Rheol Acta (2012) 51:849–866DOI 10.1007/s00397-012-0645-7

ORIGINAL CONTRIBUTION

Systems analysis of hybrid, multi-scale complex flowsimulations using Newton-GMRES

Zubair Anwar · Arvind Gopinath ·Robert C. Armstrong

Received: 15 January 2012 / Revised: 4 June 2012 / Accepted: 3 July 2012 / Published online: 20 July 2012© Springer-Verlag 2012

Abstract We present a methodology to analyze thestationary states and stability of complex fluid flowsby using hybrid, discrete, and/or continuum multi-scalesimulations. Building on existing theories, our schemeextracts dynamical and equilibrium characteristics fromcarefully chosen time integrations of these multi-scaleevolution equations. Two canonical problems are pre-sented to demonstrate the ability and accuracy of theformalism. The first is an investigation of flow-inducedtransitions seen in homogeneous, hard- rod liquid crys-tal suspensions subjected to a linear shear flow. In thesecond problem, we study the phenomenon of draw res-onance, a dynamical instability in an isothermal fiber-spinning process, by using a multi-scale hybrid simu-lation that incorporates both stochastic and continuummodels.

Keywords Hybrid simulations · Newton-GMRES ·Systems level analysis

Z. Anwar (B) · R. C. ArmstrongDepartment of Chemical Engineering, MIT,Cambridge, MA, USAe-mail: zubair.anwar@alum.mit.edu

R. C. Armstronge-mail: rca@mit.edu

A. GopinathMartin Fisher School of Physics, Brandeis University,Waltham, MA, USAe-mail: arvindg@brandeis.edu

Present Address:Z. AnwarGoogle, 1600 Amphitheater Parkway,Mountain View, CA 94043, USA

Introduction

The area of viscoelastic flow modeling has maturedconsiderably over the last few decades, with many re-cent studies focusing on three-dimensional flows andrheological descriptions that originate from polymerkinetic theories (Bird et al. 1987; Keunings 2004). Agreat deal of effort is currently devoted to enablingstudies at very high temporal and spatial resolutionsthrough the development and improvement of robustnumerical methods for continuum as well as hybridsimulations. In particular, detailed molecular dynamicsimulations (Jabbarzadeh et al. 2000, 2002) that canbe used to inform sophisticated coarse-grained modelshave recently become feasible.

One of the many aims of these studies on viscoelasticcomplex flows is the accurate prediction of dynamicallydriven flow instabilities. Standard approaches for thishave focused on steady-state solvers or time integrationof the transient nonlinear equations (Keunings 2004).The former approach, while simple in principle, can bedifficult to implement for complicated constitutive de-scriptions, and it suffers from various mathematical clo-sure approximations that limit the applicability of thetheoretical model. The latter approach is based on inte-grating appropriate dynamical evolution equations ortime-steppers. Both closed-form constitutive equations(Smith et al. 2000, 2003) as well as stochastic descrip-tions from kinetic theory (Somasi and Khomami 2001,2000) have been used to construct these dynamicalmodels. The work with stochastic descriptions presentsconsiderable improvement as it mostly steers clear ofclosure approximations. Nevertheless, this approachalso needs considerable analytical and numerical effort.Consider, for example, the simulation of polymers in a

850 Rheol Acta (2012) 51:849–866

flow. First, unlike in a traditional simulation of a closedset of deterministic equations where the base flow stateis steady, the stochastic simulation involves fluctuating(in the microscopic sense) base states. Second, analysisof the stability of such a microscopically fluctuatingbase states requires appropriate stochastic differentialequations for the evolution of small perturbations.

In this article, we describe and demonstrate a schemethat builds on previous theoretical work—notably thatof Shroff and Keller (1993), Kevrekidis et al. (2004),Kevrekidis and coworkers (Theodoropoulos et al. 2000;Kevrekidis et al. 2003), as well as work done inour group (Anwar and Armstrong 2008; Anwar 2008;Gopinath et al. 2004)—to enable the formulation offixed point problems for hybrid, multi-scale simula-tions. While our earlier work described the efficientcomputation of steady-state properties of various com-plex flows (Anwar and Armstrong 2008), herein wemove further in complexity and focus on the applica-bility of the method for mapping out bifurcation mapsand discerning system-level information. We begin byinvestigating flow-induced transitions seen in hard-rodsuspensions subjected to linear shear flows. This en-ables the testing of both the accuracy as well as abilityof the code to capture bifurcations. Second, we presentresults for the onset of draw resonance and recoveryof stability at high draw ratios for the isothermal fiber-spinning process. This is done by constructing a hybridsimulation that includes (a) a stochastic descriptionof Hookean dumbbell configuration fields discretizedwith the discontinuous Galerkin method and (b) thefiber-spinning model equations, discretized with theGalerkin finite element method.

The novelty of the method lies in combining tra-ditional Newton approaches with time integrationschemes that essentially treat the time evolution equa-tions or simulations as black boxes. Stability character-istics of computed steady states are determined fromestimates for the leading eigenvalues of the dynamicalproblem obtained by integrations over specified timeintervals. Standard methods for performing bifurcationanalysis, such as arc-length continuation, can then beapplied directly to obtain critical points of the hybridsimulation.

While explicit closed-form ordinary differentialequations exist for the two problems presented in thispaper (Doi–Hess model with a spherical-harmonic se-ries expansion and Brownian dynamics), these equa-tions are only used to construct black box time-steppers, which are then called by the numericalmethod to perform detailed bifurcation analysis. Thenumerical method never uses the explicit dynamicalequations for the stability and bifurcation analysis, and

it only requires multiple integrations from appropri-ately chosen initial conditions over a short time intervalto be returned by the simulation.

Theory and implementation

In building hybrid simulators, one typically obtains alarge system of ordinary differential equations that de-scribe the detailed dynamics of the system under study.The equations could result either from an appropriatediscrete implementation of a deterministic equation orfrom a stochastic description of the system dynamics.Let us denote these microscopic unknowns by x(t) andsuppose that they satisfy the (black box) time-steppercode

dxdt

= f (x; μ), x ∈ Rm, and m � 1 (1)

such that x represents the instantaneous state of thesystem with μ being a control parameter. The actionof the time-stepper, i.e, the result of the black boxsimulation on integrating from time to to some finitetime, to + t, gives the time−t map of the system and canbe expressed as T t

d , such that x(t0 + t) = T td x(t0). The

subscript d corresponds to detailed, and it is of courseimplicit that the initial condition at to is known.

We wish to start from the microscopic descriptionembodied in Eq. 1 and formally determine an appro-priate macroscopic, coarse description with reduceddimensionality. Specifically, we seek equations for anappropriately complete set of coarse variables X ob-tained from x via a restriction operator R and thussatisfying the equations

dXdt

= F(X; μ), X ∈ RM, and M � m. (2)

This restriction operator could, in principle, involve av-eraging over many microstructural states or ensemblesas generated, for instance, in stochastic descriptions.Successful identification of the set X satisfying Eq. 2suggests that physically relevant coarse-grained vari-ables such as moments or averages may be identifiedwith functionals of X, and conversely, these functionalscan also serve as coarse variables (Theodoropouloset al. 2000). In the framework of dynamical systemstheory, the existence of Eq. 2 with M � m implies theexistence of a slow manifold (Theodoropoulos et al.2000) in the dynamics described by Eq. 1. This manifoldembodies a closure of Eq. 1 such that all componentsof the coarse variable(s) vary slowly on this manifoldwhen observing the dynamics of the detailed time-

Rheol Acta (2012) 51:849–866 851

stepper.1 The theory of singularly perturbed systemsthen posits the existence of a change of variable, x �→u = (u1, u2) in Eq. 1 such that

du1

dt= g1(u1, u2)

du2

dt= 1

εg2(u1, u2)

u1 ∈ RN, u2 ∈ R

m−N (3)

with the parameter ε reflecting the ratio of characteris-tic slow and fast time scales. For ε � 1 correspondingto a separation of time scales, we have at leading orderg2(u1, u2) = 0, so that one may write u2 = h(u1). In thislimit, the variable u2 is slaved to the coarse variable u1,the evolution of which follows

du1

dt= g1(u1, h(u1)) (4)

completing the identification with Eq. 2. It is not neces-sary that the functions g1, g2, and h be known explicitlyor in closed form; only that they exist.

In a manner similar to the definition of thedetailed time map, T t

d , we now define the coarsemap for the coarse time-stepper (integration routine),T τ

c . The integration here corresponds to integratingEq. 2 for a duration τ defined as the time horizon(Theodoropoulos et al. 2000). This interval is chosensuch that it captures the true macroscopic dynamicswithout making the computation of T τ

d too expensive.At a minimum, τ should be large enough so that fast dy-namics have decayed, and the solution has approachedthe slow manifold. A third operator links the detailedand coarse time-steppers, L, the non-unique lifting op-erator that maps the macroscopic description in termsof coarse variables X onto an appropriate microscopicdescription x. Consistency requires this operator bespecified such that lifting from the macroscopic to themicroscopic and then restricting back to the macro-scopic has no effect on the macroscopic state.

To summarize, given an initial macroscopic con-dition X(t0), the coarse time-stepper consists of thefollowing steps: (1) construct a single or an ensembleof microscopic states such that x(t0) = LX(t0), (2) usethe detailed time-stepper Eq. 1 to evolve the micro-scopic states for a short macroscopic time τ to generatex(t0 + τ) = T τ

d x(t0), and (3) obtain the restriction ofthe evolved microscopic state such that X(t0 + τ) =Rx(t0 + τ). In essence, the coarse time-stepper can bedefined as T τ

c = RT τd L. If the detailed time-stepper

1Similar assumptions also underpin the theory of inertial man-ifolds (Mass and Pope 1992), approximate inertial manifolds(Gear et al. 2005), and many singularly perturbed systems thatarise in engineering modeling.

Eq. 1 has an equivalent coarse description Eq. 4, thenindependent of the initial condition provided by the lift-ing operator, the solution should quickly approach theslow manifold for τ � ε, at which point the dynamicsof Eq. 1 should approximate the real dynamics of Eq. 4.

A crucial component is the ability to recast the time-stepper as a fixed point solver for the nonlinear system:

X − �T(X; μ) ≡ G(X; μ) = 0 G : RM → R

M, (5)

where G is continuously differentiable everywhere inR

N . A single integration of Eq. 2 yields

�T(X; μ) = X +∫ T

0F(X(t′); μ) dt′. (6)

The key idea is to be able to evaluate G(X; μ) throughcalls to the time-stepper rather than a closed expressionfor F(X; μ), for then we can apply Newton’s methodto converge to steady states of the system. Doing sorequires, at the kth step, the solution of the linearNewton equation for step sk

GX(Xk; μ)sk = −G(Xk; μ), (7)

where Xk is the current approximate solution andGX is the Jacobian of the system. Since the systemis inevitably large, we employ methods of large-scalecomputational linear algebra. We use the iterative lin-ear solver GMRES belonging to the general class ofKrylov sub-space methods (Saad 2003) that approx-imately solve linear systems of the form Ay = b byminimizing the norm of the residual r = b − Ay. Themain advantage of these methods is that they are al-ways implemented as matrix-free methods, since onlymatrix-vector products, rather than details of the matrixitself, are needed for implementation. For hybrid sim-ulations this implies function evaluations of the coarsetime-stepper and circumventing the need to computethe Jacobian GX . Moreover, such methods also performbest if the eigenvalues of GX are in a few tight clusters,which is an essential assumption in formulating thecoarse time-stepper.

The stability of a macroscopic steady state X∗ canbe determined from the eigenvalues (σi, i = 1, . . . , N)of the unavailable Jacobian FX(X∗; μ). Note that theJacobian GX(X∗; μ) also has N eigenvalues that can beexpressed as νi = 1 − eσiT . For a stable steady state, thevalues of σi must all lie in the left half of the complexplane. Hence, the stability criterion is |1 − νi| < 1.

To trace out solution branches, i.e, to continue solu-tions, we add an additional arc-length parameter s and

852 Rheol Acta (2012) 51:849–866

rewrite the augmented equation set H(X(s), μ(s), s) asfollows:

H =⎛⎝ G(X(s), μ(s))

dXds

· (X − X j) + dμ

ds(μ − μ j) − (s − s j)

⎞⎠ , (8)

where (X j, μ j) represents a solution previously calcu-lated during continuation at arc length s j.2 Note thatthe Newton step in the implementation is never takenin full, and a sufficient decrease in the residual is appro-priate when computing the Newton step using GMRES.

The hybrid codes are then wrapped in a Newton-GMRES solver. Calculations were performed usingpublic domain legacy codes for Newton’s method thatuse Krylov sub-space iterations. Specifically, medium-sized problems with n ∼ O(100), were solved on MAT-LAB by using the freely available code nsoli.mby Kelley (2003). In contrast, larger problems imple-mented in FORTRAN were solved using the SNESsolver in the PETSc library (Balay et al. 2001).

Confined, sheared liquid crystalline suspensions

Flow-induced alignment of Brownian suspensions com-prised of high aspect ratio, rod-like macromolecules oc-cur in the manufacturing of liquid crystalline polymers.For dilute, unbounded, homogeneous suspensions (inthe absence of external fields), these idealized rods areuniformly dispersed in the solvent with a disordered,and isotropic equilibrium steady state. Several effectscan lead to alignment of these rods, including entropyat higher concentrations, presence of flow, competi-tion between the imposed flow, and the excluded vol-ume driven tendency to align and molecular diffusivity.These effects influence both the transient behavior andthe ultimate alignment, if any, exhibited by the system.We probe this behavior by identifying a slow manifoldand then exploit the dynamics in the manifold to clas-sify solutions.

The model

We consider a homogeneous rigid rod solution con-fined to a thin slit and forced to move in a two-dimensional distinguished plane identified with the z =0 surface. The rigid rods are of length L, width d, with

2Assuming that the long-term dynamics of the time-stepper aredictated by p slow moments, where p � M, it can be shownthat the GMRES iteration converges in at most p + 1 iterations,whereas for continuation, the dimension of the slow sub-space ofmoments increases from p to at most p + 2 (Kelley et al. 2004).

the aspect ratio r ≡ L/d � 1. The imposed velocityfield is v = � y ex with � being the shear rate. We writethe orientation of a fore-aft symmetric test rod in termsof the polar angle θ ∈ (0, π) as u ≡ cos θ ex + sin θ ey.Assuming constant rotational diffusivity Dr, the distri-bution function, f (u, t) satisfies in the mean-field limit

∂t f + ∂u · (ut f ) = Dr∂u · (∂u f + f∂uVev) (9)

with Vev being the excluded volume potential measuredin units of kb T and ut the rate at which u changes due tothe shear flow. This system can be rewritten as follows:

∂τ f + P �s[ f ] = ∂2θ f + U�ev[ f ], (10)

where the flow term

�s[ f ] = − f sin 2θ + cos2 θ(∂θ f ) − (∂θ f ),

and the excluded volume term chosen to be of theMaier–Saupe form (Forest et al. 2004) is

�ev[ f ] = 2 cos 2θ (−α1∂θ f + 2 fα2)

+ 2 sin 2θ (α2∂θ f + 2 fα1).

In Eq. 10, τ ≡ tDr is a scaled time, α1 ≡ 〈sin 2θ〉, α2 ≡〈cos 2θ〉 and P ≡ �/Dr is a dimensionless shear ratesimilar to the classical Peclet number used in fluid me-chanics. The scaled potential strength U is proportionalto the concentration of rods, c, and the excluded areafor two rods. We further define the structure tensor,S ≡ 〈uu〉 − (1/2)δ that yields the scalar order parameter- S ≡ (α2

1 + α22)

1/2, and the angle the director makeswith ex, α = (1/2) tan−1 (α1/α2). Guided by the fore-aftgeometric symmetry and periodicity, we use the Fourierexpansion,

f = fo +M∑1

am(τ ) cos (2mθ) +M∑1

b m(τ ) sin (2mθ),

with the constant, isotropic solution fo = π−1. It isseen readily that α1 = b 1π/2 and α2 = a1π/2. Takingsuitable inner products of Eq. 10 yields a set of ordinarydifferential equations for the Fourier coefficients ai andbi. Expressed as a single system, this reads

dτ x = f (x, μα, μ∗), (11)

where x = {am, b m}, μα denotes the set of parametersheld constant, and μ∗ is the distinguished parameterthat is varied. Our aim is, thus, to compute stableand unstable steady solutions of Eq. 11, x∗ satisfy-ing f (x∗, μα, μ∗) = 0 over the range of interest, μ∗ =[μ∗

0, μ∗f ].

Rheol Acta (2012) 51:849–866 853

Computational analysis

Since the dynamical equations of the model are knownexplicitly, we can use a standard bifurcation packagelike AUTO with an explicit or numerical Jacobianof Eq. 11. Instead, we formulate a nonlinear systemsimilar to Eq. 5 by integrating the dynamical systemof Eq. 11 as a black box computational routine overshort time horizons. It is important to note that we donot build a coarse time-stepper for our model system.Instead, we use time-stepping of Eq. 11 to identify thecoarse or slow manifold on which the dynamical sys-tem closes. Time-stepping function calls to the modelsystem provide information about the action of theassumed unknown Jacobian on the Newton solutionstep, i.e., a matrix-vector product (see Eq. 7). Thisinformation is used by the GMRES method to identifya sequence of dominant eigendirections or eigenvectorsfor the model. Sub-space convergence then indicatesthat a slow manifold of the form of Eq. 2 exists forthe model system. For the model problem, we assumea time horizon of T = 0.05 and expansion of M = 10to trace out the continuous solution branch. We usea modified predictor–corrector continuation algorithm,where corrector iterations were replaced with Newton-

GMRES iterations. Absolute and relative tolerancesof 10−6 were chosen to terminate both Newton andpredictor iterations.

We test the computational model near the isotropic-nematic critical point of U = 2 by comparing the num-ber of GMRES iterations needed for convergence atthe last Newton step (see Fig. 1 (II)). For a dimension-less shear rate, P ∈ [0.01, 1], the system converges withless than eight iterations (for a system of 20 ordinarydifferential equations (ODEs) for Eq. 11). Similar re-sults are obtained for model with a Fourier expansionfor M = 20 (system of 40 ODEs for Eq. 11). The eigen-values and stability for each computed solution wereapproximated using Ritz values (eigenvalues for thesub-space) returned by the Arnoldi procedure as imple-mented within Newton-GMRES. Since Ritz values onlyapproximate the leading eigenvalues, the remainingeigenvalues were computed by explicitly increasing thesize of the sub-space.

Results

We first touch on the characteristics of the base statesgiven by the limiting cases P = 0 and U = 0 (Fig. 1).When P = 0 and U < 2, the isotropic solution is the

Fig. 1 Accuracy andconvergence tests. I a &b Comparison with knownresults for limiting cases.a Equilibrium bifurcationdiagram for dimensionlessshear rate P = 0. The solidcurve is the uniaxial nematicbranch. b Variation of thestructure (order) parameter Sthat characterizes the degreeof alignment with thedimensionless shear rate P inthe absence of anintermolecular potential.Associated distributionfunctions are shown in theinset. II Number of functionevaluations at the lastNewton step of thetime-stepper-basedNewton-GMREScomputation as function ofdimensionless shear rate, P atdimensionless potentialstrength, U = 1.9. Functionevaluations correspond to thenumber of GMRES iterationsneeded for convergence ofthe linear system, Eq. 7.Similar results are obtainedfor M = 20

0 1 2 3 4 5 6 7

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 π/4 /2 /20.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

6.9805 3.3579 0.16636

3π ππ

0 /4 /2 3 /20

0.4

0.8

1.2

1.6

2

f(θ)

U = 6.9966U = 3.6455U = 2.0054

ππππ

0 0.5 1 1.5 2 2.5 3 3.5 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Potential U

(a) P = 0

θf (θ)

θ

Stru

ctur

e fa

ctor

Stru

ctur

e fa

ctor

(b) U = 0

Shear P

10−2 10−1 1000

1

2

3

4

5

6

7

8

9

10

Dimensionless shear rate

Num

ber

of fu

nctio

n ev

alua

tions

U = 1.9

(I)

(II)

854 Rheol Acta (2012) 51:849–866

only equilibrium solution and is stable. At U = UIN =2, an aligned nematic solution branch bifurcates fromthe isotropic solution via a supercritical pitchfork bi-furcation. Since there is no intrinsically preferred di-rection, rotational symmetry implies that the solution isuniaxial3 about the director, n. We define cos φ = u · nand write the canonical solution corresponding to α1 =0 that can then be used to generate all other solutions(Anwar 2008)

f0(φ) = exp (Uα2 cos (2φ))∫ π

0 exp (Uα2 cos (2φ′)) dφ′ (12)

with

α2 = SN0 =

∫ π

0 cos (2θ) exp (Uα2 cos (2θ)) dθ∫ π

0 exp (Uα2 cos (2θ)) dθ. (13)

The pitchfork bifurcation is mathematically transparentupon asymptotic expansion for 0 < SN

0 � 1,

(U − 2) ≈ (SN

0

)2 + 5

6

(SN

0

)4 + 19

24

(SN

0

)6 + 143

180

(SN

0

)8.

(14)

Note that for P = 0, the evolution of modes startingwith an initial state with b k(τ = 0) = 0 results in b k(τ >

0) = 0, suggesting that in the absence of shear, solu-tions with modes b k≥1 = 0 form an invariant sub-space(Gopinath et al. 2004, 2006). This result can be usedto find scaling relationships of these coefficients withk and U . The base nematic state can be written in termsof the set (aN

k , 0) for k ≥ 1, where N denotes nematic.We set b k = 0 and seek a steady nematic solution toobtain equations for the first two modes:

−4aN1 + πU(aN

1 (2π−1 − aN2 )) = 0, aN

2 = 2

π

(1 − 2

U

)

with the value of aN1 to be obtained Eq. 13. Subsequent

modes satisfy the recursive relationship

4k(πUaN1 )−1aN

k = (aNk−1 − aN

k+1)

thus implying that as U → ∞, aN1 , aN

2 , aN3 and aN

4 alltend to the value corresponding to perfect alignment,namely 2/π . When U = 0, only one stable stationary

3Consider the director rotated by a small angle ϕ so that n goesto n + ϕ × n. Since f0 = f0(u · n) and f R

0 = f0(u · (n + ϕ × n))

both satisfy Eq. (9), it follows that at linear order, (ϕ · (n ×∂ f0∂n )) = 0, being the linearized operator corresponding to the

right side of Eq. 9. Steady solutions, thus, have a neutral eigen-value λR = 0, characterizing the soft mode (rotation of director)with corresponding eigenvector ϕ · (n × ∂ f0/∂n). In terms of theFourier coefficients, setting all the bk = 0 for k ≥ 1 is equivalentto pinning the director and removing this rotational degeneracy.

aligned solution branch is found, S increases with in-creasing shear rate, and the director orientation ap-proaches the shearing direction ex.

Weak shear 0 < P � 1 unravels the bifurcation, andthe isotropic branches on either side of U = 2 areperturbed as shown in Fig. 2. We observe two so-called paranematic states, a steady, stable branch for0 < U � UIN and a steady, weakly aligned but unstablebranch for U � UIN. The computationally evaluatedstability of these states are confirmed by a classicalnormal mode, small amplitude linear stability analysis(Anwar 2008). A close up of the region around UIN =2 for non-zero values of P suggests that shear flowopens up a gap close to the critical point in which nopersistent perturbations of the isotropic state exist. Thisobservation is consistent with the analytically obtainedexpression for the structure factor characterizing theparanematic states

S ≈ P4

1

|U − 2|(

1 + P2

8

1

|U − 2|2)

+ O(P3). (15)

An important feature of the solution branches evi-dent in Fig. 2a is the loss of steady solutions at a limitpoint U = UL(P) that depends on the precise value ofP . As P → 0, the two solution curves (ascending anddescending) on either side of the limit point superim-pose and become indistinguishable.

To understand what happens at the limit point, wecomputed the eigenspectrum in its vicinity, an exampleof which is depicted in Fig. 2c, d. Evaluation of suchspectra for various U and fixed but small P shows thata single eigenvalue crossing the imaginary axis and goesfrom a negative to a positive value at the limit point. Be-yond the limit point, for U > UL(P), the only solutionsare periodic, non-sinusoidal solutions. These tumblingsolutions (Anwar 2008) correspond to the director con-tinuously rotating in the x-y plane and are characterizedby oscillatory structure factors, S with an O(SN

0 (U))

mean and O(P) periodic variations. Careful compu-tations reveal that the period Tp = ω−1

T ∼ O(P−1) ata fixed concentration. At the same time, when P iskept fixed, and U → [UL(P)]+, the periods become in-creasingly larger. The eigenspectrum is consistent withan infinite period bifurcation (Strogatz 2000). This wasconfirmed by analyzing the amplitude and frequency ofthe time-periodic states as a function of the distancefrom criticality (U − UL) for various small P . As thecritical point is approached, the frequency is found todecrease as |U − UL|ρ with ρ ≈ 0.5.

This result is a consequence of the restricted twodimensionality of the system. It is important to notethat allowing the rods to move out of plane yields a

Rheol Acta (2012) 51:849–866 855

Fig. 2 The effect of weakshear flow P > 0 on theequilibrium bifurcation mapfor various values of thepotential U . Critical pointsare tracked by followingchanges in the orderparameter, S. a For shear rateP � 1, the pitchforkbifurcation at potential U = 2unravels to yield two steadybranches, one stable and oneunstable, that meet atU = UL(P). For U > UL(P)

stable but periodic solutionsexist with O(1) amplitude.b For shear rate 0 < P � 1,the area around the criticalpoint UIN exhibits a gap.c The first few leadingeigenvalues with real partsclosest to zero evaluated forP = 10−2 and U = 2.4.d Close-up of the dominanteigenvalues depicted in c

Real (λ)−400 −300 −200 −100 0

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

Im(λ

)

(c)

−40 −20 0 20 40

−0.04

−0.02

0

0.02

0.04

0.06

Real (λ)

Im(λ

)

(d)

40 0.5 1 1.5 2 2.5 3 3.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

10-2 10-1 10-1 10-2

(a)

1.7 1.8 1.9 2 2.1 2.2 2.3 2.40

0.05

0.1

0.15

0.2

0.25

0.3

0.35

(b)

U = 2 .4114

UL= 0

S

U

= 10 − 2

U

S

homoclinic bifurcation instead (Nayak 1998) andmodifies the stability of these tumbling solutions.

For 0 � P < 1, computations reveal that two criticalpoints emerge in the bifurcation map in addition tothe saddle point obtained for small shear. One of thesecritical points represents a saddle point, while the otherrepresents a Hopf bifurcation point, as indicated by acircle and square, respectively. The new critical pointsemerge between 0.46 < P < 0.48 and are connected bya branch of stable solutions. The size of the branchincreases with increasing P until P = 0.8, when thesaddle point collides with the saddle point from smallshear leaving the Hopf bifurcation point. In Fig. 3a,we observe that for P = 0.6, the critical points areat U = 2.4442, 2.4244, and 2.4259. The eigenspectrumin the vicinity of these points is shown in Fig. 3b atU = 2.4441, 2.4245, and 2.4252, respectively. It canbe observed that only a single eigenvalue is close tocrossing the imaginary axis for the first and secondcritical point, while a pair of eigenvalues is close tocrossing the imaginary axis for U = 2.4252. In contrastwith small and moderate shear, the bifurcation diagramfor P ∼ O(1) in Fig. 3c shows branches of stable andunstable stationary solution connected by a Hopf bi-furcation point. Since a Hopf bifurcation leads to si-nusoidal time-periodic solutions, this is consistent withthe observation of wagging solutions for large shearrates. This bifurcation point marks the transition from

a flow-aligning state to a wagging state for the nematicdirector.

Our results match numerical results by Maffettoneand Crescitelli (1995) and (Zhou and Wang 2007) thatwere obtained using the package AUTO, with explicitcalculation of the Jacobian of the ODE set. The accu-racies of the two approaches are comparable.

Draw resonance in isothermal fiber spinning

The fiber-spinning process involves drawing a moltenpolymer as a liquid fiber following extrusion from anorifice to form continuous synthetic fibers. Small di-ameters of the order of 100 microns and smaller areachieved by stretching the molten polymer before so-lidifying in air or in a quench bath and wounding ona spool (Osswald and Hernández-Ortiz 2006). A majorconcern during fiber spinning is the instability known asdraw resonance that manifests as periodic fluctuationsin the diameter of the drawn fiber and an accompanyingoscillation in the tension at the take-up point. Kaseand Matsuo (1965) and Matovich and Pearson (1969)were the first to study the onset of draw resonancein inelastic fluids. Further progress was made by thework of Fisher and Denn (1976) who analyzed the spin-ning of a shear-thinning White–Metzner fluid to under-stand the data of Ishihara and Kase (1975). Alternate

856 Rheol Acta (2012) 51:849–866

0.55¬ 2.4386,0.53802,0.55

¬ 2.4071,0.42163P = 0.55

¬ 2.4077,0.40975P = 0.55

¬ 2.4319,0.54148,0.48

¬ 2.3802,0.40105, P = 0.48¬ 2.3803,0.39827, P = 0.48

¬ 2.4301,0.5426,0.46

¬ 2.4442,0.53423P = 0.6

¬ 2.4244,0.43742P = 0.6

¬ 2.4259,0.41714P = 0.6

¬ 2.4577,0.52228P = 0.7

¬ 2.4543,0.47058P = 0.7

¬ 2.4594,0.4303P = 0.7

¬ 2.4898,0.44096P = 0.8

0.50

0.45

0.40

Str

uctu

re fa

ctor

, S

Dimensionless potential, U

2.38 2.40 2.42 2.50 2.52

0

0

5

10U = 2.4252

Real part

0

0

5

10U = 2.4441

Real part

Imag

inar

y

par

t

0 100

0

5

10U = 2.4245

Real part

Imag

inar

y

par

tIm

agin

ary

p

art

0

2U = 2.4441

Real part

Imag

inar

y

par

t

0

0

2 U = 2.4245

Real part

Imag

inar

y

par

t

0

0

1U = 2.4252

Real part

Imag

inar

y

par

t

0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

2.5433, 0.45844 2.7331, 0.50779

3.1432, 0.56298 3.7886, 0.60997

U

S

2.482.462.44

(b)(c)

(a)

Fig. 3 a Steady solutions and collision of critical points seen formoderately strong shear P . As before we track the order parame-ter, S as the potential U varies for different P . The open circlesdenote saddle node bifurcations, while f illed squares denoteHopf-Poincare’ bifurcations. b Leading eigenvalues evaluated atthe critical points and close-up of the ones crossing zero for shear

rate P = 0.6. At the turning point, a single eigenvalue crosseszero, while at the Hopf point, we have a complex conjugate paircrossing zero. c Steady branches for P > 1 (values shown) shownalong with the P = 0 curve. Only one critical point is seen on thebranches at which periodic solutions bifurcate

constitutive equations have also been used, the mul-timode Phan-Thien–Tanner model (Phan-Thien 1978)and the Oldroyd-B model of Gupta et al. (1986) beingtwo examples.

In the following, we aim to obtain the onset ofdraw resonance and the transition to second region ofstability at high draw ratios for an Oldroyd-B polymeric

fluid. In particular, the fiber-spinning process for thisparticular constitutive fluid is modeled using a hybridsimulation that couples the conservation of mass andmomentum in the isothermal fiber-spinning model toa stochastic simulation for the Brownian configurationfields of the Hookean dumbbell model from polymerkinetic theory.

Rheol Acta (2012) 51:849–866 857

The model

A schematic diagram of the fiber-spinning process isshown in Fig. 4, where the origin of the cylindrical co-ordinate system is placed just downstream of the pointof maximum extrudate swell. This position correspondsto a point where the velocity and stress profiles areassumed to be independent of the radial position. Tofurther simplify analysis, it is assumed that the fiberis isothermal and that no shear or normal stresses acton the boundary of the fiber. To develop the spinningequations, we now consider a differential element ofthe fiber, as shown in Fig. 4b with local radius R(z, t),unit outward normal vector n, and total stress tensorπ . The axi-symmetric fiber is drawn along the axial zdirection with the cross section being A(z, t) at any po-sition z. The drawing length is L, measured from a po-sition beyond the maximum extrudate swell. The originof the cylindrical coordinate system coincides with z =0, thus for 0 ≤ z ≤ L, the velocity and stress profilesare assumed to be independent of r. For isothermalspinning of a very slender fiber in air, one can write

∂ A∂t

= −vz∂ A∂z

− A∂vz

∂z, G − ∂vz

∂z= 0, (16)

and

∂ A∂z

[−3

∂vz

∂z+ 3(1 − β)G + (τp,zz − τp,rr)

]

+ A[−3

∂2vz

∂z2+ 3(1 − β)

∂G∂z

+ ∂

∂z(τp,zz − τp,rr)

]

= 0. (17)

Variables z, vz, t, A, and stress have been made di-mensionless with respect to the length L, the initialvelocity v0 at z = 0, time scale L/v0, initial area, andη0v0/L, where η0 ≡ ηs + ηp and ηs and ηp are the sol-vent and polymer contributions to the zero shear rateviscosities, respectively. Finally, τp,zz and τp,rr are thepolymer contributions to the stress components τzz andτrr, respectively, and β ≡ ηs/η0.

To complete the equations set, we specify the poly-mer contribution to the stress. We consider bothclosed (mesoscopic) and stochastic formulations. First,we consider the closed form Oldroyd-B constitutiveequation

τ + λ1τ (1) = −η0

[γ̇ + +λ2

∂ γ̇

∂t

], (18)

Fig. 4 a Schematic of thefiber-spinning process.b Definition sketch

858 Rheol Acta (2012) 51:849–866

where λ1 and λ2 are the relaxation and retardationtimes and τ (1) is the upper convected derivative of τ . Bydecomposing the stress tensor into a Newtonian solventcontribution and polymer contribution and defining theDeborah number De ≡ λ1v0/L, we obtain[De

(∂

∂t+ vz

∂

∂z

)+ 1

]τp,zz

= −2G[(1 − β) − De τp,zz

], (19)

[De

(∂

∂t+ vz

∂

∂z

)+ 1

]τp,rr

= G[(1 − β)G − De τp,rr

](20)

Equations 19 and 20 constitute hyperbolic, partialdifferential equations that must be solved for appropri-ate boundary conditions. This is our time-stepper forthe Oldroyd-B model that is called for guided short-time horizon bursts. The kinetic theory equivalent tothe Oldroyd-B model is the Hookean dumbbell. Themolecular configuration of the Hookean dumbbell isrepresented by continuous, Eulerian fields that evolveaccording to the following stochastic differential equa-tions (i = 1, . . . , NF)4

dQz,i =[−vz

∂ Qz,i

∂z+ GQz,i − 1

2DeQz,i

]dt

+√

dtDe

dWz,i (21)

dQr,i =[−vz

∂ Qr,i

∂z− G

2Qr,i − 1

2DeQr,i

]dt

+√

dtDe

dWr,i. (22)

Numerical procedure

Given an initial condition for A, τp,zz, and τp,rr, wefirst solve for vz and G, followed by an update ofA, τp,zz, and τp,rr over one time step. The process isthen repeated for a specified time horizon. The poly-mer contribution to the stress tensor is obtained eitherthrough closed equations for the Oldroyd-B model or acoarse time-stepper for the Hookean dumbbell modelto evolve the molecular conformation. Equations aresolved by dividing the computational domain z ∈ [0, 1]

4Configuration variables Qz,i and Qr,i are functions of bothspace and time. The standard Wiener processes Wz,i and Wr,i arefunctions of time only.

into N elements and time domain t ∈ [0, TF] into NT

steps.The finite element method is used to discretize mass

and momentum balances and the equation for G. Con-tinuous, quadratic basis functions φi are used for A andvz, and continuous linear basis functions ψi are used forG. The polymer contributions to the stress tensor, τp,zz

and τp,rr, and configuration fields, Qz, i and Qr, i, arediscretized in space using the discontinuous Galerkinmethod with local linear discontinuous basis functions(Anwar 2008).

For a given velocity field vz and corresponding ve-locity gradient G, Eqs. 21 and 22 can be used to evolvean ensemble of NF configuration fields in time over theentire computational domain. In a standard transientsimulation, these fields are initialized from the equi-librium distribution of Hookean dumbbells, with thesame value for each field over the entire computationaldomain. The polymer contribution to the stress tensoris then obtained at the end of the simulation by usingKramers expression for the stress tensor.

It is necessary to be able to initialize the con-figuration fields such that they are consistent with aspecified initial macroscopic field for τp,zz and τp,rr.Such an initialization is then followed by evolutionof the configuration fields for a short time horizon,and subsequent averaging to obtain the updated stressfields. A crucial step in the above procedure is thelifting step for constructing an initial ensemble ofconfiguration fields consistent with the macroscopicstress. This is accomplished by assuming that all fieldsare equivalent, that is, they follow the same Weinerprocess but vary in space to be consistent with thespatial profile of the macroscopic stress. For the firstlifting step (corresponding to the initial guess of thesolution), each configuration field is given by

Qz,i =√

1 − De1 − β

τp,zz,

Qr,i =√

1 − De1 − β

τp,rr. (23)

This lifting constitutes a modified Kramers expressionwhere all configuration fields are identical (i.e., a Diracdelta function as the distribution). For all subsequentlifting steps, configuration fields are sampled from aGaussian, with mean equal to the macroscopic stressand standard deviation from the last iteration.

Reduction of statistical error is accomplished in stan-dard fashion by calculating the evolution of an ensem-ble of configuration fields, Q̂z,i and Q̂r,i at quiescentconditions that have the same initial condition and are

Rheol Acta (2012) 51:849–866 859

subjected to the same random noise as the originalconfiguration field variables Qz,i and Qr,i. The expres-sion then used to obtain the macroscopic stress field isgiven by

τp,zz = 1 − β

De1

N f

N f∑i=1

(Q̂z,i Q̂z,i − Qz,i Qz,i

),

τp,rr = 1 − β

De1

N f

N f∑i=1

(Q̂r,i Q̂r,i − Qr,i Qr,i

). (24)

A brief description of the implementation can befound in Appendix of this paper. For more details ofthe numerical discretization and implementation of theweak Galerkin procedure, the reader is referred to thethesis by Anwar (2008).

Boundary conditions

The transient simulation of the fiber-spinning processis carried out by decoupling the solution of the momen-tum equation from the update of the cross-sectionalarea of the fiber and the polymer contribution to thestress tensor.

The boundary conditions at z = 0 are unclear a prioridue to the unknown behavior of the upstream flow,including the die swell. It is customary5 to specify thevelocity and area at z = 0, and, hence in this work, weuse vz(0) = 1 and A(0) = 1, which sets the flow rate.In principle, these boundary conditions should only bevalid at steady state, but following a previous work byFisher and Denn (1976), we assume that these hold atall times for both transient and steady fiber spinning.The downstream boundary conditions are easier tospecify, as the velocity at the take-up device can befixed. Here, it is assumed that the fiber is quencheddirectly onto the take-up roll at z = 1. By defining thedraw ratio DR as the ratio of the take-up velocity to theinitial velocity, the boundary condition for the velocityis then vz(1) = DR.

We next specify boundary conditions for the polymercontributions to the stress tensor. As such, both τp,zz

and τp,rr or their difference cannot be specified at z = 0since that would fix the tension in the fiber. For a fixedflow rate, the take-up velocity and force cannot both bespecified; for a fixed take-up velocity, we must acceptwhatever tension is necessary to draw the fiber. It istherefore customary to set the ratio τrr/τzz at z = 0.While this ratio is −0.5 over the entire filament for a

5In an experimental situation, the flow rate is generally given butthe initial area is unknown because of the extrudate swell anduncertainty as to the precise location of the coordinate origin.

Newtonian fluid, for a viscoelastic fluid, it depends onthe tension in the fiber.6

At the boundary z = 0, we evolve the fully devel-oped versions of the equations for the polymer contri-butions to the stress tensor for the Oldroyd-B modeland stochastic differential equations for N f Hookeandumbbell configurations for the inlet velocity gradient.These are given by

De∂τ 0

p, zz

∂t= −2(1 − β)G − τ 0

p, zz + 2DeGτ 0p, zz,

De∂τ 0

p, rr

∂t= (1 − β)G − τ 0

p, rr − DeGτ 0p, rr (25)

and

dQ0z, i =

[GQ0

z, i − 1

2DeQ0

z, i

]dt +

√dtDe

dWz, i,

dQ0r, i =

[−G

2Q0

r, i − 1

2DeQ0

r, i

]dt +

√dtDe

dWr, i (26)

where the ensemble {Q0z, i} {Q0

r, i} is consistent with thepolymer contributions to the stress tensor, τ 0

p, zz andτ 0

p, rr at z = 0 and subjected to the same Wiener processas {Qz, i} {Qr, i}. The transient state of τ 0

p, zz and τ 0p, rr or

{Q0z, i} {Q0

r, i} is then used as a Dirichlet boundary condi-tion. With this formulation, we find that the ratio τrr/τzz

varies from −0.5 to 0 over the length of the fiber, andapproaches zero with increasing viscoelasticity, that is,decreasing β or increasing De.

Results

We denote the finite element discretization of the fieldvariables as a vector of unknowns

u = [A, vz, G, τp,zz, τp,rr]T

for u ∈ �M, and construct the time-stepper, �O, TF , toevolve the vector u over a specified time horizon TF ata fixed DR. The complementary time-stepper �H, TF isthen constructed for the Hookean dumbbell model andthe associated lifting and restriction procedures. Wenext formulate the nonlinear problems for the Oldroyd-B and Hookean dumbbell models FO(u; DR) = u −�O, TF ,, and FH(u; DR) = u − �H, TF that when solved,yield the steady state, u∗, for a specified draw ratio,

6Finite element calculations of Keunings et al. (1983) have shownthat this ratio monotonically goes from −0.5 to 0 at the pointwhere the stresses become radially uniform, as the tension inthe fiber is increased. Also, it has been shown (see Denn 1974for instance) that the fiber-spinning equations presented here areinsensitive to this stress ratio as long as it is in the range from−0.5 to 0.

860 Rheol Acta (2012) 51:849–866

DR. The main advantage of this approach is that theformulation for the Hookean dumbbell model, FH isconstructed without invoking closure even though aclosed equation may be written for FO.

Initial conditions for a transient simulation or theinitial guesses for obtaining the steady state are chosento correspond to the Newtonian solution

A(z) = e−z ln DR, vz(z) = ez ln DR, G(z) = ln DR ez ln DR,

with τp,zz, and τp,rr set to

τp,zz(z) = −2(1 − β)G(z), and τp,rr(z) = (1 − β)G(z).

The accuracy of the method to identify the right sta-bility parameters is easily checked. The eigenvalues (νi

for i = 1, . . . , M) of the Jacobian ∂FH/∂u|∗u or ∂FO/∂u|∗uare related to the eigenvalues (σi for i = 1, . . . , M) ofthe Jacobian of the closed model, by νi = 1 − eσiTF .We can then compare the eigenvalues νi to the exacteigenvalues σi since we have a closed equation for theOldroyd-B model. Specifically, these can be obtainedby writing

Mdudt

= F(u; DR), M =

⎡⎢⎢⎣

A 0 0 0 00 0 0 0 00 0 0 Tzz 00 0 0 0 Trr

⎤⎥⎥⎦ (27)

such that M, A, Tzz, and Trr are appropriate finite el-ement mass matrices for the mass balance and the con-stitutive equations (Anwar 2008). Linearization yieldsthe generalized eigenvalue problem

Mv = σF∗u

∣∣∣∗uv (28)

where σ is an eigenvalue of the dynamical system and vits corresponding eigenvector. A steady state is then astable solution if �{σi} < 0. The corresponding stability

criterion for the time-stepper formulation, therefore,requires that the eigenvalues νi must lie within a unitdisk centered at (1, 0) in the complex plane.

To emphasize, Newton-GMRES is employed so thatexplicit computation of the Jacobians is never required;only matrix-vector multiplications are needed that canbe performed at low cost by calling the time-steppersm times, where m � M. Since the Newton-GMRESsolver uses the iterative Arnoldi procedure to convergeto stationary states, the Ritz values returned by this pro-cedure provide good estimates to νi. In particular, thesevalues are very good approximations of the leadingeigenvalues of the Jacobian matrices, which are exactlythe eigenvalues of interest in a stability calculation.

In order to test the numerical discretization, thetime-steppers for the Oldroyd-B and Hookean dumb-bell models were evolved for TF = 10.0 with �t = 10−4.A one-dimensional mesh discretized into 20 elementsof equal length was used; for the Hookean dumbbellsimulation, 500 configuration fields were employed.The draw ratio was set to 20; and the Deborah numberand β were set to 0.01 and 0.2, respectively. The spatialprofiles for A, vz, G, τp,zz, and τp,rr are shown in Figs. 5and 6. The solid lines represent the solution obtainedfrom the Oldroyd-B time-stepper, whereas the circlesrepresent the solution for the Hookean dumbbell time-stepper. Since good agreement between the two resultsis seen, subsequent calculations to compute the station-ary states and perform bifurcation analysis were donewith �t = 10−4 and NF = 500.

Given the good agreement between the two time-steppers, �O, TF and �H, TF , the steady state for thefiber-spinning process was then obtained for a range ofβ and De by solving Eq. 27 with a Newton-GMRESsolver at DR =20 and time horizon of TF = 0.05. Thesteady-state velocity profiles thus obtained are shown

Fig. 5 Comparison ofsteady-state spatial profile ofarea, velocity, and velocitygradient for the Oldroyd-Band Hookean dumbbelltime-steppers. The solid linesand circles representOldroyd-B and Hookeandumbbell results,respectively.β, De, DR = 0.2, 0.01, 20

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

20

40

60

Rheol Acta (2012) 51:849–866 861

0 0.2 0.4 0.6 0.8 1

4

6

8

10

12

14

16

18

20

22

24

0 0.2 0.4 0.6 0.8 1

(a)

(b)

Fig. 6 Comparison of steady-state spatial profiles of τp,zz (a),and τp,rr (b) for the Oldroyd-B and Hookean dumbbell time-steppers. The length of the error bars is equal to two standarddeviations of standard error in the stochastic simulation. Thesolid lines and circles represent Oldroyd-B and Hookean dumb-bell results, respectively (β, De, DR) = (0.2, 0.01, 20)

in Fig. 7. A Newtonian initial guess was sufficient toconverge to a solution for β = 0.8 and De = 0.03 andalso for β = 0.2, and De = 0.001. The remaining solu-tions were obtained by using continuation in β or De.

To check consistency of these solutions, steady stateswere also obtained by solving the Oldroyd-B evolver atselected values of β and De. The steady-state profilesfor area, velocity, velocity gradient G, and the polymercontribution to the stress tensor were compared withthose obtained from applying Newton-GMRES to theHookean dumbbell time-stepper. We found excellentagreement between the results of the Oldroyd-B andHookean dumbbell time-steppers. Moreover, the ve-locity profiles were predicted to be in very good agree-ment with the computational results obtained by Guptaet al. (1986).

0 0.2 0.4 0.6 0.8 10

5

10

15

20

0.2=0.4=0.6=0.8=

0 0.2 0.4 0.6 0.8 10

5

10

15

20De=0.005De=0.01De=0.001De=0.02De=0.03

(a)

(b)

Fig. 7 a Steady-state velocity profile obtained from the Newton-GMRES solver and the Hookean dumbbell time-stepper forvarying De at (β, DR) = (0.2, 20). b Steady-state velocity profileobtained from the Newton-GMRES solver and the Hookeandumbbell time-stepper for varying β at (De, DR) = (0.03, 20)

The steady states thus obtained were used to solvethe generalized eigenvalue problem and determinethe eigenspectrum and stability of the steady state.Figure 8a shows that all eigenvalues lie in the left halfof the complex plane at (β, De, DR) = (0.8, 0.001, 20),indicating a linearly stable steady state. These eigen-values can also be used to facilitate comparison withthe approximate Ritz values obtained from the Arnoldiprocedure by mapping them to within a unit circlecentered at (1, 0) in the complex plane. The result-ing eigenspectrum and the Ritz values are plotted inFig. 8b, c for the Oldroyd-B and Hookean dumbbelltime-steppers. In obtaining the Ritz values, the timehorizon for the Oldroyd-B and Hookean time-stepperswas set to TF = 0.05. It can be observed that thereis very good agreement between the Ritz values andthe leading transformed eigenvalues, which are exactlythe eigenvalues of interest in a stability calculation.Since the Ritz values lie within the unit circle, thetime-stepper formulation returns the correct stabilityinformation for the fiber-spinning process.

862 Rheol Acta (2012) 51:849–866

0 0.5 1 1.5 2

−1

−0.5

0

0.5

1

Real part

Imag

inar

y pa

rt

−1500 −1000 −500 0−1500

−1000

−500

0

500

1000

1500

Imag

inar

y pa

rt

Real part

0 0.5 1 1.5 2

−1

−0.5

0

0.5

1

Imag

inar

y pa

rt

Real part

(a)

(b)

(c)

Fig. 8 a Eigenvalues of the dynamical equations for the Oldroyd-B formulation at (β, De, DR) = (0.8, 0.001, 20). b Comparison ofeigenvalues (box) of the dynamical equations and Ritz values (+)for the Oldroyd-B time-stepper. The Ritz values were obtainedwith a time-stepping horizon of TF = 0.05 at (β, De, DR) =(0.8, 0.001, 20). c Comparison of eigenvalues (box) of the dynam-ical equations and Ritz values (+) for the Hookean dumbbelltime-stepper. The Ritz values were obtained with a time-steppinghorizon of TF = 0.05 at (β, De, DR) = (0.8, 0.001, 20)

In this instance, we found that increasing the timehorizon results in increased clustering of the eigen-values, which is beneficial for GMRES iterations.The benefit of increased clustering should be weighedagainst the cost of running the time-stepper over alonger time horizon. For example, with the Hookeandumbbell time-stepper, increasing the time horizon toTF = 0.1 reduces the number of calls to the time-stepper by 10 % when compared with the number of

calls to the time-stepper for TF = 0.05. The doubling ofthe time-stepping horizon is, therefore, not justified forthe modest decrease in number of time-stepper calls,since the agreement between the Ritz values and theleading eigenvalues is equally good for TF = 0.05.

Finally, the transformed eigenvalues and the Ritzvalues for the Hookean dumbbell time-stepper areshown in Fig. 9a for (β, De, DR) = (0.2, 0.01, 20) witha time horizon of TF = 0.05. This case represents in-creased viscoelastic effects and we see that, once again,there is good agreement between the leading eigenval-ues and Ritz values, although not as good as in previouscases. However, in this particular case, increasing thetime horizon to TF = 0.06 results in better agreementas shown in Fig. 9b. We, therefore, use this time horizonfor tracing out solution branches and identifying criticalpoints.

To obtain bifurcation maps, we choose the drawratio as the bifurcating parameter. In particular, weset β = 0.2 and obtain the bifurcation diagrams atthree different values of De to compare with previousstudies. The computed bifurcation diagrams are shownin Fig. 10 for De = 0.001, De = 0.01, and De = 0.011,where we plot the average velocity as a function of draw

0 0.5 1 1.5 2−1

−0.5

0

0.5

1

Imag

inar

y pa

rt

0 0.5 1 1.5 2−1

−0.5

0

0.5

1

Real part

Imag

inar

y pa

rt

Real part

(a)

(b)

Fig. 9 Comparison of eigenvalues (box) of the dynamical equa-tions and Ritz values (+) for the Hookean dumbbell time-stepperat β, De, DR = 0.2, 0.01, 20. a TF = 0.05. b TF = 0.06

Rheol Acta (2012) 51:849–866 863

30

18 20 22 24 26 28 306

6.5

7

7.5

8

8.5

9

9.5

24.99, 8.10

20 30 40 50 60

10

15

20

25

25.81, 8.45

59.17, 22.62

(b) De = 0.01

(c) De = 0.011

18 20 22 24 26 28

6

6.5

7

7.5

8

8.5

21.39, 6.67

(a) De = 0.001

vz

vz

vz

DR

DR

DR

Fig. 10 Bifurcation diagrams for the fiber-spinning process atβ = 0.2, (curves correspond to De = 0.001 (a), De = 0.01 (b),and De = 0.011 (c), respectively) where the draw ratio has beenused as the continuation parameter. The solid and dotted curvesrepresent stable and unstable steady states, respectively. Thesolid squares mark Hopf bifurcations

ratio. It can be observed that the transition from stableto unstable steady states occurs through a Hopf bifur-cation, where a pair of complex conjugate eigenvalueswith non-zero imaginary part cross the imaginary axisfor the dynamical problem. In the case of the time-stepper formulation, this corresponds to a pair of com-plex conjugate eigenvalues leaving the unit circle.

As De is increased from 0.001 to 0.011, the tran-sition from stable to unstable steady states occurs atincreasing value of the draw ratio. This is in agreement

with previous studies with the Maxwell model (Denn1974) which corresponds to β = 0. Additionally, thecritical draw ratio at De = 0.001 is greater than theNewtonian fluid critical draw ratio of 20.21, which isalso in agreement with the linear stability analysis fora Maxwell fluid. Most importantly, however, we findthat at De = 0.011 and a high draw ratio of 59.17, thefiber-spinning process regains stability to infinitesimaldisturbances. Similar results were obtained by Fisherand Denn (1976) in their analysis of the draw resonancewith the White–Metzner fluid. In particular, the criticaldraw ratios of 25.81 and 59.17 are in very good agree-ment with their results.

Conclusion

We presented a scheme that combines traditional New-ton approaches with time integration schemes to read-ily perform system-level analysis of multi-scale, hybrid(possibly black box type) simulations. Transient simu-lations recently carried out by Koppol et al. (2007) foran axi-symmetric contraction–expansion flow and flowcalculations with a stochastic description for Hookeandumbbells (Ramírez and Laso 2004) suggest that de-tailed, large-scale, accurate hybrid simulations are fea-sible. A natural extension of our work would, thus, beto study flows with larger degrees of freedom and morecomplex geometries. For instance, as a first step, onecan consider the flow of FENE chains with up to sixbeads and contrast it to properties of Kramers chainmodels.

As a word of caution, we note that the choice of thetime horizon should be guided by the physics of theproblem. An efficient scheme will ensure that the num-ber of function calls to the time-stepper or time horizonis not increased considerably. For calculations we pre-sented, typically one or two additional function evalua-tions of the time-stepper were needed when comparedwith the calculation of the same steady state without thearc-length equation. This is not true generally; an exam-ple being tracing out the loci of periodic solutions thatemanate from a infinite period bifurcation point. Sincecomputation of a periodic solution requires adding aphase constraint to the original nonlinear system inorder to find the unknown period, this will necessitateevolving the time-stepper over large time horizons.

Comparison of results from stochastic and deter-ministic approaches to the draw resonance problemdemonstrate both the utility as well as the accuracyof our methodology. Our analysis, however, is not in-tended to compare the benefits of either approach; wemerely state that the lack of closed form equations,

864 Rheol Acta (2012) 51:849–866

which are amenable to a traditional Jacobian basedstudy, is not a barrier to systems level analysis. Theseparation of time scales demonstrated by many physi-cally relevant complex fluid flows allows for reductionin the number of degrees of freedom or dynamicalmodes that need to be followed for long-time dynamicsof the problem. Our scheme provides a simple andcomputationally easy way to automatically determinethis reduced set of dynamical modes so that long-timedynamics are accessible by observing the behavior ofthe system in this slow manifold. The next step wouldbe to then extend this to other physically relevantproblems in a way that is also computationally efficient.

Appendix: Numerical discretization

The partial differential equations for the fiber-spinningprocess are solved by discretizing the computationaldomain z ∈ [0, 1] into N elements and time domaint ∈ [0, T f ] in NT steps. We then use the finite elementmethod to discretize spatially the mass and momen-tum balances and the equation for the velocity gra-dient G. Continuous, quadratic basis functions φi areused for A and vz, whereas continuous linear basisfunctions ψi are used for G. In contrast, the polymercontributions to the stress tensor, τp, zz and τp, rr, andconfiguration fields, Qz, i and Qr, i, are discretized inspace by using the discontinuous Galerkin method withlocal linear discontinuous basis functions ψDG, l. Fortime discretization, all terms are treated explicitly toyield the following algorithm.

1. Given Ak, τ kp, zz, τ k

p, rr, and the initial guess vkz and

Gk, update the velocity and velocity gradient bysolving the following nonlinear problem for theweak formulation

∫ 1

0

(∂ Ak

∂z

[−3

∂vk+1z

∂z+ 3(1 − β)Gk+1

+ (τ kp, zz − τ k

p, rr)

]

+ Ak

[−3

∂2vk+1z

∂z2+ 3(1 − β)

∂Gk+1

∂z

+ ∂

∂z

[τ k

p, zz − τ kp, rr

]])φi dz = 0

∫ 1

0

(Gk+1 − ∂vk+1

z

∂z

)ψi dz = 0 (29)

2. Given the new velocity field update the area asfollows∫ 1

0Ak+1φi dz

=∫ 1

0Akφi dz−�t

∫ 1

0

(vk+1

z∂ Ak

∂z+ Ak ∂vk+1

z

∂z

)φi dz

(30)

3. Update the polymer contributions to the stress ten-sor for the Oldroyd-B model by using the followingweak formulations over each element [z j, z j+1] ofthe domain∫ z j+1

z j

Deτ k+1p, zzψDG, l dz

=∫ z j+1

z j

Deτ kp, zzψDG, l dz

+ �t∫ z j+1

z j

(−2(1 − β)Gk+1 − τ k

p, zz

+ 2DeGk+1τ kp, zz

− vk+1z

∂τ kp, zz

∂z

)ψDG, l dz + Z1

×

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

For vk+1z

∣∣∣z j

> 0

Z1 = �t De(ψDG, lv

k+1z (τ k,ex

p, zz − τ k,inp, zz)

)∣∣∣z j

For vk+1z

∣∣∣z j+1

< 0

Z1 = −�t De(ψDG, lv

k+1z (τ k,ex

p, zz − τ k,inp, zz)

)∣∣∣z j+1

(31)

∫ z j+1

z j

Deτ k+1p, rrψDG, l dz

=∫ z j+1

z j

Deτ kp, rrψDG, l dz

+�t∫ z j+1

z j

((1 − β)Gk+1 − τ k

p, rr − DeGk+1τ kp, rr

− vk+1z

∂τ kp, rr

∂z

)ψDG, l dz + Z2

Rheol Acta (2012) 51:849–866 865

×

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

For vk+1z

∣∣∣z j

> 0

Z2 = �t De(ψDG, lv

k+1z (τ k,ex

p, rr − τ k,inp, rr)

)∣∣∣z j

For vk+1z

∣∣∣z j+1

< 0

Z2 = −�t De(ψDG, lv

k+1z (τ k,ex

p, rr − τ k,inp, rr)

)∣∣∣z j+1

(32)

For the Hookean dumbbell model, instead, we liftτ k

p, zz and τ kp, rr to a consistent ensemble of molecular

configurations {Qkz, i, Qk

r, i} that are then evolvedusing the following weak formulations∫ z j+1

z j

Qk+1z, i ψDG, l dz

=∫ z j+1

z j

Qkz, iψDG, l dz

+ �t∫ z j+1

z j

[−vk+1

z

∂ Qkz, i

∂z+ Gk+1 Qk

z, i

− 1

2DeQk

z, i

]ψDG, l dz

+�Wz, i

∫ z j+1

z j

ψDG, l dz + Z3

×

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

For vk+1z

∣∣∣z j

> 0

Z3 = �t(ψDG, lv

k+1z (Qk,ex

z, i − Qk,inz, i )

)∣∣∣z j

For vk+1z

∣∣∣z j+1

< 0

Z3 = −�t(ψDG, lv

k+1z (Qk,ex

z, i − Qk,inz, i )

)∣∣∣z j+1

(33)

∫ z j+1

z j

Qk+1r, i ψDG, l dz

=∫ z j+1

z j

Qkr, iψDG, l dz

+�t∫ z j+1

z j

[−vk+1

z

∂ Qkr, i

∂z− Gk+1

2Qk

r, i

− 1

2DeQk

r, i

]ψDG, l dz

+�Wr, i

∫ z j+1

z j

ψDG, l dz + Z4

×

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

For vk+1z

∣∣∣z j

> 0

Z4 = �t(ψDG, lv

k+1z (Qk,ex

r, i − Qk,inr, i )

)∣∣∣z j

For vk+1z

∣∣∣z j+1

< 0

Z4 = −�t(ψDG, lv

k+1z (Qk,ex

r, i − Qk,inr, i )

)∣∣∣z j+1

(34)

For the Hookean dumbbell simulation, anotherensemble of equilibrium configurations {Q̂k

z, i, Q̂kr, i}

are also initialized that have the same initialconfiguration as {Qk

z, i, Qkr, i}. These are then sub-

jected to the same Wiener process and evolvedaccording to a similar weak form with vk+1

z andGk+1 equal to zero. The evolved configurations forthe flow field and quiescent state are then used toevaluate the updated stress field, τ k+1

p, zz and τ k+1p, rr .

4. Repeat steps 1–3 with Ak+1, τ k+1p, zz, τ k+1

p, rr , vk+1z , and

Gk+1 until t = T f .

In the discontinuous Galerkin method, the weakform of the advection term has been replaced with thesum of an integral over the element [z j, z j+1] and ajump term that is only evaluated at inflow boundaries.Specifically, terms labeled τ k,ex

p, rr are evaluated upstreamof a given element, while terms labeled τ k,in

p, rr are evalu-ated within the element itself. If a boundary, z j or z j+1 isnot an inflow boundary, the jump term at that boundaryis equal to zero. Also, in the weak formulation forthe Hookean dumbbell configuration, the same Wienerprocess (�Wz, i or �Wr, i) is applied to all points inspace. In writing the weak formulation for the dumbbellconfiguration, the Wiener process has been scaled by√

�t/De.

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