2-D lid-driven cavity flow of nematic polymers: an unsteady ...
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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
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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.
This journal is ª The Royal Society of Chemistry 2010
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.
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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
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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.
This journal is ª The Royal Society of Chemistry 2010
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
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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)
This journal is ª The Royal Society of Chemistry 2010
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.
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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
This journal is ª The Royal Society of Chemistry 2010
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
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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
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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.
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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
This journal is ª The Royal Society of Chemistry 2010
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
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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.
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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
This journal is ª The Royal Society of Chemistry 2010
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.
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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
This journal is ª The Royal Society of Chemistry 2010
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.
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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,
This journal is ª The Royal Society of Chemistry 2010
Fig. 11 The orientational ellipsoid graphics of the defect merger in Fig. 10 at the lid entrance corner.
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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
This journal is ª The Royal Society of Chemistry 2010
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.
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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
This journal is ª The Royal Society of Chemistry 2010
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.
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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
This journal is ª The Royal Society of Chemistry 2010
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
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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.
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