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MULTISCALE MODEL. SIMUL. c© 2005 Society for Industrial and Applied MathematicsVol. 4, No. 3, pp. 709–731

FENE DUMBBELL MODEL AND ITS SEVERAL LINEAR ANDNONLINEAR CLOSURE APPROXIMATIONS∗

QIANG DU† , CHUN LIU† , AND PENG YU†

Abstract. We present some analytical and numerical studies on the finite extendible nonlinearelasticity (FENE) model of polymeric fluids and its several moment-closure approximations. Thewell-posedness of the FENE model is established under the influence of a steady flow field. Wefurther infer existence of long-time and steady-state solutions for purely symmetric or antisymmetricvelocity gradients. The stability of the steady-state solution for a general velocity gradient is illumi-nated by the analysis of the FENE-P closure approximation. We also propose a new linear closureapproximation utilizing higher moments, which is shown to generate more accurate approximationsthan other existing closure models for moderate shear or extension rates. An instability phenomenonunder a large strain is also investigated. This paper is a sequel to our earlier work [P. Yu, Q. Du,and C. Liu, Multiscale Model. Simul., 3 (2005), pp. 895–917].

Key words. finite extendible nonlinear elasticity, non-Newtonian fluids, moment closure, finiteextendible nonlinear elasticity-P, Fokker–Planck equation, stability analysis

AMS subject classifications. 76A05, 76M99, 65C30

DOI. 10.1137/040612038

1. Introduction. Non-Newtonian viscoelastic fluids comprise a large class ofsoft materials, such as polymeric solutions, liquid crystal solutions, electro(magneto)-rheological fluids, and fiber suspensions. The stress endured by a viscoelastic fluidelement depends upon the history of the deformation experienced by that element,which is further attributable to the coupling between the flow-induced evolution ofmolecular configurations and macroscopic rheological response. This coupling callsnaturally for the development of multiscale methods to analyze the flow of rheologi-cally complex fluids. The multiscale models of viscoelastic fluids bridge directly themicroscopic scale of kinetic theory and the macroscopic scale of continuum mechan-ics. Serving as the link between the two scales is the macroscopic stress carried byeach material element which is determined by the distribution of the molecular con-figurations within that element. On the macroscopic level, the theory of continuummechanics yields a force-balance law expressed, for instance, by the incompressibleNavier–Stokes equation, except for the inclusion of an additional stress term account-ing for the contribution from the coarse-grained microstructure. On the microscopiclevel, the kinetic theory gives a Fokker–Planck equation for the probability densityfunction (PDF) f( �Q, t), where �Q denotes the set of variables defining the coarse-grained microstructure [2, 5]. For example, dilute solutions of linear polymers can bedescribed in some detail by a freely jointed bead-rod Kramers chain or, on a coarserlevel, by a freely jointed bead-spring chain. In this paper, we discuss an even coarsermodel of the single dumbbell, namely two beads connected by an elastic spring. Inthis case, the configuration variable �Q simply represents the vector connecting thetwo beads of the dumbbell.

∗Received by the editors July 21, 2004; accepted for publication (in revised form) March 16,2005; published electronically August 22, 2005. This research was supported in part by grants NSFDMS-0196522, NSF ITR-0205232, and NSF-DMS 0405850.

http://www.siam.org/journals/mms/4-3/61203.html†Department of Mathematics, Pennsylvania State University, University Park, PA 16802 (qdu@

math.psu.edu, [email protected], [email protected]).

709

710 QIANG DU, CHUN LIU, AND PENG YU

Thus, the multiscale approach to modeling dilute solutions of flexible polymers(treated as dumbbells) involves solving the coupled PDEs consisting of the macro-scopic momentum equation, the incompressible Navier–Stokes equation, and a Fokker–Planck equation describing the PDF f = f(�x, �Q, t) of the dumbbell orientation �Q onthe microscopic level. The coupled system [2] reads

∂�u

∂t+ (�u · ∇)�u + ∇p = ∇ · τp + ν � �u,(1.1)

∇ · �u = 0,(1.2)

∂f

∂t+ (�u · ∇)f + ∇�Q · (∇�u�Qf) =

2

ζ∇�Q · (∇�QΨ( �Q)f) +

2kT

ζ��Q f.(1.3)

In the Navier–Stokes equation (1.1), p is the hydrostatic pressure, ν is the fluid vis-cosity, and τp is a tensor representing the polymer contribution to stress,

τp = λ

∫(∇�QΨ( �Q) ⊗ �Q)f(�x, �Q, t)d�Q,(1.4)

where Ψ( �Q) is the elastic spring potential to be specified later, and λ is the polymerdensity constant. This induced elastic stress τp in (1.4) can be derived from the leastaction principle by looking at the variation of the following elastic part of the actionfunction [5, 26]:

λδ

(∫ T

0

∫Ω

∫R3

KTf ln f + Ψf d �Qd�xdt

)=

∫ T

0

∫Ω

∫R3

τp : ∇δ�x d�Qd�xdt,(1.5)

where δ represents the infinitesimal variation of the corresponding functions. In theFokker–Planck equation (1.3), ζ is the friction coefficient of the dumbbell beads, T isthe temperature, and k is the Boltzmann constant. The terms in (1.3) can be roughlyexplained as follows. The second and third terms on the left-hand side of (1.3) stemfrom the fact that the polymers are convected and stretched by the macroscopicflow, the first term on the right-hand side of (1.3) accounts for the inner force of thedumbbell due to the spring potential, and the last diffusion term on the right-handside of (1.3) models the random collisions of the solvent particles with the polymers.This interpretation becomes clearer when we associate the Fokker–Planck equation(1.3) with the stochastic differential equation (SDE) [10]

d�Q + �u · ∇ �Qdt =

(∇�u�Q− 2

ζ∇�QΨ

)dt +

√4kT

ζd �Wt,(1.6)

where �Wt is the standard Brownian motion. This SDE defines Brownian dynamics ofthe connector vector �Q such as the fact that the probability distribution of dumbbellpolymers evolves according to the Fokker–Planck equation (1.3). The Lagrangianversion of (1.6) is given in [9], and for a discussion of various choices of the noiseterm Wt for variance reduction purposes, readers are referred to [13].

We note that for the kind of complex fluids we are considering here, the mechanicaltransport and deformation are only imposed by the macroscopic level. To illustratethis point, we explain the origin of the particular transport terms in (1.3), ∂f

∂t +

(�u · ∇)f + ∇�Q · (∇�u�Qf). For a given material point �X, we define the characteristic

�x( �X, t) such that

�xt = �u(�x( �X, t), t), �x( �X, 0) = �X.(1.7)

FENE MODEL AND CLOSURE APPROXIMATIONS 711

Denoting F the deformation tensor ∂�x

∂ �Xand �q = F−1 �Q the inverse stretching of the

molecular orientation by the flow field, we can verify that

∂f

∂t+ (�u · ∇)f + ∇�Q · (∇�u�Qf) =

D

Dtf(�x( �X, t), F�q, t).(1.8)

Here we have used the indentity [8, 21]

Ft + �u · ∇F = ∇�uF ,(1.9)

which is a direct consequence of the chain rule.The simplest spring potential is given by the Hookean law Ψ(�Q) = HQ2/2,

where Q = | �Q| and H is the elasticity constant. In fact, exploiting this simplepotential, one can derive a macroscopic differential constitutive equation for thepolymeric stress τp from the Fokker–Planck equation, which leads to the so-calledOldroyd-B fluids. However, for a more practical finite extendible nonlinear elasticity(FENE) potential that takes into account a finite-extensibility constraint, Ψ(�Q) =−(HQ2

0/2) log(1 − (Q/Q0)2), there does not exist an exact macroscopic constitutive

equation for τp, and thus the FENE model represents a truly multiscale model. HereQ0 is the maximum dumbbell extension. The FENE spring force law reads

∇�QΨ =H �Q

1 − (Q/Q0)2.(1.10)

The goal of this paper is to study several analytical and numerical issues related tothe FENE model.

For one-dimensional shear flows, the well-posedness of the system coupling theNavier–Stokes equations and the SDE (1.6) with the FENE potential has been estab-lished by Jourdain, Lelievre, and Le Bris [14] using the specific simple structure ofshear flow. To extend such analysis to higher dimension, a crucial step has been doneby E, Li, and Zhang [6] to analyze the local well-posedness of the coupled system.On the other hand, there is much less analytical work on the Fokker–Planck equationitself. Lin, Liu, and Zhang [19] recently proved well-posedness of the flow system cou-pled with the Fokker–Planck equation for Hookean potential (the existence of globalclassical solutions when the initial datum are near the equilibrium configurations).Li, Zhang, and Zhang [20] have established local existence of the same system whenthe force law satisfies polynomial growth condition.

It is important to note that the added complexity with the FENE potential affectsmostly the analysis of the underlying Fokker–Planck equation, such as the issue ofboundary conditions as well as the singularity of coefficients near the boundary ofthe configuration domain. Hence, in this paper, we present the well-posedness of theFokker–Planck equation alone, with the fluid velocity being steady and homogeneous.In other words, we suppress the convective term �u · ∇f in (1.3), and the velocitygradient ∇u is treated as a constant matrix. We note that, independently from ourwork here, a preliminary analysis of the coupled system with the FENE potentialwas made recently [27]. The approach used there, however, is different from whatwe employ here, and the well-posedness results established in [27] were also undermore restrictive assumptions on the maximum extension rate than what are requiredhere. The major difficulty is to select the appropriate functional space such that thedegeneracy of the FENE potential at the boundary �Q = Q0 can be treated naturally.Upon choosing the suitable space, we show finite-time existence of the solution to


the Fokker–Planck equation for general velocity gradients and show global-in-timeexistence of the solution in the case where the velocity gradients are purely symmetricor antisymmetric. We mention that such a “decoupled” study for the SDE (1.6) hasbeen carried out by Jourdain and Lelievre [11] with a detailed discussion on thedependence of the existence and uniqueness of the solution on the finite-extensibilityparameter. The approach there is, of course, very different.

Simulating the coupled micro-macro system numerically is a demanding task, inthat at each spatial point �x, the configuration space of �Q needs to be resolved or atleast sampled. Recently, Lozinski and Chauviere [22] have introduced a fast solverto tackle the Fokker–Planck equation directly with a novel time splitting scheme.However, a more widely used method to solve the coupled system is the so-calledCONNFFESSIT approach [15, 23] that simulates the SDE corresponding to theFokker–Planck equation to achieve a Monte Carlo sampling of the configuration space.The latter approach has the advantage that the complexity increases only linearlywith the number of beads of the multibead polymer chains. We will briefly reviewthe Monte Carlo simulation of the FENE model and present a finite-difference schemeused later in this paper which has several attractive features, such as simplicity andpositivity preserving. Our focus regarding numerical simulation of the FENE model,however, is on the so-called moment-closure approach.

To illustrate the idea of moment-closure approximations, we multiply the Fokker–Planck equation in two dimensions by �Q ⊗ �Q and perform integration by parts onthe probability space | �Q| < Q0. Denoting by A the conformation tensor 〈 �Q ⊗ �Q〉,where the brackets represent the ensemble average over the space | �Q| < Q0, i.e.,

〈 �Q⊗ �Q〉 ≡∫|�Q|<Q0

�Q⊗ �Qf(�x, �Q, t)d�Q, we get

∂A

∂t+ (�u · ∇)A− (∇u)A−A(∇u)T = −4

ζ

∫H �Q⊗ �Q

1 −Q2/Q20

f(x, �Q, t)d�Q +4kT

ζ.(1.11)

If we select the conformation tensor as the only state variable, to close the equation,it is necessary to find an approximate relation to represent the ensemble average term

〈 H �Q⊗�Q1−Q2/Q2

0〉 on the right-hand side in terms of A. If done properly, this closure approach

reduces the need to resolve the probability space and thus results in dramatic savingscomputationally. Certainly, the challenge is to find a good closure approximation,which is often not an easy task. Since no closure model is expected to give quantitativeagreement with the original FENE model for all ranges of velocity gradients, a goodmodel would be one that agrees quantitatively with the FENE model for certainregimes of the physical parameters and at least reproduces the qualitative behaviorof the FENE model for other choices of the physical parameters.

A particularly simple closure approximation is the FENE-P closure based on anad hoc preaveraging assumption,⟨

H �Q⊗ �Q

1 −Q2/Q20

⟩≈ H〈 �Q⊗ �Q〉

〈1 −Q2/Q20〉

=HA

1 − trA/Q20

.(1.12)

The FENE-P model has been used widely in qualitative studies of FENE fluids (see,for example, [28] and the references therein). But it is known that quantitative pre-diction of polymeric stress using FENE-P can deviate significantly from the originalFENE model, even for considering only the steady state [28]. Furthermore, the time-dependent behavior of the FENE-P model is qualitatively different from that of the


FENE model, in that the hysteretic behavior of FENE fluids is lost [16]. As opposedto the ad hoc closure, a more systematic strategy to derive closure approximation isbased on making an ansatz on the class of functions that the PDF solution to theFokker–Planck equation can take. The merit of this approach lies in the fact that anyclosure relation derived this way is backed up by some underlying PDF and thus iskept from producing totally unrealistic predictions. Moreover, the resulting systemmay still keep certain energy law. On the other hand, the success of such methodsclearly depends on how well the assumed class can capture the actual solution to theFokker–Planck equation. An application of this strategy to the FENE model is givenby Lielens et al. [16] and Lielens, Keunings, and Legat [17] and is named the FENE-L

closure. They assume that the PDF f = f( �Q) takes a factored form,

f( �Q) = ψl(Q)ψo( �Q/Q),(1.13)

where ψl(Q) and ψo( �Q/Q) are the distribution of the dumbbell’s length and unitorientation vector, respectively. The length distribution is further assumed to take anL-shaped form assembled from the Dirac-δ and Heaviside functions, i.e.,

Qψl(Q) =1 − β

α(1 −Hα(Q)) + βδα(Q),(1.14)

where (α, β) ∈ [0,√b]× [0, 1] are parameters to be determined from the second-order

moments A = 〈Q⊗Q〉 and a fourth-order moment 〈Q4〉. Here b = HQ20/kT . Leaving

the orientation distribution ψo( �Q/Q) unspecified, one may express the ensemble aver-

age 〈 H �Q⊗�Q1−Q2/Q2

0〉 as an explicit function of α and β, which are in turn expressed in terms

of the selected moments. One thus arrives at an explicit system of equations describingthe evolution of state variables A and 〈Q4〉. The use of singular distributions in theansatz (1.14) is motivated by the observation in numerical simulations that the truesolution to the FENE Fokker–Planck equation can become rather singular for largeshear or extension rates. Therefore, the FENE-L model has the advantage of beingable to capture PDFs far away from the equilibrium. Its capability to qualitativelyrecover the hysteretic behavior of the original FENE fluids has also been shown.

We have recently proposed a closure model [25] based on restricting the PDFs to aclass viewed as a perturbation to the equilibrium distribution. Taking two dimensionsas an example, the ansatz reads

f( �Q) =1

Jc

[1 −

(Q

Q0

)2]c/2

(1 + βQ1Q2 + γ(Q21 −Q2

2)),(1.15)

where �Q = (Q1, Q2), and Jc is scaling constant such that∫f( �Q)d�Q = 1. When

the three parameters c, β, and γ take the values HQ20/kT , 0, and 0, respectively,

the ansatz reduces to the equilibrium distribution of the FENE model. It is this ex-plicit incorporation of the equilibrium distribution that distinguishes this model fromFENE-P and FENE-L. It is shown in [25] that excellent agreement is obtained withthe FENE model for shear flow with small shear rates and for coupled simulation oftwo-dimensional driven-cavity flow. However, it is also observed that the approxima-tion deteriorates as the distortion on the PDF by the flow becomes stronger. In orderto improve the closure approximation at larger shear or extension rates, we proposein this paper an improved closure that allows systematic inclusion of higher-order mo-ments. The higher moment-closure relation is derived based on a similar framework,


except that the dependence of the ensemble average term needing closure on momentsis linear, while the relation is nonlinear in all the previous closure models. We willshow that, for moderate shear and extension rates, the closure based on higher-ordermoments produces much better stress predictions than the existing closure models,and increasing the order of moments involved results in better quantitative agreementwith the FENE model. However, the deficiency of the model is that the moment equa-tions become unstable for large shear or extension rates. This can be attributed tothe linear nature of the closure relation which fails to control the growing eigenvalueof the linear system as the shear or extension rate increases. Stability analysis of thismodel as well as the FENE-P model will be carried out. In summary, the new highermoment-closure model outperforms the FENE-P or FENE-L models for moderateshear or extension flow rates but fails for large flow rates, whereas the FENE-P orFENE-L models do not.

The rest of the paper is organized as follows. We first address the well-posednessof the Fokker–Planck equation with a constant velocity gradient in section 2. Then wediscuss its numerical simulations in section 3. The stability analysis of the FENE-Pmodel is presented in section 4, and a higher-order moment-closure approximation ispresented in section 5. A conclusion is given in section 6. We note that the resultspresented in the various sections of this paper are centered around the FENE modeland its difficulties in both analysis and computing, in particular the singularity of theFENE force law. Our attempts in proving existence, building higher-order closuremodels, analyzing stability or instability, and developing suitable numerical schemesall illustrate the influence of this difficulty and also shed light on how it may be dealtwith.

2. Well-posedness of the Fokker–Planck equation. As mentioned before,we constrain ourselves with studying the Fokker–Planck equation alone, decoupledfrom the flow calculation. Upon applying a standard scaling [9], the Fokker–Planckequation (1.3) reduces to

∂f

∂t( �Q, t) + ∇�Q · (κ�Qf) =

1

2(∇�Q · (∇�QΨf) + ��Qf),(2.1)

where the FENE potential is such that

∇�QΨ =�Q

1 −Q2/b.(2.2)

Here κ = ∇u is the steady homogeneous velocity gradient. We assume incompressibleflow field u; hence κ is traceless. The Fokker–Planck equation is defined on the openball Ω = { �Q ∈ R

n : Q2 < b}, where n = 2, 3. Typical values of b are 20, as in [22],and 50, as in [9, 16].

We first note that the operator Af = 12 (∇�Q · (∇�QΨf) +��Qf) on the right-hand

side of the Fokker–Planck equation has some nice mathematical properties. More

specifically, A as an operator on the functional space {f : eΨ(�Q)/2f ∈ L2(Ω)} is self-adjoint and coercive on the orthogonal of its null space [4]. However, for the solution fto represent a PDF, (2.1) needs to be conservative, and a major difficulty is to assigna meaning to the formal natural boundary condition(

1

2(∇�QΨf + ∇�Qf) − κ�Qf

)· �n |∂Ω = 0,(2.3)


for ∇�QΨ is singular at the boundary of ∂Ω. To circumvent this problem, we introduce

the following transformation:

f( �Q, t) = e−Ψ(�Q)/2g( �Q, t).(2.4)

It is easy to check that g( �Q, t) satisfies

∂g

∂t+ ∇�Q · (κ�Qg) −

∇�QΨ · κ�Q2

g =1

4

(��QΨ − 1

2|∇�QΨ|2

)g +

1

2��Q g.(2.5)

Terms involving ∇�QΨ · ∇�Qg have been cancelled due to our choice of the factor 2

in (2.4), and this is inherently linked to the properties of the self-adjoint operator Amentioned above. The cancellation of ∇�QΨ · ∇�Qg also makes it easier to define

the proper space in which the existence of solution to (2.5) is proved. We notethat in the analysis of the fully coupled system given by Zhang [27], approximations(regularizations) were made on the singular coefficients.

In terms of the new function g, the original no-flux boundary condition (2.3)becomes (

1

4∇�QΨe−Ψ/2g + e−Ψ/2

(1

2∇�Qg − g

))· �n |∂Ω = 0.(2.6)

Since e−Ψ/2 = (1 − Q2/b)b/4, the singularity caused by ∇�QΨ in (2.3) has been re-moved, as long as b > 4. Hereafter, we make the assumption that b > 4, and thisassumption is reasonable for practical purposes since, according to [23], b is roughlythe number of monomer units represented by a bead; thus it generally exceeds 10.But we mention that this constraint on b is not optimal mathematically, for it isknown that the solution to the SDE associated with (2.1) exists and has trajectorialuniqueness if and only if b ≥ 2 [11].1 Now our goal is to establish the finite-time exis-tence of the solution g to (2.5), and through the transformation (2.4) we then obtainthe solution to the Fokker–Planck equation (2.1) which automatically satisfies thenatural boundary condition, given certain regularity properties on g near the domainboundary. Thus, ideally, we would not have to impose any extra boundary condi-tions on (2.5) for g. However, as we exploit the transformation (2.4) to suppress theboundary singularity, we pay a price of losing the probability structure of the originalFokker–Planck equation. In particular, to prevent boundary terms from arising whencarrying out integration by parts in the following analysis, we need to impose an extrahomogeneous Dirichlet boundary condition on g. This does not affect the quality ofthe result because we are establishing the existence of the solution under more re-strictive conditions. But, philosophically, it is not completely satisfactory because thereformulation of the original problem as in (2.5) where we request g ∈ H1

0 (Ω) is onlypartially consistent with the Fokker–Planck equation (2.1).

Let V = {g :∫Ω| g1−Q2/b |2d�Q < ∞}, which is equipped with the norm ‖g‖V =∫

Ω| g1−Q2/b |2d�Q.

It is easy to verify that V is weakly compact and the set of smooth functionswith compact support on Ω is dense in V . We further consider the functional space

1The well-posedness of the Fokker–Planck equation with the FENE law has been established ina recent paper [1] for b > 2 with a smoothed velocity using similar but more complicated techniquesas here. But we were unaware of this work at the time of writing this paper.


H10 (Ω) ∩ V with the norm ‖g‖H1 + ‖g‖V . Given any finite time T , we will seek a

weak solution to (2.5) in X = L∞([0, T ];L2(Ω)) ∩ L2([0, T ];H10 (Ω) ∩ V ). We state

the weak formulation of (2.5): g ∈ X is a weak solution to (2.5) if and only if for allh ∈ H1

0 (Ω) ∩ V we have, in the sense of distributions,

d

dt(g, h) =

∫Ω

1

4

(��QΨ − 1

2|∇�QΨ|2

)ghd�Q− 1

2

∫Ω

∇�Qg · ∇�Qhd�Q

+

∫Ω

(1

2∇�QΨg −∇�Qg

)· κ�Qhd�Q ∀t ∈ [0, T ],(2.7)

and g satisfies the initial condition

g(0, �Q) = g0( �Q) ∀ �Q ∈ Ω,(2.8)

where (g, h) denotes the L2 inner product∫Ωghd�Q. The initial condition makes sense

because any solution to (2.7) will automatically be continuous from [0, T ] to V (aftera modification on a zero measure set) [24].

To study the well-posedness of (2.7), we use the Galerkin method. Given a se-quence of linearly independent smooth functions with compact support on Ω, h1, h2,. . . , hm, . . . , which is total in the separable space H1

0 (Ω) ∩ V , for each m, we definean approximate solution gm of (2.7) as follows:

gm =

m∑i=1

cim(t)hi(2.9)

and

d

dt(gm, hj) =

∫Ω

1

4

(��QΨ − 1

2|∇�QΨ|2

)gmhjd�Q− 1

2

∫Ω

∇�Qgm · ∇�Qhjd�Q

+

∫Ω

(1

2∇�QΨgm −∇�Qgm

)· κ�Qhjd�Q, j = 1, . . . ,m,(2.10)

gm(0) = g0m,(2.11)

where g0m is the orthogonal projection in H10 (Ω) ∩ V of g0 on the space spanned by

h1, h2, . . . , hm.It is easy to see that gm is well defined. Moreover, we have the following theorem.Theorem 2.1. For any given finite time T and b > 4, there exists a solution to

(2.7) satisfying the initial condition (2.8), which is the weak limit point of {gm}.Proof. The system (2.10) constitutes a linear system of ODEs with constant

coefficients for the scalar functions cim(t). Together with the initial condition, thislinear system defines uniquely the cim on the whole interval [0, T ]. Obviously, theapproximate solutions gm thus obtained belong to the solution space X.

To pass to the limit, we obtain a priori estimates independent of m for the func-tions gm. We multiply (2.10) by cjm and sum over j = 1, . . . ,m. We get

d

dt‖gm‖2

L2 =

∫Ω

1

4

(��QΨ − 1

2|∇�QΨ|2

)g2md�Q− 1

2

∫Ω

|∇�Qgm|2d�Q

+

∫Ω

(1

2∇�QΨgm + ∇�Qgm

)· κ�Qgmd�Q.(2.12)


Here we have used the fact that∫Ω∇�Qgm ·κ�Qgmd�Q = 0 because κ is traceless and gm’s

satisfy zero boundary conditions. Performing integration by parts on∫Ω��QΨg2

md�Q,one may easily verify that the above can be rewritten as

d

dt‖gm‖2

L2 = −1

2

∫Ω

∣∣∣∣12∇�QΨgm + ∇�Qgm

∣∣∣∣2

d�Q +

∫Ω

(1

2∇�QΨgm + ∇�Qgm

)· κ�Qgmd�Q.

(2.13)

The right-hand side is majorized by 12

∫|κ�Qgm|2d�Q ≤ 1

2 κ2b

∫|gm|2d�Q, where κ is

twice (in two dimensions) or triple (in three dimensions) the largest element (in mag-nitude) of κ. Integrating from 0 to s, 0 < s < T , we have

‖gm(s)‖2L2 ≤ ‖g0m‖2

L2eκ2bs/2.(2.14)

Hence

sups∈[0,T ]

‖gm(s)‖2L2 ≤ ‖g0m‖2

L2eκ2bT/2 ≤ ‖g0‖2

L2eκ2bT/2.(2.15)

Hence we have a uniform bound on gm in L∞([0, T ];L2(Ω)).On the other hand, if we majorize the right-hand side of (2.13) by

−1

4

∫Ω

∣∣∣∣12∇�QΨgm + ∇�Qgm

∣∣∣∣2

d�Q +

∫|κ�Qgm|2d�Q

and integrate from 0 to T , we get a uniform bound on∫ T

0

∫Ω| 12∇�QΨgm+∇�Qgm|2d�Qdt

independent of m. However, what we need are uniform bounds on∫ T

0

∫Ω|∇�QΨgm|2d�Qdt

(or equivalently∫ T

0

∫Ω| gm1−Q2/b |2d�Qdt) and on

∫ T

0

∫Ω|∇�Qgm|2d�Qdt separately. To this

end, we return to (2.12). The right-hand side of (2.12) can be reorganized into∫Ω

(1

4��Q Ψ − 1

8|∇�QΨ|2 +

1

2∇�QΨ · κ�Q

)g2md�Q− 1

2

∫Ω

|∇�Qgm|2d�Q,(2.16)

where we have used again the fact that κ is traceless. Direct computation shows

1

4��Q Ψ − 1

8|∇�QΨ|2 +

1

2∇�QΨ · κ�Q =

1

2

d2 (1 −Q2/b) + ( 1

b − 14 )Q2 + (1 −Q2/b) �Q · κ�Q

(1 −Q2/b)2,

(2.17)

where d = 2, 3 is the dimensionality. For Q2 = b, since b > 4, we have

d/2(1 −Q2/b) + (1/b− 1/4)Q2 + (1 −Q2/b) �Q · κ�Q = 1 − b/4 < 0.

Thus, there exist r > 0 and ε > 0, such that d/2(1 − Q2/b) + (1/b − 1/4)Q2 +

(1 − Q2/b) �Q · κ�Q < −ε for |Q| > r. If we split the first integral in (2.16) into twointegrals over the domain |Q| < r and |Q| > r, respectively, and integrate (2.12) from0 to T , we get

ε

∫ T

0

∫r2<Q2<b

∣∣∣∣ gm1 −Q2/b

∣∣∣∣2

d�Qdt ≤ C

∫ T

0

‖gm(s)‖2L2ds−

∫ T

0

‖∇gm(s)‖2L2ds

− (‖gm(T )‖2L2 − ‖gm(0)‖2

L2),(2.18)


where C is a constant independent of m. In conjunction with (2.15), this shows

at the same time that both∫ T

0

∫Ω| gm1−Q2/b |2d�Qdt and

∫ T

0

∫Ω|∇�Qgm|2d�Qdt are uni-

formly bounded independent of m; hence we have the uniform bound on gm inL2([0, T ], H1

0 (Ω) ∩ V ).Finally, using the fact that the spaces V and H1

0 (Ω)∩ V are weakly compact, wecan subtract a subsequence of gm that converges weakly in X to a weak solution of(2.7). Since the equation of g is linear, this procedure of passing to the limit is trivial.For details of similar arguments, one may see, for example, [24].

Remark 1. Equation (2.13) can be estimated in a different way from the aboveproof of the theorem, namely, using the Cauchy–Schwarz inequality and a weightedPoincare inequality∫

Ω

e−Ψf2d�Q ≤ 1

c

∫Ω

e−Ψ|∇f |2d�Q + α

(∫e−Ψf d �Q

)2

(2.19)

for any f such that e−Ψ/2f ∈ H10 (Ω). Here c > 0 is a constant, Ψ may be any

function such that ∇2Ψ ≥ cI, and α−1 =∫Ωe−Ψd�Q < ∞. We now outline a proof

of the weighted Sobolev imbedding inequality due to R. Varadan which might be of useto those interested in the analytical study of Fokker–Planck equations. First, let usdefine the operator

L = eΨ∇(e−Ψ∇) = Δ −∇Ψ∇(2.20)

over the domain e−Ψ/2f ∈ H10 (Ω). We notice that L is the dual operator of the

original Fokker–Planck operator. Now consider the equation

ut = Lu, u(·, 0) = f(·),(2.21)

and define the quantity:

D(t) =

∫Ω

e−Ψ|u|2 d�Q,(2.22)

and

H(t) =

∫e−Ψ|∇u|2 d�Q.(2.23)

We see that

d

dtD(t) =

∫e−Ψuut d�Q = −H(t)(2.24)

by the original equation and integration by parts. Now we compute

d

dtH(t) =

∫e−Ψ∇u∇ut d�Q(2.25)

=

∫e−Ψ∇u∇Lud�Q(2.26)

=

∫e−Ψ∇uL∇u + e−Ψ∇u[∇L]u d�Q(2.27)

=

∫−e−Ψ|∇2u|2 + e−Ψ∇u(−∇2Ψ)∇u d�Q(2.28)

≤ −cH(t).(2.29)


Hence H(t) ≤ H(0)e−ct. Integrating in time, we get

D(0) −D(∞) =

∫ ∞

0

H(t) dt ≤ H(0)

∫ ∞

0

e−ct dt =1

cH(0).(2.30)

Also, as t approaches infinity, u approaches the stationary constant solution of L.Since

∫e−Ψu d�Q is a conserved quantity in time, this constant solution will be

α∫e−Ψf d �Q. In addition,

D(∞) = α

(∫e−Ψf d �Q

)2

.(2.31)

We want to point out that although the finite-time existence has been shown in theabove without using the above estimate, the latter remains very useful and representssome of the fundamental properties of the Fokker–Planck equations.

Remark 2. By multiplying (2.7) by dgdt and performing similar energy esti-

mates, one may extend the regularity of the solution to L∞([0, T ];H10 (Ω) ∩ V ) ∩

H1([0, T ];L2(Ω)).Remark 3. Assuming the existence of classical solutions to (2.5), it is easy to

show by a maximum principle argument that the solution remains positive in Ω if theinitial condition is such. In fact, if g assumes for the first time a zero local minimum at( �Q, t), we then have g( �Q, t) = ∇�Qg(

�Q, t) = 0, ∂∂tg(

�Q, t) < 0, and ��Qg(�Q, t) ≥ 0. But

this contradicts (2.5). The uniqueness of the solution to (2.5) is another consequenceof the maximum principle. It takes more careful analysis to generalize these argumentsto the weak solution of (2.5).

In the estimate of the energy law (2.13), we did not assume any special propertiesof the velocity gradient κ. In fact, we may improve our results in the special caseswhen κ is symmetric or antisymmetric. In these cases, the energy can be shownto be dissipative, and thus the energy law enables the existence of a global-in-timesolution and the steady-state solution to (2.7). We briefly point out the techniques

to exploit the special features of κ. First, if κ is symmetric, κ�Q is then a gradientof the potential 1

2�QTκ�Q, and thus can be absorbed into the FENE potential by a

substitution Ψ = Ψ − 12�QTκ�Q. Replacing Ψ by Ψ in (2.13), the second integral

on the right-hand side of (2.13) no longer exists, and the energy is thus dissipative.Furthermore, the steady-state solution can be explicitly written out in this case.Second, if κ is antisymmetric, it is easy to check that the second integral on theright-hand side of (2.13) vanishes again. The following corollary summarizes thesefindings.

Corollary 2.1. There exists a global solution and a steady-state solution to(2.7) when κ is symmetric or antisymmetric.

Remark 4. Although we have not established the existence of the steady-statesolution to the Fokker–Planck equation for a general velocity gradient, our numericalexperiences with the Fokker–Planck equation under simple steady shear flow (where κis neither symmetric nor antisymmetric) seem to suggest that the steady-state solutionto the Fokker–Planck equation does exist even for quite large shear rate. The stabilityanalysis we carry out on a simplified FENE-P model later may shed some light on theunderstanding of the behavior of the stationary solution to the FENE model itself.

Remark 5. We have limited our analysis to the Fokker–Planck equation with aspecial given velocity gradient. The same technique (of using a transformed variableand using a weighted Sobolev space) can be combined with the methods presented in [18]


to study the local well-posedness of the fully coupled system as well. Note that similarlocal existence results were recently presented in Zhang [27] under more restrictiveassumptions on b.

Remark 6. We note that the existence of steady states is also valid for the finite-dimensional Galerkin approximations, and one may also show the convergence of suchapproximations to the steady-state solutions discussed in Corollary 2.1. Though (2.10)provides a possible numerical scheme for approximating the solution of the Fokker–Planck equation, for instance an approximation based on the spectral methods [22],we discuss an alternative difference method which is simpler while still preservingimportant properties of the Fokker–Planck equation.

3. Numerical solutions of the FENE model. For general velocity gradi-ent κ, the Fokker–Planck equation cannot be solved analytically. We now describetwo alternative strategies to numerically simulate the FENE model: deterministicsimulation of the Fokker–Planck equation directly and Monte Carlo simulation of theassociated SDE. The algorithms will be used later to compare the polymeric stressprediction of the FENE model and its various closure approximations.

Fokker–Planck simulation. Recently, Lozinski and Chauviere have proposed anefficient spectral method to numerically solve the Fokker–Planck equation with theFENE potential [22, 3]. They solve the Fokker–Planck equation in polar coordinates,and the appropriate boundary conditions are taken into account by a transformationof the unknown similar to (2.4) used in our analysis. They have shown in [3] thatfor small flow rates, the spectral method is able to capture the solution accuratelyeven with very low spectral resolution. However, getting accurate numerical resultswould require many more refined meshes when the flow rate increases. For a lowlevel of discretization, numerically oscillatory solutions may arise at large flow rates,and this is also related to the fact that the spectral method does not guarantee thepositivity of the PDF. The direct reason for this difficulty with large flow rates isthat the exact solution in this case processes huge gradients which are very localized,and a robust scheme for the Fokker–Planck equation with the FENE potential thatovercomes this problem is yet to be developed. Leaving this difficulty for futureinvestigation, we present here a simple finite-difference scheme that is appropriatefor modest flow rates. The motivation of the scheme is mainly its simplicity and itsability to preserve the unity and positivity of the PDF solution. We have used thescheme in [25] without a very detailed description, which we seek to provide here. Thefinite-difference scheme presented here is explicit, and it requires further investigationin the future to incorporate implicit time stepping to improve the scheme as far asthe stability and positivity preserving are concerned.

We use a regular grid as shown in Figure 3.1. The probability space { �Q :

| �Q| <√b} is discretized on an M × M regular grid with mesh size Δx = 2

√b/M ,

where cells outside the admissible space are marked inactive. Each active cell is as-signed a probability Pn

ij defined at the center of the cell, where n denotes the currenttime step. The discrete probabilities are updated by means of a discrete analogue ofthe Chapman–Kolmogorov equation

Pni,j = (1 − Γi−1,j

i,j − Γi+1,ji,j − Γi,j−1

i,j − Γi,j+1i,j )Pn−1

i,j(3.1)

+ Γi,ji,j−1P

n−1i,j−1 + Γi,j

i,j+1Pn−1i,j+1 + Γi,j

i−1,jPn−1i−1,j + Γi,j

i+1,jPn−1i+1,j ,

where Γk,li,j represents the transition probability from the grid point (i, j) to (k, l)

during one time step. Using the Taylor expansion around the cell (i, j), one may


(i,j+1)(i,j)(i,j-1)

(i-1,j)

(i+1,j)

Physical Boundary

(i,j)

viewzoom in

ComputationalBoundary

Fig. 3.1. Finite-difference mesh for solving the Fokker–Planck equation. The computationaldomain is made of the cells contained in the smaller disk.

easily find that the consistency of the finite-difference scheme with the Fokker–Planckequation requires⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

Γi,j−1i,j − Γi,j+1

i,j =[

12 ( Q1

1−Q2/b ) − (κ11Q1 + κ12Q2)]

ΔtΔx ,

Γi,j−1i,j + Γi,j+1

i,j = Δt/Δx2,

Γi−1,ji,j − Γi+1,j

i,j =[

12 ( Q2

1−Q2/b ) − (κ21Q1 + κ22Q2)]

ΔtΔx ,

Γi−1,ji,j + Γi+1,j

i,j = Δt/Δx2,

(3.2)

where Δt is the time-step size, and the functions on the right-hand side are evaluatedat the center of the cell (i, j). With a no-flux boundary condition (to be discussedlater), the scheme automatically conserves the sum of all Pn

i,j . Furthermore, it pre-

serves the positivity of Pni,j , as long as the transition probabilities Γi,j−1

i,j , Γi,j+1i,j , Γi−1,j

i,j ,

and Γi+1,ji,j are all in the range of [0, 1]. In view of (3.2), a sufficient condition is

h ≤ min

(∣∣∣∣[1

2

(Q1

1 −Q2/b

)− (κ11Q1 + κ12Q2)

]∣∣∣∣−1

,

∣∣∣∣[1

2

(Q2

1 −Q2/b

)− (κ21Q1 + κ22Q2)

]∣∣∣∣−1

),(3.3)

and Δt ≤ h2.Due to the singularity of the FENE force law near the boundary of the configura-

tion space Q2 = b, (3.3) poses an impractical constraint on the mesh size h. To circum-vent this problem, we choose to solve the Fokker–Planck equation in a smaller domainwith an artificial boundary at Q2 = b− ε, which ensures the existence of a finite Δxverifying (3.3). At the artificial boundary, a no-flux boundary condition is imposed,which can be implemented on the discrete level by setting those transition probabilitiesto or from grid points violating the constraint Q2 < b−ε to zero. To analyze the effectof the artificial boundary, we consider a simple case where the velocity gradient κ issymmetric. The steady-state solution to the Fokker–Planck equation (defined on the

actual domain Q2 < b) can be explicitly written as f(Q) = 1Z (1−Q2/b)b/2 exp( �QTκ�Q),

where Z =∫Q2<b

(1 −Q2/b)b/2 exp( �QTκ�Q)d�Q is the normalizing constant. Since this

solution satisfies detailed balance [7], the solution to the Fokker–Planck equation on


the artificial domain with a no-flux boundary condition is of the same form, exceptfor a different normalizing constant Z =

∫Q2<b−ε

(1 −Q2/b)b/2 exp( �QTκ�Q)d�Q. Thus,if the PDF is well concentrated around the origin, as in the case of a moderate ve-locity gradient, the difference between the solution on the restricted domain and theactual solution is negligibly small. A more rigorous analysis of this effect will be pre-sented elsewhere. We note that, however, for large κ, due to the exponential factorexp( �QTκ�Q), the PDF becomes rather singular and concentrated near the boundaryQ2 = b. Though one may choose a smaller ε, this would result in extremely small meshsize h because of (3.3). So for large κ, we perform instead Monte Carlo simulation ofthe SDE associated with the Fokker–Planck equation.

Monte Carlo simulation. To sample the probability space, a sufficiently largenumber of random walkers are simulated according to the SDE

d�Q =

(κ�Q− 1

2∇�QΨ

)dt + d �Wt.(3.4)

A concern with the brute force way of implementing this SDE (by, for example, the

Euler scheme) is that the constraint | �Q|2 < b may not be preserved. A nice algorithmthat avoids this problem is given by Ottinger [23] in the form of a predictor-corrector

scheme. Given �Qn at time tn, the predictor step is to apply the Euler–Maruyamascheme to compute an intermediate value �Q∗,

�Q∗ = �Qn +

(κ�Qn − 1

2∇�QΨ( �Qn)

)Δt + Δ �W,(3.5)

where Δ �W is a two-dimensional Gaussian random variable whose components havevariance Δt. We recall that the force term ∇�QΨ = �Q/(1−Q2/b). The corrector step

that determines �Qn+1 at time tn+1 = tn + Δt treats only the force term implicitlyand reads [

1 +Δt

4(1 − (Qn+1)2/b)

]�Qn+1 = �Qn(3.6)

+1

2

[κ�Q∗ + κ�Qn −

�Qn

2(1 − (Qn)2/b)

]Δt + Δ �W.

Notice the same random number is used in (3.5) and (3.6). The length of �Qn+1 canbe determined from a cubic equation which, for arbitrary length of the vector on theright-hand side of (3.6), has a unique solution between 0 and

√b. Thus, the scheme

automatically guarantees the finite-extensibility constraint.Figure 3.2 gives an example of the steady-state distributions obtained from the

Fokker–Planck simulation and the Monte Carlo simulation. The simulation is forshear flow with shear rate equal to 2. The Fokker–Planck equation is solved on a500× 500 grid using the algorithm described above. The Monte Carlo simulation uti-lizes 105 Brownian particles (polymers), and the histogram is constructed on 100×100bins (regular grid cells in which the numbers of Brownian particles are counted). Notethat the purpose of Monte Carlo simulation in practical computation is to producevarious ensemble averages rather than the fully detailed probability distribution. Infact, in spite of the noisy appearance of the Monte Carlo histogram, the relative differ-ence between the shear and normal stress predictions obtained by the two approachesis within only 1%. The data to be presented later in this paper are generated by


Fig. 3.2. Steady-state PDFs obtained from simulating the Fokker–Planck equation directly (left)and the Monte Carlo simulation of the associated SDE (right).

either the Fokker–Planck or Monte Carlo simulation, depending on the magnitude ofshear or extension rates.

4. Stability of FENE-P. Both the Fokker–Planck simulation and the MonteCarlo simulation are targeted at resolving or sufficiently sampling the probabilityspace. They are computational very demanding, especially when one is interested inthe coupled system of the macroscopic Navier–Stokes equation and the microscopicFokker–Planck equation. Therefore, model reduction via various moment-closure ap-proximations provides an attractive alternative to the above detailed simulation ap-proaches. In this section, we study the FENE-P closure which is widely adopted inmany qualitative numerical studies of the FENE polymeric fluids. The existence ofglobal-in-time solutions to the FENE-P model for a given homogeneous flow has beenproven by Jourdain and Lelievre in [12]. In this paper, we focus on the linear stabilityanalysis of the steady-state solution to the FENE-P model. Recall that A denotes theconformation tensor 〈 �Q�QT 〉. In the scaled form, the FENE-P model can be writtencompactly as

∂A

∂t− κA−AκT = 1 − A

1 − TrA/b,(4.1)

where TrA is the trace of A. In the one-dimensional case (A is a scalar), it is trivialto see that the ODE has a unique solution in (0, b) which is asymptotically stable.

We now study the two-dimensional case in which the components of A can berearranged into M1 = 〈Q2

1 +Q22〉, M2 = 〈Q2

1−Q22〉, and M3 = 〈Q1Q2〉. Equation (4.1)

becomes ⎧⎪⎪⎨⎪⎪⎩

∂M1

∂t = (κ11 − κ22)M2 + (κ12 + κ21)M3 + 2 − M1

1−M1/b,

∂M2

∂t = (κ11 − κ22)M1 + (κ12 − κ21)M3 − M2

1−M1/b,

∂M3

∂t = (κ12 + κ21)M1 − (κ12 − κ21)M2 − M3

1−M1/b.

(4.2)

We perform asymptotic analysis on the stability of the system obtained by linearizing(4.2) around its steady-state solution (M∗

1 ,M∗2 ,M

∗3 ), which can be solved by setting

the time derivatives in (4.2) to zero. The analysis focuses on steady shear flow forthe FENE-P model, whereas the theorem presented in the previous section could notguarantee the existence of long-time and steady-state solutions to the FENE model


under steady shear flow. However, the argument can be easily extended to the caseof extensional flow.

For simple shear flow �u = (αy, 0), the velocity gradient simplifies to κ12 = α,and κ11 = κ21 = κ22 = 0. The Jacobian of the linearized system evaluated at(M∗

1 ,M∗2 ,M

∗3 ) is ⎛

⎜⎜⎝− 1

(1−M∗1 /b)

2 0 α

− M∗2 /b

(1−M∗1 /b)

2 − 11−M∗

1 /bα

α− M∗3 /b

(1−M∗1 /b)

2 −α − 11−M∗

1 /b

⎞⎟⎟⎠.(4.3)

Our goal is to study the asymptotic behavior of the eigenvalues of this matrix asα → ∞. We need to first describe the asymptotic behavior of M∗

1 . It is easy to seethat (4.2) implies

− M∗1

1 −M∗1 /b

+ 2α2(1 −M∗1 /b)

2 = −2.(4.4)

This equation uniquely determines a solution in (0, b) which, as α increases, ap-proaches b asymptotically. Solving for the leading-order term, we obtain the rateby which M∗

1 approaches b in the large shear rate limit:

1 −M∗1 /b = (b/2)

13 |α|− 2

3 + l.o.t.(4.5)

On the other hand, omitting the lower-order terms, the eigenvalues of (4.3) λ can beshown to satisfy the algebraic equation

λ3 +α

43

(b/2)23

λ2 +2α2

bλ +

α83

(b/2)43

= 0,(4.6)

where λ = λ+1/(1−M∗1 /b). We assume that, up to the leading-order terms, (4.6) may

be factored as (λ+C1αp)(λ+C2α

q)(λ+C3αr) = 0, where p ≥ q ≥ r. Comparing with

(4.6), we get p = 4/3, q = r = 2/3, C1 = 1/(b/2)23 , and C2, C3 = (1 ±

√3i)/2(b/2)

13 .

Thus, all eigenvalues have negative real parts. The “least stable” mode correspondsto the least negative real part of the eigenvalues, and it is

Reλ ∼ − 3

(b/2)13

α23 ,(4.7)

which indicates that the steady-state solution of the FENE-P model becomes morestable as the shear rate increases. This is a somewhat surprising result because numer-ous numerical simulations have indicated that the PDF of the FENE model becomesmore singular for larger shear rates. Although it is questionable to relate the presentanalysis of the FENE-P model to the FENE model, the two models are somehow re-lated in the sense that the preaveraging assumption (1.12) becomes exact if the PDFis a delta distribution in Q, and from numerical studies the PDF of the FENE modelseems to approach a delta distribution for large flow rates. Thus assuming that theFENE-P model captures the qualitative behavior of the FENE model at large shearrates, the present analysis suggests that the singular PDFs produced by the FENEmodel at large shear rates may be rather stable to perturbations.

To validate our calculation, we compare the asymptotic result (4.7) with thenumerical evaluation of the eigenvalues of (4.3) using MATLAB. For b = 50, the


0 50 100 150 200 250 300 350 400 450 500- 35

- 30

- 25

- 20

- 15

- 10

- 5

0

Shear rate α

Leas

t neg

ativ

e R

e(λ)

NumericalAsymptotic

0 5 10 15 20 25 30 35 40 45 50- 60

- 50

- 40

- 30

- 20

- 10

0

α

Leas

t neg

ativ

e R

e(λ)

Fig. 4.1. The least negative eigenvalues of the Jacobian matrix (4.3) in shear flow (left) andin a combination of shear and extensional flow (right).

least negative real part of the calculated eigenvalue is plotted against the shear rate α(the left picture of Figure 4.1) as well as the prediction from the asymptotic formula.Excellent agreement is found. Extensional flow can be studied in the same way. Inparticular, one finds the same pattern that the steady-state solution becomes morestable as the extension rate increases, except that the least negative eigenvalue in thiscase goes to −∞ linearly. We give another example of a more general flow describedby κ11 = −α/2, κ12 = α, κ21 = 0, and κ22 = α/2. The numerical results usingMATLAB are shown on the right of Figure 4.1. The pattern that the eigenvaluedecays to −∞ as α increases is again observed.

Finally, we mention that the same analysis can be applied to the closure approx-imation proposed in [25]. The model is motivated by assuming the PDF to be aperturbation from its equilibrium state, and the moment equations are closed basedon an ansatz on the perturbed PDF. The model has the advantage of being able tofaithfully reproduce the quantitative features of the FENE model for weak flow. Infact, the moment equations are very similar to the FENE-P model:

⎧⎪⎪⎨⎪⎪⎩

∂M1

∂t = (κ11 − κ22)M2 + (κ12 + κ21)M3 + 2 − M1

1−2M1/b,

∂M2

∂t = (κ11 − κ22)M1 + (κ12 − κ21)M3 − 1+M1/b1−2M1/b

M2,

∂M3

∂t = (κ12 + κ21)M1 + (κ12 − κ21)M2 − 1+M1/b1−2M1/b

M3.

(4.8)

Thus, one may establish the linear stability of the steady-state solutions to this clo-sure approximation using the same method. However, for large velocity gradients, theasymptotic limit of M∗

1 of this model is b/2, which is an artificial and false constrainton the maximum extensibility of the dumbbell. To combine the quantitative strengthof the model proposed in [25] for the flow of small velocity gradients with the morereasonable asymptotic behavior of the FENE-P model, one may, for example, replace1 − 2M1/b in (4.8) with 1 − η(κ)M1/b, where η(κ) ∈ [1, 2] is a phenomenological pa-rameter that can be calibrated against the original FENE model. We leave discussionon such hybrid methods to our future research.

5. A higher-order linear closure model. Most existing closure models in-volve second-order or, at the most, fourth-order moments. It is a natural question toask whether including even higher moments can improve the closure approximationto the original FENE model. Based on the framework proposed in [25], we make thefollowing ansatz on the PDF:


f =

(1 − Q2

b

)b/2 N∑n=0

∑k+l=2nk≥0, l≥0

Ck,lQk1Q

l2.(5.1)

The first factor (1− Q2

b )b/2 is the equilibrium distribution of the FENE model whencethe velocity gradient κ = 0. The second factor is a Taylor polynomial of order 2Nviewed as a perturbation to the equilibrium distribution due to the nonvanishingvelocity gradient in the general case. Note that since the solution to the FENE modelis always an even function of �Q, only even powers of �Q are included in the Taylorpolynomial.

The restricted class of PDFs defined by the ansatz (5.1) is parameterized by(N + 1)2 coefficients Ck,l, which can be determined, provided that the values of allmoments up to order 2N are known. More specifically, given the moments M i,j =〈Qi

1Qj2〉 for all i + j = 2n and n = 0, 1, . . . , N , Ck,l may be determined from the

following linear system:

N∑n=0

∑k+l=2nk≥0, l≥0

Ck,l

∫ (1 − Q2

b

)b/2

Qk+i1 Ql+j

2 d�Q = M i,j , i + j = 2n, n = 0, 1, . . . , N.

(5.2)

Owing to the simplicity of (5.1), the integral∫

(1 − Q2

b )b/2Qk+i1 Ql+j

2 d�Q ≡ I(i,j),(k,l)

can be evaluated analytically. Using polar coordinates, we may rewrite I(i,j),(k,l) as∫√b

0(1 − Q2

b )b/2Q2p+1dQ∫ 2π

0cosk+i θ sinl+j θdθ, where p = (k + l + i + j)/2. One

may easily verify that the radial integral S(p) =∫√

b

0(1− Q2

b )b/2Q2p+1dQ satisfies therecursive formula S(p) = S(p− 1)bp/(b + b/2 + 1), and S(0) = b/(b + 2). A recursiveformula for the angular integral can also be easily obtained. To cast (5.2) into matrixform, we denote by I the (N + 1)2 × (N + 1)2 stiffness matrix whose ((i, j), (k, l))thcomponent is I(i,j),(k,l) (imagine a natural ordering of the double index (i, j), forexample, such that (i, j) > (k, l) if and only if i + j > k + l or i + j = k + l andi > k). Obviously, I is symmetric positive-definite. Similarly, let C and M be the(N +1)2-dimensional vectors with components Ck,l and Mk,l, respectively. In matrixform, (5.2) reads IC = M .

Next, we derive evolution equations for the moments up to order 2N based onclosure approximations as a consequence of ansatz (5.1). Multiplying the Fokker–Planck equation by Qi

1Qj2, where i + j ≤ 2N , and integrating by parts over the

probability space Q2 < b, we have

∂M i,j

∂t− (iκ11M

i,j + iκ12Mi−1,j+1 + jκ21M

i+1,j−1 + jκ22Mi,j)

= − i + j

2

∫Qi

1Qj2

1 −Q2/bf( �Q, t)d�Q +

i(i− 1)

2M i−2,j +

j(j − 1)

2M i,j−2.(5.3)

The only term requiring closure approximation is the integral∫ Qi

1Qj2

1−Q2/bf( �Q, t)d�Q

which, under the assumption that the PDF satisfies the ansatz (5.1), may be rep-resented as∫

Qi1Q

j2

1 −Q2/bf( �Q, t)d�Q =

N∑n=0

∑k+l=2nk≥0, l≥0

Ck,l

∫ (1 − Q2

b

)b/2−1

Qk+i1 Ql+j

2 d�Q.(5.4)


The integrals on the right-hand side are very similar to those of the stiffness matrix Iand can be evaluated analytically. Denoting by J the matrix whose ((i, j), (k, l))th

component is given by i+j2

∫(1 − Q2

b )b/2−1Qk+i1 Ql+j

2 d�Q, we may write (5.3) in itsmatrix form,

∂M

∂t= −JC + KM = (K − JI−1)M,(5.5)

where K is a constant (N+1)2×(N+1)2 matrix defined by (KM)(i,j) = i(i−1)2 M i−2,j+

j(j−1)2 M i,j−2 +(iκ11M

i,j + iκ12Mi−1,j+1 + jκ21M

i+1,j−1 + jκ22Mi,j). The polymeric

stress∫

Q⊗Q1−Q2/bf( �Q, t)d�Q can also be represented by the moments in a similar fash-

ion, and it provides a checkpoint to evaluate the current closure approximation incomparison with other closure models. Note that unlike the FENE-P model or theone proposed in [25], the current higher-order closure approximation leads to a linearsystem of evolution equations for the moments M .

It is interesting to remark on the similarity of the present moment-closure strategyand the general spectral Petrov–Galerkin method for numerically solving PDEs. Ifwe make the variable transformation M = IC, (5.5) becomes the evolution equationfor the parameters Ci,j of the ansatz (5.1):

∂C

∂t= I−1(K − JI−1)IC.(5.6)

From the derivation of the moment equations, this is exactly the Petrov–Galerkinformulation if the spectral solution space is defined by (5.1) and the test functions are

taken to be powers of �Q up to order 2N . The advantage of choosing the solution space(5.1) lies in its close relation to the equilibrium distribution, and thus the ability tocapture the actual solution to the FENE Fokker–Planck equation for a small velocitygradient. However, the performance of the present closure for larger velocity gradientsis not automatically guaranteed by (5.1) and needs to be evaluated using numericalsimulation.

To test the proposed closure model, we perform numerical simulations for incep-tion of shear or extension flow with various rates followed by relaxation. The FENEparameter b is chosen to be 50. First, we illustrate for moderate values of shear orextensional rates that increasing the order N of the moments involved does result inbetter approximation to the FENE model. Taking normal stress as an example, weshow the stress growth and relaxation for shear flow with shear rate 2 (left picturein Figure 5.1) and for extension flow with extensional rate 0.7 (right picture in Fig-ure 5.1) for different choices of N , and we compare them with the original FENEmodel. The same pattern is observed that we get better approximation as the orderof moments involved is increased, and perfect agreement with the FENE model is ob-tained for N = 3 in the shear case and N = 9 in the extension case. The larger orderrequired in the extension case is due to the stronger tendency for the polymer stretch-ing under an extension flow than that under a shear flow with comparable velocitygradients. We have also verified numerically that the moment-closure model with theabove orders gives a perfect agreement for any shear flow with rates less than 2 andany extension flow with rates less than 0.7. We compare the stress predictions withthe FENE-P and FENE-L closure models in Figure 5.2. For shear rate 2, both theFENE-L and the FENE-P models overpredict the steady-state value of the normalstress. For extensional rate 0.7, the FENE-L or FENE-P closure produces neither the


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

1

2

3

4

5

6

time

Nor

mal

Str

ess

Moment closure N=1Moment closure N=2Moment closure N=3FENE

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

time

Nor

mal

Str

ess

Moment closure N=6Moment closure N=7Moment closure N=8Moment closure N=9FENE

Fig. 5.1. Increasing the order of moments involved does improve the closure approximation.Left: normal stress difference in startup of shear flow with κ12 = 2. Right: normal stress differencein startup of extensional flow with κ11 = −κ22 = 0.7.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

1

2

3

4

5

6

time

Nor

mal

Str

ess

FENEClosure N=3FENE - PFENE - L

0 2 4 6 8 10 12 14 16 18 200

5

10

15

20

25

time

Nor

mal

Str

ess

FENEClosure N=9FENE - PFENE - L

Fig. 5.2. The proposed higher-order closure approximations outperform the FENE-P andFENE-L models for moderate shear or extension rates.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Shear Rate

She

ar S

tres

s

FENEClosure N=3

Fig. 5.3. Shear stress prediction by the FENE model and the proposed closure model with N = 3.

correct steady-state stress nor the correct time scale to approach the steady state.The proposed higher-order closure approximation clearly outperforms the lower-ordermethods such as the FENE-P and FENE-L models. At last, we present, for the rangeof shear rates from 0 to 2, the steady-state shear stress prediction by the proposedclosure model with N = 3 in Figure 5.3. The point worth noticing is that the slightshear-thinning effect of the FENE model is correctly reproduced by our closure modelup to the shear rate 2.

Limitation. The most severe limitation of the proposed linear closure modelis that (5.5) becomes unstable for large shear or extensional rates. Taking shear


Table 5.1

Largest real part of the eigenvalues of K − JI−1.

Shear rate 4 5 6 7 8 9 10Eigenvalue −1.7087 0.6670 1.1466 1.5795 1.9858 2.3736 2.7470

flow as an example, for N = 3, we may numerically evaluate the eigenvalues ofthe matrix (K − JI)−1 in (5.5) for increasing values of the shear rate. Table 5.1summarizes the largest real part of the eigenvalues, which changes from being negative(stable) to being positive (unstable) as the shear rate increases. The same pattern isalso observed for extension flow and for higher orders of the closure approximation.Since the FENE-P model has been shown to yield stable steady-state solutions andthe nonlinear nature of the FENE-P closure approximation has played a crucial rolein the analysis, one may speculate that the instability of the proposed method ismainly due to the linear relation between the parameters of the ansatz (5.1) on thePDF and its moments. An alternative explanation lies in the observation that inthe original Fokker–Planck equation, the infinitely large eigenvalues of the Laplacianoperator provide a control mechanism over arbitrarily large velocity gradients. Incontrast, in a linear closure approximation where only a finite number of momentsare taken, the sequence of eigenvalues of the Laplacian operator is cut finite, andthus can no longer act as a stabilizing factor against the increasing κ. There isyet another intuitive interpretation of the deficiency of the proposed closure modelfor large velocity gradients. As shown in Figure 5.4, the steady-state probabilitydistribution to the FENE model (via Monte Carlo simulation) under strong extensionflow κ11 = −κ22 = 1 displays a rather singular shape. The insufficiency of the ansatz(5.1) to incorporate such singular shapes may be a direct reason for the failure of theproposed method for large velocity gradients.

Fig. 5.4. The steady-state PDF from Monte Carlo simulation of the FENE model under ex-tension flow κ11 = −κ22 = 1.

Generally speaking, one would not expect a closure approximation to reproducethe original model at all regimes of the physical parameters. Depending on practicalinterests, a satisfactory closure model is to yield quantitative agreement with theoriginal model at a certain regime of parameters and still retain its stability for otherregimes. It is left for our future work to investigate how to improve the stability of thepresent linear closure model by introducing suitable nonlinearity or singular shapesto the ansatz (5.1).


6. Conclusion. In this paper, we have addressed several analytical and numer-ical issues of the FENE model of polymeric fluids. On the analytical side, we haveestablished the finite-time well-posedness of the Fokker–Planck equation for generalvelocity gradients κ. Although we have only been able to establish long-time asymp-totic results for special symmetric or antisymmetric κ, we speculate that this is stilltrue for general κ. The speculation is partly based on stability analysis of the FENE-Pclosure model, for which we have shown that the steady-state solution becomes evenmore stable as the shear or extension rate increases. On the numerical side, we havepresented a linear closure model allowing for systematic inclusion of higher-order mo-ments. The model has its advantage of being able to more faithfully reproduce thequantitative behavior of FENE for moderate shear and extension rates. But the modelis shown to become unstable for large shear or extension rates. The question of howto improve the stability of the model by introducing appropriate nonlinear effects orsingular PDF shapes will be a major focus of our future research.

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