasize that the stresses

stored by the microstructure during processing are the

dominant residual stresses after quench to a solid phase;47

these features will be highlighted as well in our study below.

Since our results will identify a prominent role of defect

domains, the associated stress concentrations within those

domains will be highlighted.

Our model of the nematic polymer microstructure, and in

particular our description of defects as introduced earlier, is

worthy of special mention. The classical liquid crystal director

theory due primarily to Leslie,49 Ericksen,50 and Frank51 is not

capable of resolving the microstructure defect events that arise in

a confined, recirculating cavity. Specifically, a director theory

presumes a uniform degree of order throughout the fluid, and

thereby an identifiable principal axis of orientation is assumed at

every spatial location for all time. Defects are defined based on

a winding number criterion applied to the director field around

closed contours;43 defect detection is necessarily laborious from

this metric, since one must print 2-D texture snapshots and then

manually search the texture. There are no theoretical measures

for detection of topological changes in defect degree other than



Dow

nloa

ded

by U

nive

rsity

of

Nor

th C

arol

ina

at M

oreh

ead

City

on

23 A

ugus

t 201

0Pu

blis

hed

on 2

7 Ja

nuar

y 20

10 o

n ht

tp://

pubs

.rsc

.org

| do

i:10.

1039

/B90

8502

EView Online

comparing snapshots. These limitations were overcome in

a series of developments.

It is now understood that the singularities in a director field

associated with topological defects are regularized by expanding

the descriptive variable for the microstructure from a director to

some variable that allows for variability in the degree of order,

not just the direction of order. While there have been ad hoc

attempts to generalize the director theory of liquid crystals, the

fundamental approach comes from the kinetic theory of rigid-

rod macromolecules in a viscous solvent.9,22,31 In this theory, the

descriptive variable for the microstructure is a probability

distribution function (PDF) for the orientation of the rod

ensemble. This mean-field kinetic theory couples the physical

mechanisms of rotational diffusion, an excluded-volume

potential following Onsager’s description of the isotropic-

nematic phase transition, a distortional elasticity potential due

to Marrucci and Greco,31 which generalizes the Frank elasticity

of a director field, and hydrodynamic coupling through the

stress stored by the rigid-rod dispersion. The application of

these kinetic and orientation tensor models in numerical

simulations has led to significant progress, including contri-

butions from the research groups of Rey,40,41,42 Leal,13,24,36

Dhont and Briels.8

The defect diagnostics introduced and applied in46 focus on

local metrics for the oblate and isotropic phases, which are

monitored cost-free during flow-orientation simulations. This

defect detection strategy is an alternative to the traditional focus

on topological degree of the principal axis around a closed

contour. One simply monitors the differences in eigenvalues 1 $

d1 $ d2 $ d3 $ 0 of the second-moment orientation tensor; level

set contours of the oblate metric, d1�d2, and the isotropic metric,

d1�d3, detect the oblate and isotropic phases by color-coding for

the zero level sets of each metric. These metrics are summarized

below with graphical illustrations that are then applied in the

flow simulations. Thus, any 2-D flow-nematic simulation is

afforded a cost-free means for detecting defects, tracking them,

and exploring their creation and annihilation mechanisms.

This background sets the stage for the fundamental questions

addressed in the present paper: what types of defect domains

(defined and tracked by level set metrics of the oblate and

isotropic phases) are created in a truly confined 2-D cavity flow?

What flow conditions in the cavity lead to defect creation? What

is the topological degree of the orientational distribution

surrounding each defect domain, and is the topology transient or

persistent? What new phenomena arise that are exclusive to 2-D

confinement as opposed to 2-D flows with 1-D confinement and

periodicity in the second dimension?

In this paper, we solve the coupled 2-D Navier–Stokes and

Doi–Marrucci–Greco orientation tensor model for the lid-driven

square cavity flow of nematic liquid crystalline polymers. In

order to remove the complexities induced from the corner

singularities, we ‘‘regularize’’ the driven lid by imposing

a smoothed velocity distribution along the moving lid as in

Ref. 37. We choose a Reynolds number at which the viscous

cavity flow is steady, and then compare the results with a nematic

polymer simulation at the same Reynolds number. The departure

from viscous flow reveals the hydrodynamic effects due to the

ensemble of rigid rods, and the orientational morphology and

dynamics associated with the cavity flow.


2. The LCP-flow model and numerical method

2.1. The Doi–Marrucci–Greco (DMG) model

We briefly summarize the coupled system of dynamical equations

for the orientation tensor of the nematic liquid crystal polymer

fluid and the flow field. We refer the reader to our recent paper46

for a review of the literature on the hydrodynamics of rigid-rod

macromolecular dispersions (liquid crystal polymers). In,46 we

develop a numerical algorithm for 2-D confined flow between

counter-translating parallel plates, where physical boundary

conditions are imposed at the plates (in the flow-gradient spatial

variable) and periodicity is assumed in the vorticity direction. By

contrast, in the driven cavity one must consider physical

boundary conditions in the flow-gradient direction (the top plate

is translating while the bottom of the cavity is stationary) and

physical boundary conditions on the sidewalls of the cavity (in

the primary flow direction). For the present paper, we suppress

flow and spatial heterogeneity along the vorticity direction, and

consider the coupled orientation and flow structures that emerge

in a 2-D driven cavity. We also assume the orientational distri-

bution has two principal axes in the plane of flow. Our code is

compatible with the viscous limit where the orientation dynamics

decouples and the stress tensor reduces to the Navier–Stokes

relation; this allows us to compare the driven cavity flow of

a viscosity-matched isotropic fluid with that of a nematic liquid.

We implement the defect diagnostics and tracking tools from46 to

identify the spawning of defect domains and their evolution in

the cavity, as well as the role of defect domains in generating new

flow behavior relative to a viscous fluid.

The Doi–Hess kinetic theory describes the dynamics of rigid-rod

macromolecules in a viscous solvent in terms of an orientational

probability distribution function (PDF), whereas Marrucci and

Greco31,32 extended the model to include a distortional elasticity

potential. As inRef. 46, we project the PDF onto its second

moment tensor, M, and adopt a closure approximation that yields

reasonable approximations of the full kinetic theory in various

benchmark problems (cf.15,47). The fundamental descriptive vari-

able of the rod ensemble is the so-called orientation tensor, Q:

Q ¼M� I

3; M ¼

�mm�

(2.1)

where the angular brackets indicate an average with respect to

rod PDF, m is a unit vector representing the primary axis of an

individual rigid-rod macromolecule, and I is the identity tensor.

Q and M share spectral properties, with the same eigenvectors

and eigenvalues that differ by 1⁄3 . Recall that the second moment

M is symmetric, trace 1, and positive semi-definite, with non-

negative eigenvalues 0 # d3 # d2 # d1 # 1. The orthonormal

frame of eigenvectors nI with semi-axes di therefore geometrically

determines a triaxial ellipsoid at each mesoscopic location and

time. Spheres correspond to isotropic distributions d1¼ d2¼ d3¼1/3 with all directions of orientation equally probable; prolate

spheroids (with two equal minor axes) correspond to uniaxial

distributions d1 > d2 ¼ d3, such as all stable nematic equilibria;

oblate spheroids (with two equal major axes, d1 ¼ d2 > d3)

correspond to a defect phase in which the most likely axis of

orientation lies anywhere on the circle in the plane normal to the

unique minor axis associated with the principal value d3);

full triaxial ellipsoids correspond to a biaxial orientation where

Soft Matter, 2010, 6, 1138–1156 | 1141


Dow

nloa

ded

by U

nive

rsity

of

Nor

th C

arol

ina

at M

oreh

ead

City

on

23 A

ugus

t 201

0Pu

blis

hed

on 2

7 Ja

nuar

y 20

10 o

n ht

tp://

pubs

.rsc

.org

| do

i:10.

1039

/B90

8502

EView Online

all di are distinct. Whenever d1 is simple, the associated peak

orientation direction n1 is called the major director (Fig. 1).

We note that the Q tensor model for nematic polymers

generalizes liquid crystal (LC) theory for the director n1 in

significant ways: (i) LC theory essentially assumes d1 ¼ 1 and the

other eigenvalues are 0, so that the orientational distribution is

a delta function with all rods in the direction n1. (ii) The Q tensor

admits variable degrees of order, ranging from highly focused

rod ensembles where d1 is unique and approaching unity, to

disordered phases where d1 either has multiplicity two (di ¼ d2 >

d3) or multiplicity three (di ¼ 1/3 for all values of i). (iii) The

model equations shown below include an excluded-volume

potential which captures the equilibrium isotropic-nematic phase

diagram, which is absent in the Leslie–Ericken–Frank theory of

LCs. Furthermore, the Marrucci–Greco distortional elasticity

potential generalizes the Frank elasticity potential to the tensor

level. Thus, the DMG model allows for hydrodynamics and

other applied fields to modify the full tensor Q in space and time,

including defect formation, without core singularities in the

director field of LCs. The cores of topological singularities in the

orientation tensor field are regularized by disordered phases,

either the oblate defect phase or the isotropic defect phase.

As shown in Ref. 46, these eigenvalue degeneracy conditions

for defect cores are easily monitored by local metrics, the zero

level sets of d1 � d2 and d1 � d3. The oblate defect phase is

detected when d1 � d2 ¼ 0 and d1 � d3 > 0; as in Ref. 46, we find

this is the selected disordered phase in the lid-driven cavity at

nematic volume fractions. We graphically represent these defect

domains both by color-scale level sets of d1 � d2 and by showing

snapshots of 2-D arrays of the orientation tensor ellipsoids,

where the oblate defect phases correspond to platelet shapes. We

emphasize that our defect detection and tracking diagnostics are

centered on the defect cores, which obey local and blindly

monitored conditions, replacing the onerous process of taking

snapshots of texture, then computing the topological degree of

the orientational field over a non-local two-dimensional domain.

Finally, once a positive test for defects has been taken, we can

recover the topological degree surrounding each core by printing

snapshots of the major director n1 throughout the cavity, except

in the core where n1 is not identifiable. As we shall show, the

oblate defect domains are easily detected and tracked through

extreme changes in defect topology which occur continuously as

defects are spawned, collide and annihilated. Such defect

phenomena would be impossible to resolve without the local

scalar metrics based on order parameters rather than directors.

2.2. Model equations

We non-dimensionalize the DMG model using the gap height 2h,

the nematic rotational diffusion time scale tn, and the charac-

teristic stress s0 ¼rh2

t2n

where r is the density of the nematic

polymer liquid. The dimensionless velocity, position, time, stress,

and pressure variables become:

~U ¼ tn

hU; ~x ¼ 1

hx; ~t ¼ t

tn

;~s ¼ ss0

; ~p ¼ p

s0

(2.2)

The top lid moves at constant speed v0, which defines a bulk

flow time scale t0 ¼h

v0; the average rotary diffusivity Dr of the

1142 | Soft Matter, 2010, 6, 1138–1156

rods defines another time scale tn ¼1

6Dr

, whose ratio defines the

Deborah number: De ¼ tn

t0¼ v0

6hDr. The following seven dimen-

sionless parameters arise:

Re ¼ s0tn

h;a ¼ 3ckT

s0

;Er ¼ 8h2

Nl2;mi ¼

3ckTzi

tns0

; i ¼ 1; 2; 3 (2.3)

where Re is the solvent Reynolds number; the solvent viscosity

is h; a measures the strength of entropy relative to kinetic

energy; c is the number density of rod molecules; k is the

Boltzmann constant; T is absolute temperature; Er is the

Ericksen number, which measures short-range nematic potential

strength relative to distortional elasticity strength, which

involves the persistence length l and the dimensionless volume

fraction N, which governs the strength of the Maier–Saupe

intermolecular potential; 1/mi, i ¼ 1, 2, 3 are the three nematic

Reynolds numbers, themselves dependent on the three shape-

dependent viscosity parameters due to the polymer-solvent

interaction, 3ckTzi, i ¼ 1, 2, 3. If we drop the � on all variables;

the dimensionless flow and stress constitutive equations take the

following forms.

DU

Dt¼ V,ð � pIþ sÞ (2.4)

The extra stress constitutive equation is given by

s ¼�

2

Reþ m3

�Dþ aaFðQÞ

þ aa

3Er

DQ : Q

�Qþ I

3

�� 1

2ðDQQþQDQÞ � 1

3DQ

!

þ a

3Er

�1

2ðQDQ� DQQÞ � 1

4ðVQ : VQ� VVQ : QÞ

�

þ m1

�Qþ I

3

�DþD

�Qþ I

3

�!þ m2D : Q

�Qþ I

3

�;

(2.5)

whereDU

Dt¼ vU

vtþ ðU,VÞU is the material derivative, D¼ (VU +

(VU)T)/2 is the symmetrized velocity gradient tensor (rate-of-

strain tensor), a ¼ r2 � 1

r2 þ 1parametrizes the aspect ratio r of

spheroidal molecules, and rotational Brownian motion and

short-range excluded volume effects enter through the gradient

of the Maier–Saupe potential,

FðQÞ ¼�

1�N

3

�Q�NQ,QþNQ : Q

�Qþ I

3

�(2.6)

The orientation tensor equation is

DQ

Dt¼ UQ�QUþ aðDQþQDÞ þ 2a

3D

� 2aD : Q

�Qþ I

3

��

FðQÞ þ 1

3Er

�DQ : Q

�Qþ I

3

�

� 1

2

�DQQþQDQ

�� 1

3DQ

��(2.7)



Fig. 2 Viscous steady state features for Reynolds number 4000. (a) Contour lines of the stream function, color coded by min(4) ¼ �0.0034, max(4) ¼0.1737. The magnitude of the primary vortex: 0.1737, secondary vortex on the bottom right:�3.4� 10�3, bottom left:�9.12� 10�4, upper left:�3.24�10�4. (b) First normal stress difference N1 ¼ s22 � s33 and shear stress s23. (c) Velocity components Uy and Uz.

Dow

nloa

ded

by U

nive

rsity

of

Nor

th C

arol

ina

at M

oreh

ead

City

on

23 A

ugus

t 201

0Pu

blis

hed

on 2

7 Ja

nuar

y 20

10 o

n ht

tp://

pubs

.rsc

.org

| do

i:10.

1039

/B90

8502

EView Online

whereDQ

Dt¼ vQ

vtþ ðU,VÞQ and U ¼ (VU � (VU)T)/2 is the

vorticity tensor. In the simulations presented in this paper,

we suppose that the upper lid of the square domain U moves

at constant speed. In order to take advantage of spectral

accuracy, we consider a ‘‘regularized’’ driven cavity flow

where the singularities at the upper corners are removed.

Namely, we take the horizontal speed on the upper lid of the

cavity to be


Ujy¼hy¼

0; 0;De16

h4z

z2ðhz � zÞ2!

(2.8)

At the other walls, we impose no-slip boundary conditions.

The only flow restriction we make is to suppress flow dependence

along the x-axis. The orientation tensor is assumed to be at

equilibrium along all four walls, with principal axis normal to

each wall.

Soft Matter, 2010, 6, 1138–1156 | 1143


Fig. 3 Vortex and defect shedding from the lid exit corner after startup of the cavity flow. (a) Snapshots are orientational morphology at times t ¼ 1.5,

1.75, 2.5, 2.75. (b) A blow-up of the region in the lid exit corner where the vortex and defects are spawned. The color bar code at the top corresponds to

the level sets of the oblate defect metric, with dark blue indicating a positive test for an oblate defect phase.

1144 | Soft Matter, 2010, 6, 1138–1156 This journal is ª The Royal Society of Chemistry 2010

Dow

nloa

ded

by U

nive

rsity

of

Nor

th C

arol

ina

at M

oreh

ead

City

on

23 A

ugus

t 201

0Pu

blis

hed

on 2

7 Ja

nuar

y 20

10 o

n ht

tp://

pubs

.rsc

.org

| do

i:10.

1039

/B90

8502

EView Online


Fig. 4 Topological graphics of the initial snapshots from Fig. 2. It corresponds to the projection of the principal axis of M (or Q) onto the YZ plane.

Blue rectangles surround a topological degree �1/2 defect while red circles surround a degree +1/2 defect.

Fig. 5 A blow-up around the defect and vortex shedding region of the first normal stress difference, N1 ¼ s22–s33 of the vortex and defect shedding

region at t ¼ 1.5, 1.75, 2.5, 2.75 from Fig. 3 and 4.

This journal is ª The Royal Society of Chemistry 2010 Soft Matter, 2010, 6, 1138–1156 | 1145

Dow

nloa

ded

by U

nive

rsity

of

Nor

th C

arol

ina

at M

oreh

ead

City

on

23 A

ugus

t 201

0Pu

blis

hed

on 2

7 Ja

nuar

y 20

10 o

n ht

tp://

pubs

.rsc

.org

| do

i:10.

1039

/B90

8502

EView Online


Fig. 6 A blow-up of the shear stress s23 in the defect and vortex shedding region at t ¼ 1.5, 1.75, 2.5, 2.75 from Fig. 3–5.

Dow

nloa

ded

by U

nive

rsity

of

Nor

th C

arol

ina

at M

oreh

ead

City

on

23 A

ugus

t 201

0Pu

blis

hed

on 2

7 Ja

nuar

y 20

10 o

n ht

tp://

pubs

.rsc

.org

| do

i:10.

1039

/B90

8502

EView Online

3. Numerical method

We consider a 2-D square domain: (y, z) ˛ U ¼ [�hy, hy] � [0, hz]

with hy ¼ 1, hz ¼ 2. For the time discretization, the second-order

pressure-correction scheme for the flow equation21 and second-

order semi-implicit scheme46 of the morphology equation are

adopted. This algorithm provides an advantage: at each time step,

we only need to solve a sequence of Poisson equations of the form

u� lDu ¼ f ðuÞ;ujvU ¼ 0 or

vu

vnjvU ¼ 0

((3.1)

We solve the above 2-D Poisson equations using the spectral-

Galerkin method as in Refs 38 and 46. In all simulations, we use

256� 256 Legendre polynomials in y and z, with a time step of dt

¼ 0.0001. The following parameter values are implemented in the

simulations presented below: De ¼ 1, a ¼ 0.8, m1 ¼ 10�3,m2 ¼10�3, m3¼ 5� 10�4, Re¼ 4000, a¼ 0.01 and Er¼ 100. We set the

equilibrium nematic concentration N ¼ 6, so the equilibrium

order parameter is s0 ¼ 0.809. We note that the chosen critical

flow-orientation coupling parameter a is relatively small here,

which has the effect of qualitatively preserving the stationary

viscous flow geometry of the stream function. In simulations not

presented here, at higher values of a, the quasi-stationary flow

structure (one core eddy with three corner eddies) becomes

unstable. This is analogous to the Reynolds number transition

for viscous cavity flows, where the stationary flow structure

becomes unstable. The complex flow-orientation behavior at

such parameter regimes is deferred; the results presented here are

already sufficiently complex even when the qualitative flow

geometry is relatively simple.

1146 | Soft Matter, 2010, 6, 1138–1156

4. Viscous and nematic polymer simulations

4.1. The viscous fluid cavity simulation

If we restrict to the Navier–Stokes equations (by setting mi, i ¼ 1,

2, 3, a and N to zero), our algorithm yields the solution for

a viscous fluid. Fig. 2 shows the steady stream function, and the

first normal stress difference and shear stress, and the stationary

velocity components. The flow profile consists of a core eddy in

the center of the cavity and three smaller eddies in the upper left,

lower right and left. This is quantitatively consistent with

a previous driven cavity simulation37 using the same smoothing

function adopted here for the lid velocity boundary condition.

4.2. The nematic polymer cavity simulation.

4.2.1 Flow-defect dynamics at start-up. Fig. 3–9 show the

dynamics of the initial start-up of the cavity flow. The early

dynamics shows how defects are first spawned and evolve along

with the development of the basic flow topology. At onset,

a shear layer forms at the top lid, with a vortex at the exit corner

which turns the shear layer downward along the exit wall and

into the cavity. In Fig. 3, we show four snapshots, at low values

of t, of the stream function level set contours, superimposed with

color-coded level sets of the oblate defect metric, d1 � d2. The

dark blue color signals a positive test for an oblate phase domain,

where the principal axis of orientation spreads to a circle of

directions (equivalently, isotropic in the plane of the flow, but

ordered in the vorticity direction). We find an oblate thin fila-

ment domain (A) forms along the shear layer; the domain is

stretched, and sheds an oblate domain (B). Another weak oblate



Fig. 7 The early entrainment of defects shed at the lid exit corner during the formation of the large cavity vortex, at times t ¼ 5, 6, 7, 8, 9, 10.

This journal is ª The Royal Society of Chemistry 2010 Soft Matter, 2010, 6, 1138–1156 | 1147

Dow

nloa

ded

by U

nive

rsity

of

Nor

th C

arol

ina

at M

oreh

ead

City

on

23 A

ugus

t 201

0Pu

blis

hed

on 2

7 Ja

nuar

y 20

10 o

n ht

tp://

pubs

.rsc

.org

| do

i:10.

1039

/B90

8502

EView Online


Fig. 8 The second moment ellipsoid graphics, color-coded by the oblate defect metric (dark blue indicates an oblate defect phase), associated with Fig. 7

at t ¼ 5, 6, 7, 8, 9, 10.

Dow

nloa

ded

by U

nive

rsity

of

Nor

th C

arol

ina

at M

oreh

ead

City

on

23 A

ugus

t 201

0Pu

blis

hed

on 2

7 Ja

nuar

y 20

10 o

n ht

tp://

pubs

.rsc

.org

| do

i:10.

1039

/B90

8502

EView Online

domain (C) forms below the drop (B), which then intensifies and

splits into two oblate drops (C and D). These three defect

domains (B, C and D) are subsequently swept away with the

vortex (Fig. 7 below). This phenomenon is typical of defect

spawning throughout the cavity: an oblate domain is created in

a region with a shear layer, and the oblate domain is typically

stretched by the flow, sometimes splitting the domain into

satellite oblate domains, and then the satellite domains are swept

into the cavity by the circulating flow topology.

Once we have a positive test from the oblate defect metric, then

we can print the major director texture surrounding this domain,

and deduce the more traditional topological degree associated

with the oblate domains. Here, we implement the enhanced

topological tools developed inRef. 46 for 2-D nematic polymer

morphologies. Fig. 4 is the more traditional orientational

texture, where the principal axis of M (where defined) is projected

onto the YZ plane, from which the topological degree or winding

number is deduced. From the t ¼ 1.5 snapshot, we observe that

a pair of defects of degree +1/2 (B) and �1/2 (A) have formed,

with total degree 0 preserved from the initial data here and for all

1148 | Soft Matter, 2010, 6, 1138–1156

time. As the oblate strip is stretched and rotated along with the

flow geometry, at t ¼ 2.5 another pair of (C: +1/2, D: �1/2)

topological defects have formed, which continue at t¼ 2.75 to be

swept by the flow. In Fig. 5 and Fig. 6, we give the rheological

texture snapshots, of the first normal stress difference and shear

stress respectively, associated with these early transient flow and

orientational textures.

The preference of the oblate degeneracy over the isotropic

phase appears to be universal in 2-D confined shear-dominated

flows at relatively high nematic concentrations and for uniform

initial equilibrium distributions. This has been detailed in our

recent study46 of confined flow between two counter-translating

steady plates with periodicity in the second space dimension. The

only 2-D simulations where isotropic domain cores are found, to

our knowledge, involve initial data or boundary conditions that

seed isotropic phases.6,44 The same result is almost surely true of

the analogous 2-D simulations of the Leal group.24 Although

they did not implement the oblate metric, they show topological

defects are accompanied by strong order parameter changes, and

they show isotropic phases do not form. The only consistent



Fig. 9 The topological degree surrounding the four defect domains of Fig. 7 and 8. Again, blue rectangles encircle degree�1/2 defects, while red circles

surround +1/2 defects.

Dow

nloa

ded

by U

nive

rsity

of

Nor

th C

arol

ina

at M

oreh

ead

City

on

23 A

ugus

t 201

0Pu

blis

hed

on 2

7 Ja

nuar

y 20

10 o

n ht

tp://

pubs

.rsc

.org

| do

i:10.

1039

/B90

8502

EView Online

explanation is that oblate defects form at the core of their

topological degree �1/2 defects.

Isotropic (not shown since the test fails for isotropic phases)

and oblate defect metrics detect the formation of defect domains,

their shape and size, and track defect domain propagation,

deformation, and domain splitting completely irrespective of the

traditional topological defect metrics. The orientation tensor

ellipsoid textures contain richer information than the principal

axis, and clearly show the regularized behavior around the

topological singularity resulting from the dual projection of the

principal axis from the ellipsoid and then onto the YZ plane.

Fig. 7–9 show defect transport of the 3 oblate domains (B, C

and D) which carry charges +1/2, +1/2 and –1/2, respectively.

The thin lid oblate domain (A) retains a charge�1/2 defect at the

far right, or exit tip. Notice that the pair of +1/2 defect domains

(B and C) lie within the core vortex that has formed, whereas the

�1/2 oblate domain (D) has migrated along the wall in the outer

bands of the core vortex. Remarkably, these four defect domain

and topology features persist for as long as the simulations

run, creating a background stationary defect orientational


morphology: the �1/2 charge defect at the exit tip of the lid

oblate domain (A); the pair of rotating +1/2 charge defects in the

central eddy (B and C); and the �1/2 charge defect (D) that

migrates down the exit wall and becomes stationary in the lower

right corner between the central eddy and the lower right corner

eddy. This background orientational morphology persists ad

infinitum while the rest of the cavity undergoes a continuous

stream of defect generation, propagation and merger events. We

turn to these transient defect phenomena for the remaining

figures and discussion.

4.2.2. Long timescale flow-orientation features. As indicated

above, after ten time units (106 time steps), the lower right corner

develops an oblate drop-shaped defect domain (D in Fig. 7) and

corner vortex which are stationary throughout the subsequent

dynamics. The defect sits near the top of the corner vortex

toward the interior of the cavity, and actually splits the vortex

into a primary eddy and a very small eddy. This structure fluc-

tuates mildly as defect domains intermittently pass by, but the

flow-defect structure is essentially blocked from the complex

Soft Matter, 2010, 6, 1138–1156 | 1149


Fig. 10 Snapshots of the stream function contours and orientational morphology at times t¼ 179.25, 182.5, 188, 188.5, 189, 189.25. The central vortex

and lower corner eddies have formed and remain essentially unchanged (t ¼ 179.25 is shown), while the defect morphology in this sequence below

exhibits a merger at the left corner while other defects rotate by within the central eddy. (a) Left: snapshots of orientational morphology superimposed by

the contours of stream function f at t ¼ 179.25. The magnitude of the primary vortex: 0.1989, secondary vortex on the bottom right: �5.12 � 10�4,

bottom left: �7.97 � 10�4, upper left: �2.54 � 10�4. Right: the labels of all oblate defect domains at t ¼ 179.25. (b) Sequences of the defect morphology

around the top left cavity corner.

Dow

nloa

ded

by U

nive

rsity

of

Nor

th C

arol

ina

at M

oreh

ead

City

on

23 A

ugus

t 201

0Pu

blis

hed

on 2

7 Ja

nuar

y 20

10 o

n ht

tp://

pubs

.rsc

.org

| do

i:10.

1039

/B90

8502

EView Online

defect dynamics that unfolds in the remainder of the cavity. This

structure is evident in Fig. 10, 13 and 14.

There are additional long-lived defect-flow morphologies that

appear to be metastable: they form and persist for long periods of

1150 | Soft Matter, 2010, 6, 1138–1156

time, then get destroyed during certain ‘‘bursts’’ in defect

generation and merge, but reform and persist for tens of

normalized time units until the next burst in defect activity. One

such metastable defect is the structure illustrated in Fig. 10–12,



Fig. 11 The orientational ellipsoid graphics of the defect merger in Fig. 10 at the lid entrance corner.

Dow

nloa

ded

by U

nive

rsity

of

Nor

th C

arol

ina

at M

oreh

ead

City

on

23 A

ugus

t 201

0Pu

blis

hed

on 2

7 Ja

nuar

y 20

10 o

n ht

tp://

pubs

.rsc

.org

| do

i:10.

1039

/B90

8502

EView Online

associated with the top left corner of the cavity where the lid

enters: one circular oblate drop (B) that resides between the small

stationary eddy in the corner and the lid, and a thin strip (A)

aligned from the left wall to the top lid between the corner eddy,

the central eddy, and the lid shear layer. Fig. 10(a) shows four

rotating oblate defect domains (C1, C2, C3 and C4) within the

central eddy, the lower right stationary oblate domain, the 45�

degree thin oblate strip (E) in the top left corner, whereas the

only defect changes are with the small circular drop (B) in the top

left corner. Next, the oblate drop domain (B) is absorbed by the

lid oblate strip domain (A), captured quite clearly in the snapshot

sequence of Fig. 11. This sequence illustrates the power of the

oblate level set metric to track complex defect events. Next,

Fig. 12 projects the information from each snapshot of Fig. 11

onto the orientation ellipsoid and major director textures,

showing the dynamics of the defect topology in the upper left

corner. Looking at the oblate domains, it is clear that the small

oblate drop domain is attracted to the thin lid oblate domain: this

is expected at t ¼ 179.25 since the small drop (B) has charge +1/2

while the thin lid has charge �1/2 (A) with the core at the right

exit tip. Therefore, even though the oblate domains merge to


form one continuous oblate strip, the +1/2 and �1/2 charges are

too separated to merge and annihilate, and the lid oblate strip

now has both a +1/2 and a �1/2 topological defect at the left and

right tips from t ¼ 182.5 to t ¼ 188.5, respectively. After a few

time units, t ¼ 189, another oblate domain (F) of charge �1/2 is

swept into the top right corner, then merges with the left tip (B)

of the lid oblate layer. Finally, at t ¼ 189.25, the two opposite

charge 1/2 defects (BF) merge and annihilate, and the �1/2

topological degree remains at the right exit corner (A).

4.2.2.1. Defect elasticity. If we follow the dynamics around t

¼ 179.25 of Fig. 10 and the tens of time units afterward, we

observe the spatial coupling of oblate defect domains through the

Marrucci–Greco distortional elasticity potential. As the defects

that are entrained in the central vortex rotate around the cavity,

every passage by the top left corner leads to a coupling of

a rotating defect drop with the defects in the corner. Fig. 13 (time

evolves left-right, then down) shows a time sequence in which the

passing defect drop interacts with the top left corner defect

structure. The corner drop (B) and strip (E) persist, but the strip

is apparently pulled and deformed by the passing defect (C1); as

Soft Matter, 2010, 6, 1138–1156 | 1151


Fig. 12 The topological signature of the defect merger in Fig. 10 and 13. At t¼ 179.25, the lid oblate domain has a degree +1/2 defect (red circle) at the

left tip and a degree �1/2 defect (blue rectangle) far to the right tip. The drop domain rising from below has degree �1/2, which is carried toward the lid

and the two localized opposite charge defects merge, leaving a trivial topology while the lid retains the �1/2 defect at the right tip.

Dow

nloa

ded

by U

nive

rsity

of

Nor

th C

arol

ina

at M

oreh

ead

City

on

23 A

ugus

t 201

0Pu

blis

hed

on 2

7 Ja

nuar

y 20

10 o

n ht

tp://

pubs

.rsc

.org

| do

i:10.

1039

/B90

8502

EView Online

the passing defect (C1) eventually gets swept far enough away, it

decouples from the corner defect domains (B), and the drop-strip

(B–E) structure rebounds back into the corner.

We close with Fig. 14 to convey additional representative

defect dynamical phenomena. Note in snapshots t ¼ 276.5 and

277, a triangular array (A, B and E) of defects forms in the lid

entry corner; the thin vertical strip (B) subsequently merges with

the elongated horizontal oblate domain (A), while the domain

(A, B and E) couples to another defect drop (C1) passing by in

outer bands of the rotating central eddy. In this same time

sequence, we observe a new defect spawning event, which occurs

along the left wall, between the corner eddies and the central

eddy. From t ¼ 276.5 to 279, the new defocusing domain (F)

starts to form along the lower left wall, then intensifies into

a strip oblate defect domain that splits in two (F1 and F2), with

the top half (F1) entering the left corner and the bottom half (F2)

getting swept by the outer bands of the central eddy while

reducing in domain size to a small drop. A short time later the

defect domain (F1) from below moves into the lid entry corner

and merges with (B) on the left of the lid domain (A and B). The

1152 | Soft Matter, 2010, 6, 1138–1156

lid oblate domain (A) then returns to a uniform strip by t ¼ 282,

with a total of 4 satellite drop defect domains (C1, C4, E and F2)

now rotating in the central eddy. This time sequence illustrates

several defect generation, splitting, and merger events, with

a high density defect structure at t¼ 278.5 for example and a low

density defect structure at t ¼ 282. Bursts of defect domain

transitions of this nature continue to occur in simulations out to

t ¼ 400, or O(107) timesteps, so that we are confident the

simulation has converged to a highly transient and heteroge-

neous space-time attractor.

5. Concluding remarks

A new algorithm for nematic polymers allows for physical

boundary conditions on four sidewalls of a square cavity, which

we implement to simulate a lid-driven cavity flow. The purpose of

this study is to gain insight into the defect morphology and

dynamics associated with spatially confined, 2D recirculating

flows of nematic polymers, where the flow is physically trapped.

There is a significant amount of literature on monodomain



Fig. 13 The defect elasticity phenomenon. The tilted defect domain in the lid entry corner interacts with a passing defect drop, getting successively

‘‘pulled’’ by the passing defect. When the separation increases, they decouple and the strip rebounds back to the lid entry corner. t ¼ 219, 219.5, 220,

220.5, 221, 221.5, 222, 222.5, 224, 227 are presented left to right and then down.

Dow

nloa

ded

by U

nive

rsity

of

Nor

th C

arol

ina

at M

oreh

ead

City

on

23 A

ugus

t 201

0Pu

blis

hed

on 2

7 Ja

nuar

y 20

10 o

n ht

tp://

pubs

.rsc

.org

| do

i:10.

1039

/B90

8502

EView Online

dynamics, and on 1-D heterogeneous flows of LCPs, where

locally disordered oblate defect phases are now understood to

arise where local monodomains execute different limit cycles and


their respective principal axes get far out of phase. A small set of

2-D flow-nematic simulations exist, by the Leal group24 and our

group,46 with physical boundary conditions at top and bottom

Soft Matter, 2010, 6, 1138–1156 | 1153


Fig. 14 Another defect spawning ground along the left wall between the top and bottom corner eddies and the central vortex is shown. An oblate defect

strip forms along the wall, splits in two with one piece entering the corner and the other getting swept by the flow. The lid oblate domain absorbs the

defect domain from the corner. t ¼ 276.5, 277, 277.5, 278, 278.5, 279, 279.5, 280, 280.5, 281, 281.5, 282 are presented left to right and then down.

1154 | Soft Matter, 2010, 6, 1138–1156 This journal is ª The Royal Society of Chemistry 2010

Dow

nloa

ded

by U

nive

rsity

of

Nor

th C

arol

ina

at M

oreh

ead

City

on

23 A

ugus

t 201

0Pu

blis

hed

on 2

7 Ja

nuar

y 20

10 o

n ht

tp://

pubs

.rsc

.org

| do

i:10.

1039

/B90

8502

EView Online


Dow

nloa

ded

by U

nive

rsity

of

Nor

th C

arol

ina

at M

oreh

ead

City

on

23 A

ugus

t 201

0Pu

blis

hed

on 2

7 Ja

nuar

y 20

10 o

n ht

tp://

pubs

.rsc

.org

| do

i:10.

1039

/B90

8502

EView Online

plates but with periodicity in the transverse direction. Thus the

effects due to two-dimensional physical confinement are simul-

ated here for the first time.

By decoupling the orientational stress contributions, the same

code yields the viscous cavity flow. We identify a Reynolds

number with a stationary cavity flow, consisting of a central

vortex and three corner vortices; a vortex does not form at the

lid exit corner. Coupling the orientational stress and corres-

ponding dynamics and gradient morphology, we find the

nematic polymer simulation is unsteady at the same Reynolds

number of the viscous stationary simulation: the flow is essen-

tially stationary, whereas the orientational distribution is

unsteady. The stream function contours are qualitatively similar

for the viscous and nematic polymer flow, but the three LCP

corner vortices are 78%, 15% and 87% of the strength of the

viscous vortices. There is a persistent defect domain structure

common to all snapshots with �1/2 degree defects sitting near

the lid exit corner and in the lower exit corner, and two

+1/2 degree defects that continuously rotate within the central

vortex. Superimposed on this stationary defect background,

there is an additional transient sea of defects, where defects are

continuously spawned in shear layers, swept throughout the

cavity by the flow, interact, merge, and regenerate. The impli-

cation of these results for mold and cavity filling of nematic

polymers for materials applications is that one can expect

intermittency in the location, number, and area (or volume)

fraction of defects in the orientational distribution.

References

1 S. Albensoeder and H. C. Kuhlmann, Accurate three-dimensional lid-driven cavity flow, J. Comput. Phys., 2005, 206, 536–558.

2 F. Auteri, N. Parolini and L. Quartapelle, Numerical investigation onthe stability of singular driven cavity flow, J. Comput. Phys., 2002,183, 1–25.

3 B. R. Bird, C. F. Curtiss, R. C. Armstrong, and O. Hassager.Dynamics of Polymeric Liquids. Wiley, John & Sons, Incorporated,1994, vol. I and II.

4 O. R. Burggraf, Analytical and numerical studies of the structure ofsteady separated flows, J. Fluid Mech., 1966, 24, 113–151.

5 P.-G. de Gennes and J. Prost. The Physics of Liquid Crystals. OxfordUniversity Press, New York, 1993.

6 G. de Luca and A. D. Rey, Dynamic interactions between nematicpoint defects in the extrusion duct of spiders, J. Chem. Phys., 2006,124, 144904.

7 M. M. Denn, Fifty years of non-newtonian fluid dynamics, AIChE J.,2004, 50, 2335–2345.

8 J. K. G. Dhont and W. J. Briels, Isotropic-nematic spinodaldecomposition kinetics, Phys. Rev. E: Stat., Nonlinear, Soft MatterPhys., 2005, 72(3), 031404.

9 M. Doi, Molecular dynamics and rheological properties ofconcentrated solutions of rodlike polymers in isotropic and liquidcrystalline phases, J. Polym. Sci. Polym. Phys. Ed., 229–243.

10 M. Doi and S. F. Edwards. The Theory of Polymer Dynamics. OxfordUniversity Press, London, 1986.

11 D. A. Edwards, H. Brenner, and D. T. Wasan. Interfacial TransportProcess and Rheology. Butterworths/Heinemann, London, 1999.

12 V. Faraoni, M. Grosso, S. Crescitelli and P. L. Maffettone, The rigid-rod model for nematic polymers: An analysis of the shear flowproblem, J. Rheol., 1999, 43, 829–843.

13 J. J. Feng, J. Tao and L. G. Leal, Roll cells and disclinations insheared nematic polymers, J. Fluid Mech., 2001, 449, 179–200.

14 J. M. Floryan and L. Czecheowski, On the numerical treatment ofa corner singularity in the vorticity field, J. Comput. Phys., 1995,118, 222–228.


15 M. G. Forest, S. Heidenreich, S. Hess, X. Yang and R. Zhou,Robustness of pulsating jet-like layers in sheared nano-roddispersions, J. Non-Newtonian Fluid Mech., 2008, 155, 130–145.

16 M. G. Forest and Q. Wang, Monodomain response of finite-aspect-ratio macromolecules in shear and related linear flows, Rheol. Acta,2003, 42, 20–46.

17 M. G. Forest, Q. Wang and R. Zhou, The flow-phase diagram of Doi-Hess theory for sheared nematic polymers, ii: finite shear rates, Rheol.Acta, 2004, 44, 80–93.

18 M. G. Forest, H. Zhou and Q. Wang, A model study of the spinningof thermotropic liquid crystalline polymers: fiber performancepredictions and bounds on throughput, Adv. Polym. Technol., 1999,18, 314–335.

19 M. G. Forest, R. Zhou and Q. Wang, Kinetic structure simulations ofnematic polymers in plane couette cells, II: In-plane structuretransitions, Multiscale Model. Simul., 2005, 4, 1280–1304.

20 A. M. Grillet, B. Yang, B. Khomami and E. S. G. Shaqfeh, Modelingof viscoelastic lid driven cavity flow using finite element simulations,J. Non-Newtonian Fluid Mech., 1999, 88, 99–131.

21 J. L. Guermond, P. Minev and J. Shen, An overview of projectionmethods for incompressible flows, Comput. Methods Appl. Mech.Eng., 2006, 195, 6011–6045.

22 S. Hess, Fokker-Planck-equation approach to flow alignment inliquid crystals, Z Naturforsch, 1976, 31a, 1034–1037.

23 G. Kiss and R. S. Porter, Rheology of concentrated solutions ofpoly(g-benzylglutamate), Phys. Fluids, 1978, 65, 193–211.

24 D. H. Klein, C. J. Garcia-Cervera, H. D. Ceniceros and L. G. Leal,Ericksen number and deborah number cascade predictions ofa model for liquid crystalline polymers for simple shear flow, Phys.Fluids, 2007, 19, 023101.

25 R. Kupferman, M. N. Kawaguchi and M. M. Denn, Emergence ofstructure in a model of liquid crystalline polymers with elasticcoupling, J. Non-Newtonian Fluid Mech., 2000, 91, 255–271.

26 R. G. Larson, Constitutive equations for polymer melts and solutions,Butterworth-Heinemann, Boston, 1988.

27 R. G. Larson. The structure and rheology of complex fluids. OxfordUniversity Press, 1999.

28 R. G. Larson and D. W. Mead, The Ericksen number and Deborahnumber casades in sheared polymeric nematics, Liq. Cryst., 1993,15, 151–169.

29 R. G. Larson and H. C. Ottinger, Effect of molecular elasticity on out-of-plane orientations in shearing flows of liquid crystalline polymers,Macromolecules, 1991, 24, 6270–6282.

30 M. Rubinstein and R. H. Colby. Polymer physics. Oxford UniversityPress, 2003.

31 G. Marrucci and F. Greco, The elastic constants of Maier-Saupe rod-like molecule nematics, Mol. Cryst. Liq. Cryst., 1991, 206, 17–30.

32 G. Marrucci and F. Greco, Flow behavior of liquid crystallinepolymers, Adv. Chem. Phys., 1993, 86, 331–404.

33 P. Pakdel and G. H. McKinley, Cavity flows of elastic liquids: purelyelastic instabilities, Phys. Fluids, 1998, 10, 1058–1070.

34 A. D. Rey and M. M. Denn, Dynamical phenomena in liquid-crystalline materials, Annu. Rev. Fluid Mech., 2002, 34, 233–266.

35 W. W. Schultz, N. Y. Lee and J. P. Boyd, Chebyshev pseudospectralmethod of viscous flows with corner singularities, J. Sci. Comput.,1989, 4, 1573–7691.

36 G. Sgalari, G. L. Leal and J. J. Feng, The shear flow behavior of lcpsbased on a generalized doi model with distortional elasticity, J. Non-Newtonian Fluid Mech., 2002, 102, 361–382.

37 J. Shen, Hopf bifurcation of the unsteady regularized driven cavityflow, J. Comput. Phys., 1991, 95, 228–245.

38 J. Shen, Efficient spectral-Galerkin method I. direct solvers forsecond- and fourth-order equations by using Legendre polynomials,SIAM J. Sci. Comput., 1994, 15, 1489–1505.

39 R. Srinivasan, Accurate solutions for steady plane flow in the drivencavity I. Stokes flow, Z. Angew. Math. Phys., 1995, 46, 524–545.

40 T. Tsuji and A. D. Rey, Effect of long range order on sheared liquidcrystalline materials, part I: Compatability between tumblingbehavior and fixed anchoring, J. Non-Newtonian Fluid Mech., 1997,73, 127–152.

41 T. Tsuji and A. D. Rey, Orientational mode selection mechanisms forsheared nematic liquid crystalline materials, J. Non-Newtonian FluidMech., 1998, 57, 5609–5625.

42 T. Tsuji and A. D. Rey, Effect of long range order on sheared liquidcrystalline materials: Flow regimes, transitions, and rheological

Soft Matter, 2010, 6, 1138–1156 | 1155


Dow

nloa

ded

by U

nive

rsity

of

Nor

th C

arol

ina

at M

oreh

ead

City

on

23 A

ugus

t 201

0Pu

blis

hed

on 2

7 Ja

nuar

y 20

10 o

n ht

tp://

pubs

.rsc

.org

| do

i:10.

1039

/B90

8502

EView Online

diagrams, Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat.Interdiscip. Top., 2000, 62, 8141–8151.

43 G. E. Volovik and O. D. Lavrentovich, Topological dynamics ofdefects: boojums in nematic drops, Zhurn. Eksp. Teor. Fiz., 1983,85, 1997–2010.

44 K. Xu. Ph. D. thesis. University of North Carolina at Chapel Hill,Department of Mathematics, preprint, 2009.

45 X. Yang, Z. Cui, M. G. Forest, Q. Wang and J. Shen, Dimensionalrobustness & instability of sheared, semi-dilute, nano-roddispersions, Multiscale Model. Simul., 2008, 7, 622–654.

46 X. Yang, M. G. Forest, W. Mullins and Q. Wang, Dynamic defectmorphology and hydrodynamics of sheared nematic polymers intwo space dimensions, J. Rheol., 2009, 53(3), 589–615.

1156 | Soft Matter, 2010, 6, 1138–1156

47 X. Yang, M. G. Forest, W. Mullins and Q. Wang, Quench sensitivityto defects and shear banding in nematic polymer film flows, J. Non-Newtonian Fluid Mech., 2009, 159, 115–129.

48 X. Zheng, M. G. Forest, R. Lipton, R. Zhou and Q. Wang, Exactscaling laws for electrical conductivity properties of nematic polymernano-composite monodomains, Adv. Funct. Mater., 2005, 15, 627–638.

49 F. M. Leslie, Continuum theory for nematic liquid crystals,Continuum Mech. Thermodyn., 1992, 4, 167–175.

50 J. L. Ericksen, A thermodynamic view of order parameters of liquidcrystals, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo,1984, pp. 24–36.

51 F. C. Frank, I. Liquid crystals. On the theory of liquid crystals,Discuss. Faraday Soc., 1958, 25, 19–28.



2-d lid-driven cavity ﬂow of nematic polymers: an unsteady

Documents