J. Non-Newtonian Fluid Mech. 133 (2006) 14–27

Asymptotic basis of the L closure for finitely extensibledumbbells in suddenly started uniaxial extension

Ludwig C. Nitsche∗, Weidong Zhang, Lewis E. WedgewoodDepartment of Chemical Engineering, University of Illinois at Chicago, 810 South Clinton Street, Chicago, IL 60607, USA

Received 28 May 2005; received in revised form 29 September 2005; accepted 1 October 2005

Abstract

This paper applies the recent L closure [G. Lielens, P. Halin, I. Jaumain, R. Keunings, V. Legat, J. Non-Newtonian Fluid Mech., 76 (1998), 249–279], which was originally developed for FENE dumbbells with fixed friction, to two alternative dumbbell models of polymers in a dilute solutionundergoing uniaxial extension: (A) a linear-locked dumbbell with fixed friction and (B) a FENE dumbbell with variable friction. The simplifiedbox-spike representation of the probability density function (PDF) – used for Model B to close conformational averages of nonlinear quantities interms of a reduced set of state variables (moments of the PDF) – is justified through detailed asymptotic analysis (singular perturbations combinedwith multiple scales) of the Smoluchowski equation, in the limit of large extensibility parameter at fixed elongation rate. Both dumbbell models Aa ns comparef luchowskie ionless) limio©

K scaleA

1

c“fasrttnililg

an bewskiF tohemepesed byrf the

s (cf.by de-PDF

beenond

alpen-

upondtly.

ved inhing

0d

nd B are actually more amenable to the L closure than the FENE to which it had previously been applied. The resulting closure relatioavorably with corresponding integrals of asymptotic or numerical PDF’s (the latter obtained via atomistic SPH simulations of the Smoquation). Example calculations show the L closure to yield reasonably accurate stress–extension curves, even for a moderate (dimenstf extension (L = 4).2005 Elsevier B.V. All rights reserved.

eywords: FENE dumbbell; Linear-locked dumbbell; L closure; Variable friction; Dilute polymer solution; Rheology; Singular perturbations; Multiples;tomistic SPH

. Introduction

For the rheology of dilute solutions of FENE dumbbells, re-ent work by Lielens and coworkers[8,16] has proposed a newL closure” whereby the one-dimensional probability densityunction (PDF) is approximated with a simple class of shapes:

rectangular central section terminated with a delta functionpike. The two-parameter family of shapes defined by thisansatzepresents a canonical subspace of all possible PDF’s, for whichhe second and fourth moments (state variables) can be relatedo the relevant geometric parameters (location and relative mag-itude of the delta function). When the Smoluchowski equation

s averaged against appropriate spatial weights to extract evo-ution equations for the state variables, the resulting nonlinearntegrals are evaluated using the simplified PDF, thereby re-ating them to the state variables via the intermediaries of theeometric parameters. Thus, the evolution equations are closed

∗ Corresponding author. Tel.: +1 312 996 3469; fax: +1 312 996 0808.E-mail address: lcn@uic.edu (L.C. Nitsche).

consistently within the two-parameter state space, and csolved much more easily than: (1) solving the Smoluchoequation for the full PDF and then (2) integrating this PDobtain the moments at each time. The success of the schinges on the fidelity with which the canonical family of sharepresents the actual progression of the PDF as governthe Smoluchowski equation. Lielens et al.[8] formulated theicanonical subspace using knowledge of the time evolution oactual PDF, which was obtained from stochastic simulation[12,7]). The three-dimensional case has been addressedcoupling the orientational and length dependences of thein a multiplicative fashion, and the rectangular core hasfurther simplified (FENE-LS) by concentrating it into a secdelta function[9].

Lielens et al.[8] were originally motivated by canonicdistribution subspaces applied to the rheology of fiber sussions[23]. Systematic selection of state variables baseda maximum entropy principle[5] and applications to liquicrystalline polymers[6] have been developed more recenHysteresis loops in the stress–extension plane are obsertransient elongation, because different PDF’s in the stretc

377-0257/$ – see front matter © 2005 Elsevier B.V. All rights reserved.oi:10.1016/j.jnnfm.2005.10.004

L.C. Nitsche et al. / J. Non-Newtonian Fluid Mech. 133 (2006) 14–27 15

versus relaxing phases can be characterized by the same valueof the second moment, while leading to different stresses[26,28,20,16]. At least a second-order closure is required tocapture this feature. A one-parameter family of PDF’s will nec-essarily lock the stress and extension together in a single-valueddependence.

To determine under which circumstances the L closurerepresents a rigorous approximation of the PDF, we applysingular perturbation theory to the Smoluchowski equation, inthe asymptotic limit of a large extensibility parameter. For asudden start of elongation, a narrow peak in probability densityat or near the limit of extension is the inner solution (boundarylayer), and the initially wide and ever broadening central core ofprobability is the outer solution. This disparity of lengthscalesis accompanied by a separation of timescales between the slowrate of arrival of probability at the end and its rapid rearrange-ment within the boundary layer—as treated with the method ofmultiple scales. While the flow is on, the boundary layer doesnot move, which predetermines one parameter (location of thedelta function) in the L closure, and thereby reduces it to firstorder. Interestingly, the FENE dumbbell with fixed friction (towhich the L closure has hitherto been applied) does not exhibitan asymptotically narrow peak, so the delta function spike isnot rigorously justified as an approximation. Lielens et al.[8]addressed this issue by augmenting the canonical subspace ofPDF’s with an additional parameter to describe a terminal spiket delsd ckedd ithv lyv eryd e Lc

1

byt ingl uldc gript y thc y exp f thc gest ventw elast hf nablt t twc fer-e ands pico sub-s ndedso s, bya rnal

(conformational) degrees of freedom: orientation and length.Macroscopic properties of the polymer solution – in particu-lar viscosities in various types of viscometric flows and normalstress differences – are calculable from the PDF. Dependingupon the particular dumbbell model and any assumptions orapproximations employed to solve for the PDF—the resultingrheology includes non-Newtonian (shear thinning) behavior andtransients such as hysteresis loops in the stress–extension plane[8,9,16,20,17].

Kinetic theory of dilute polymer solutions is mainlybased upon an advection–diffusion or Smoluchowski equation(Fokker–Planck equation in the strongly damped limit[3]),which balances local accumulation of probability in each con-formational state with fluxes due to the spring force, the hydro-dynamic force exerted by the fluid on the beads and diffusion[1,12]. This paper treats a one-dimensional dumbbell model,wherein both beads are confined to the axis of a uniaxial elon-gational flow (cf.[7,8,20]). In the absence of vorticity, rotarymodes play only a minor role, or else they can be decoupled atleast approximately[9,17].

To motivate the mechanistic issues and closure approxima-tions that figure in the relevant literature, it is useful first todescribe the qualitative behavior of the PDF, which has been elu-cidated by Keunings[7], Lielens et al.[8,9] and Szeri[17] usingstochastic simulations for a FENE dumbbell with the Warnerform of the spring law. Here, we elaborate on their observationsb f thisp ped( tweenB ther nalfl e oft en-snt ngedi herei equi-l thec

DFa anyG it ofe urea asedu inals thee sianm sisd andm tived bute rmi-n ads,a evanta ns eta

hat was widened into a rectangle. However, two other moo have the desirable asymptotic structure: (A) the linear-loumbbell with fixed friction and (B) the FENE dumbbell wariable friction. In the former case we obtain a uniformalid asymptotic solution for the full PDF, which allows a vetailed consideration of the assumptions underlying thlosure.

.1. Dumbbell models for dilute polymer solutions

The rheology of a dilute polymer solution is determinedhe interaction of a single polymer chain with the surroundiquid. A micro-mechanical model of a polymer molecule shoapture two main features: (1) spatial extent of the frictionalhat the macromolecular chain exerts upon the fluid, wherebhain straddles spatial variations in the flow field and thereberiences deforming hydrodynamic forces; (2) resistance ohain to deformation by the surrounding flow, which chanhe macroscopic rheology from that of the Newtonian solhen added up over all dissolved chains. The ubiquitous

ic dumbbell model[1,20,17,12,25,7,19,14,4,30]captures boteatures in a crude but effective manner that is readily ameo analytical and numerical modeling. The beads represenenters of hydrodynamic friction which can be pulled in difnt directions by the flow field. The resistance to unravelingtretching of a real coiled macromolecule is primarily of entrorigin, for which the spring law represents an enthalpictitute, and should be finitely extensible to prevent unboutretching of the dumbbell in strong flows[1,24,26,28]. The statef the dumbbell at any instant is described in statistical termprobability density function defined on the space of inte

e-e

-

eo

ased upon the asymptotic analysis that is the subject oaper. Starting at equilibrium, the PDF will be a mound-sharoughly Gaussian) peak, which represents a balance berownian fluctuations tending to extend the dumbbell and

estoring force of the spring. A sufficiently strong elongatioow (abruptly started) overpowers the central, linear regimhe spring, and pulls probability outward. The probability dity away from the ends eventually decays to zeroas if there wereo stop in the spring, and whatever probabilitywould have ex-

ended beyond the limit of extension at any time gets rearranto a thin boundary layer at or near the limit of extension. Ts a separation of timescales, whereby the boundary layeribrates to the incoming probability density much faster thanentral peak drains off into the boundary layer.

At the final steady state, the actual finitely extensible Pccumulates all of the probability at the end, whereasaussian PDF must necessarily extend beyond the limxtension[7,17]. Thus, it seems surprising that simple clospproximations (e.g., FENE-P) or averaging schemes bpon a Gaussian PDF actually do yield an accurate termtress[1]. But transient stresses hinge on the details ofvolution of the PDF, and are poorly predicted by Gausodels or closures[7]. The comprehensive asymptotic analyeveloped here, which combines singular perturbationsultiple scales: (1) formalizes and quantifies the qualitaescription given above and (2) motivates the very simpleffective L closure representation of the PDF (rectangle teated with a delta function near the limit of extension) that let larger extensions, to linear closure relations between relverages of the PDF and the second moment (see Lielel. [8,9]).

16 L.C. Nitsche et al. / J. Non-Newtonian Fluid Mech. 133 (2006) 14–27

1.2. Spring and drag laws

In this paper we consider two structural features of dumbbellmodels that have a major impact on the rheology of a dilutesolution, as calculated for the one-dimensional case:

(1) Finitely extensible spring law: We will compare the Tannermodel – a Hookean spring with rigid stops at the limit ofextension[21,18,1]– with the FENE dumbbell[1,28,7].

(2) Drag law: We will compare a constant friction factor forthe beads with an empirical, linear configuration-dependentfriction law [19,15]that represents the increasing frictionalgrip that the surrounding liquid has on the polymer chain asthe latter gets stretched[2,4,14].

Among the four possible combinations involving these twofeatures, which are treated in the Ph.D. thesis of Zhang[29],only the FENE dumbbell with fixed friction does not share theseparation of inner and outer timescales described above, andis not amenable to the same kind of asymptotic analysis. Thispaper is confined to the simplest and most complicated cases:

• Model A: Linear-locked spring; fixed drag.• Model B: FENE spring; linear dependence of drag upon ex-

tension.

T or-d dedl ac-t d acc

1

e ext actei con-f . Thl senco cheh hosa nt: (c

1eter-

l ath-e emba rageT equt tionf e oi isioni disc anyn on-s imu

L, the effect of the Peterlin approximation is to soften the con-straint to limit only the root-mean-square end-to-end distance toL, thereby allowing the tails of the PDF to extend to infinitely.An additional complication of the approximate model is that thesecond-moment equation is rendered nonlinear and now admitsmultiple steady states[2] in contrast to the exact model[28],which predicts a single steady state. A similar Peterlin-like ap-proximation was applied to a dumbbell with variable drag[15].Again, the effect was to close the second-moment equation; how-ever, again the mathematical ansatz led to nonphysical multiplesteady states[28].

Averaging schemes linearize the Smoluchowski equation byreplacing the nonlinear, configuration-dependent diffusion ten-sor with an averaged version that is independent of configu-ration, and thus yield a Gaussian PDF as the solution. Theensemble-averaged diffusion tensor is calculated by integrat-ing the configuration-dependent diffusion tensor against a PDFwhich is also assumed to be Gaussian, and the distinction be-tween averaging methods lies in the source of this PDF and itsrelation to the Gaussian PDF actually being solved for.

1.3.2. Equilibrium averaging (EA)For dumbbells with hydrodynamics interaction (a form of

variable drag), Zimm[31] decoupled the PDF being solved forfrom the PDF used to obtain the averaged diffusion tensor. Forthe latter he employed the PDF for equilibrium in the absenceo qua-t all-d sis-t rag.

1sis-

t en-s s witht wskie ernalc omem ethods mep

1h re-

t thana ap-p qua-t asedi r beings outn lu-c satzl

1state

v he

he configuration-dependent friction term can vary by aner of magnitude or more, depending upon the fully exten

ength of the chain. Any correction for hydrodynamic interions would be comparatively minor, and is here neglecteording to the freely draining dumbbell assumption.

.3. Closure approximations and averaging schemes

To obtain a macroscopic characterization (mean-squarension, stress, etc.) the Smoluchowski equation is contrnto its second-moment form by a suitable integration overormations, based upon the premise of a Gaussian PDFeads to an ODE for the second-moment tensor. In the pref a nonlinear spring and/or variable drag, two main approaave emerged to deal with the resulting nonlinear terms, wverages cannot be closed in terms of the second momelosure approximations and (ii) averaging schemes.

.3.1. Peterlin closure (PC)For a FENE dumbbell with a constant drag coefficient, P

in [13] made a convenient, but not rigorously justifiable, mmatical ansatz in the second-moment equations: the ensverage of a ratio was replaced by a ratio of ensemble avehe effect of the ansatz was to close the second-moment

ion and lead to a closed-form rheological constitutive equaor the “approximate” model. Convenience came at the pricnconsistency: interchanging the order of averaging and divs not rigorously correct even for a Gaussian PDF, and therepancy is typically not small in numerical practice or inontrivial asymptotic limit. While the exact FENE model ctrains the end-to-end distance to be less than some max

-

-d

isesei)

les.a-

f

-

m

f flow. This procedure leads to a linear second-moment eion and gives results that are reasonable in the limit of smisplacement flows. However, equilibrium averaging incon

ently removes all of the flow dependence in the variable d

.3.3. Consistent averaging (CA)As an improvement upon equilibrium averaging, the con

ent averaging method[11,27]bases the averaged diffusion tor upon a Gaussian PDF whose second moment coincidehe second moment being solved for when the Smoluchoquation is contracted to an ODE. This enhanced level of intonsistency leads to qualitatively improved predictions of saterial properties; however, the consistent-averaging m

till fails quantitatively, even predicting the wrong sign of soroperties.

.3.4. Gaussian closure (GC)The most successful, and chronologically latest, approac

urned to the idea of closure, but with a sounder justificationn ad hoc ansatz[25,32]. To estimate the ensemble averagesearing in the second-moment form of the Smoluchowski e

ion for a Hookean dumbbell, a Gaussian PDF was used—bn a consistent manner on the same second-moment tensoolved for. The conformational integrals had to be carriedumerically. Although the solution to the full, nonlinear Smohowski equation is not actually Gaussian, this simple aneads to excellent predictions for material functions.

.3.5. The L closure (LC)For expressing the nonlinear averages in terms of the

ariables, the L closure[8,9,16] proceeds consistently by t

L.C. Nitsche et al. / J. Non-Newtonian Fluid Mech. 133 (2006) 14–27 17

same basic premise as the Gaussian closure, but replaces theGaussian PDF with the characteristic box-spike shape that moreaccurately represents the actual PDF, while also being moretractable analytically. Rheological predictions of the schemenear the initial equilibrium and the terminal steady state havebeen improved by augmenting and modifying the canonicalsubspace as follows: (i) replacing the flat central core with aGaussian distribution and (ii) widening the delta function spikeinto a rectangle[8]. But such complications seem to propelthe approach of closure to the point of diminishing returns.In this paper we justify the assumptions of the L closure withreference to the asymptotic structure of the Smoluchowskiequation.

2. Governing kinetic equation

Posed in dimensionless form, and reduced to half of theoriginal domain by symmetry, the Smoluchowski equationgoverning the probability density in one-dimensional confor-mation space (extensionx of the spring) of a finitely extensibledumbbell confined to the axis of a uniaxial elongational flowis [7,19,1]

∂P

∂t+ ∂J

∂x= 0, J = VP − D

∂P

∂x(0 < x < L) (1)

T area

M

M

H uste iablyt nsia emp thed0 aint

J

A ,r thisc

ead

3. Asymptotics for the linear-locked dumbbell with fixedfriction

For the linear Smoluchowski equation

∂P

∂t+ ∂

∂x

{(γ − 1

2

)xP − 1

2

∂P

∂x

}= 0, 0 < x < ε−1 (5)

the initial condition is equilibrium with no flow (γ = 0). For asufficiently long spring (ε � 1) the Gaussian profile

P(x, 0) ∼√

2

πe−x2/2 (6)

has only exponentially small errors in the normalization factorand no-flux condition(4). The final steady-state PDF is

P(x, ∞) ={∫ ε−1

0e(γ−1/2)x2

dx

}−1

e(γ−1/2)x2

=(

2γ − 1

ε

)(1 − ε2

2γ − 1+O(ε4)

)e(γ−1/2)(x2−ε−2)

(7)

3.1. Exact outer solution (Hookean dumbbell)

After the elongational flow is turned on, one can obtain anexact solution to Eq.(5) by assuming that the PDF maintains itsGx jobo d.

s,w lly)b

P

w

〈S lc

T

�

A hen�

3

valentf lp sub-s iabled

he advective and diffusive coefficients for the two modelss follows[20,19,1,15]:

odel A: V (x) =(

γ − 1

2

)x, D(x) = 1

2(2)

odel B: V (x) =(

γ − 1

2

1

1 + kx

1

1 − x2/L2

)x,

D(x) = 1

2

1

1 + kx(3)

ere,γ is an appropriately reduced elongation rate, which mxceed 1/2 in order for the flow to stretch the spring apprecoward its limit of extension. Illustrative numerical calculation this paper will be restricted toγ = 1. The factor 1+ kx

ppearing in the denominators for Model B represents anirical estimate for the increase in the friction coefficient ofumbbell as it becomes extended. The constantk ranges from.02 to

√2/2[15]. No-flux conditions at both ends of the dom

ake care of symmetry and finite extensibility, respectively.

= 0 atx = 0, L (4)

lthough the FENE factor (1− x2/L2)−1 does, in principleender the no-flux condition superfluous at the right end,ondition is retained in the numerical solutions.

The Smoluchowski equation(1) admits simplification in thsymptotic limit of a long dumbbell (L → ∞), for which weefine the small perturbation parameterε = L−1 � 1.

-

aussian form. This will satisfy the symmetry condition(4) at= 0 but not the no-flux condition at the end—which is thef the inner solution to which the Gaussian form is matche

Enforcing normalization of total probability at all timehich includes the portion of the PDF extending (unphysicaeyond the stop of the spring, we use the trial solution

[O](x, t) = 1

�(t)

√2

πexp

{−1

2

[x

�(t)

]2}

(8)

herein�(t) is directly related to the second moment:

x2〉 = �(t)2 (9)

ubstituting Eq.(8) into the PDE(5), we find the ODE and initiaondition for�(t).

d�

dt=(

γ − 1

2

)�(t) + 1

2�(t), �(0) = 1 (10)

he solution is

(t) ={(

1 + 1

2γ − 1

)e(2γ−1)t − 1

2γ − 1

}1/2

(11)

dvection and diffusion are of comparable importance w= ord(1), but advection dominates when� = ord(ε−1).

.2. Contraction to the second moment

Consistent averaging and the Gaussian closure are equior Model A, and recover the exact solution(8). The generarocedure is briefly reviewed here by way of preparing forequent complications associated with the FENE and varrag terms of dumbbell Model B.

18 L.C. Nitsche et al. / J. Non-Newtonian Fluid Mech. 133 (2006) 14–27

We start by integrating the PDE(5)againstx2 over the infiniteinterval 0< x < ∞, which ignores finite extensibility.∫ ∞

0x2∂P

∂tdx = −

∫ ∞

0x2 dJ (12)

The time derivative is pulled outside the first integral, and theflux integral is done by parts to yield an equation directly for thesecond moment of the PDF:

d〈x2〉dt

= −x2J |∞0 +∫ ∞

02xJ dx (13)

The symmetry and no-flux conditions(4) dispose of the upperand lower limits. We now explicitly write the advective and dif-fusive contributions to the fluxJ, and integrate by parts againfor the diffusive term, to obtain

d〈x2〉dt

= 2〈xV (x)〉 + 2

⟨d

dx[xD(x)]

⟩(14)

This is the general starting point for arbitrary forms of the ad-vective and diffusive terms.

With a linear velocity and constant diffusivity, the conforma-tional averages can be closed exactly to obtain

d〈x2〉dt

= (2γ − 1)〈x2〉 + 1, 〈x2〉 = 1 at t = 0 (15)

T eO tf toa

t mer

3

traino le ite ale stopoa

uters dt

A

I

A

For future reference we note the derivative formula

A′(t) = 1

[�(t)]2ε√

2π

[(2γ − 1)�(t) + 1

�(t)

]exp

{ −1

2ε2[�(t)]2

}

(19)

For matching with the subsequent inner solution, we can de-rive one particular relation betweenA(t) and the end value ofthe outer solution directly from the Smoluchowski equation(5).Noting that

A′(t) = −∫ ε−1

0

∂P

∂tdx =

[(γ − 1

2

)xP − 1

2

∂P

∂x

]ε−1

0(20)

we find

P[O]end(t) = ε

γ − 12

{A′(t) + 1

2

∂P

∂x

∣∣∣∣end

}(21)

By the time the end probability builds up to appreciable mag-nitude,∂P/∂x is O(ε2), and so negligible. Before then, the endprobability is negligible. Thus, we have

P[O]end(t) ≈ ε

{A′(t)γ − 1

2

}(22)

3.4. Stretched boundary-layer coordinate

ilityf ap

x

R

ε

O pen-e ameo

3

ibra-t totala ores singa

P

w

τ

T tt in a

here are two routes for calculating�(t). First, one can solve thDE (15) for 〈x2〉(t), and then take�(t) =

√〈x2〉(t) on a pos

acto basis. Alternatively, the ODE(15)can be transformed inn ODE directly for�(t),

d�

dt=[

d〈x2〉d�

]−1

[(2γ − 1)〈x2〉 + 1] = 1

2�[(2γ − 1)�2 + 1]

(16)

hereby recovering Eq.(10). Both approaches give the saesult for�(t).

.3. General features of the outer solution

The Gaussian outer solution does not satisfy the consn maximum extension. This peak is ord(1) in height whixtends over an ord(1) width inx, which characterizes the initiquilibrium distribution. By the time it has spread near thef the spring (x = ε−1), its height has decayed to O(ε), which iss far as we shall go in matching to the inner solution.

As will be seen below, the most important feature of the oolution is the amountA(t) of probability that extends beyonhe stop of the spring:

(t) =∫ ∞

ε−1P [O](x, t) dx = 1 −

∫ ε−1

0P [O](x, t) dx (17)

f the outer solution adopts the Gaussian form(8) then

(t) = erfc

{1

�(t)ε√

2

}(18)

t

Only the stop in the spring prevents the outflux of probabrom the end,x = ε−1. The steady-state solution(7) suggestseak of width of ord(ε). Thus, we define the inner coordinateη,

= ε−1 + εη, −∞ < η ≤ 0 (23)

ewriting the PDE(10) in terms ofη we find

2∂P [I]

∂t+ ∂

∂η

{[(γ − 1

2

)P [I] − 1

2

∂P [I]

∂η

]+ ε2ηP [I]

}= 0

(24)

n the inner lengthscale the advective flux toward the (imtrable) end is balanced by a diffusive counterflux of the srder.

.5. Two-timescale expansion for the inner solution

The scaling of the time derivative suggests a rapid equilion to a pseudo-steady, piled-up boundary layer, with theccumulated probability being determined by what arrives mlowly from the outer solution. This idea can be formalized utwo-timescale expansion for the inner solution:

[I] (η, t; ε) ∼ ε−1P[I]0 (η, τ, T ) + εP

[I]1 (η, τ, T )

+ ε3P[I]2 (η, τ, T ) + · · · (25)

ith the respective fast and slow time variables

= ε−2t, T = t (26)

he two-timescale expansion begins at orderε−1, in order thahe inner solution can accumulate ord(1) probability with

L.C. Nitsche et al. / J. Non-Newtonian Fluid Mech. 133 (2006) 14–27 19

peak of ord(ε) width, and ascends in powers ofε2, since this isthe only form in whichε appears in the inner PDE.

Substituting the expansion(20) into the inner PDE written inboth time variables{

∂P [I]

∂τ+ ∂

∂η

[(γ − 1

2

)P [I] − 1

2

∂P [I]

∂η

]}

+ ε2{

∂P [I]

∂T+ ηP [I]

}= 0 (27)

and collecting terms of like powers ofε, we obtain a hierarchyof PDE’s inη andτ, with the slow time variableT appearing asa parameter.

∂P[I]0

∂τ+(

γ − 1

2

)∂P

[I]0

∂η− 1

2

∂2P[I]0

∂η2 = 0 (28)

∂P[I]1

∂τ+(

γ − 1

2

)∂P

[I]1

∂η− 1

2

∂2P[I]1

∂η2

= −∂P[I]0

∂T−(

γ − 1

2

)∂

∂η

[ηP

[I]0

](29)

The boundary condition forP (I)0 asη → −∞ is obtained by

integrating Eq.(28) over η. In order to avoid accumulation ofprobability on the short timescale we must set

t theh in

on-ined

s be

the

To avoid secular accumulation of probability at first order wemust require that

A′(T ) =[(

γ − 1

2

)P1 − 1

2

∂P1

∂η

]−∞

(34)

In other words, the slow-timescale rate at which probability ac-cumulates within the leading-order inner solution equals the in-coming flux at first order. The latter is of ord(1), which matchesthe advective flux from the leadingouter solution near the bound-ary layer (see Eq.(22)). Thus, we should cut off from the outersolution – a Gaussian peak that expands on the slow-timescale– whatever extends beyond the limit of extension, and rearrangethat amount of probability into a “piled-up” boundary layer usingthe pseudo-steady exponential solution(32). This justifies iden-tifying the inner, slow-timescale functionA(T ) with the outerquantity defined by Eq.(18).

Instead of integrating the second-order equation in order todetermine theT-dependent coefficients appearing inP

[I]1 , we

shall be content with the simpler normalization argument, whichshould give an equivalent result. The quasi-steady solution atfirst order is

P[I] ,∞1 (η, T ) =

[(λ2

2A(T )

)η2 + (2A′(T ))η

+(

2A′(T ) − A(T )

)]eλη + B(T ) (35)

N n in-t est

η

3

m-p nnerl t acw

P

w

0 = ∂

∂τ

∫ ∞

0P

(I)0 (η, τ, T ) dη =

[(γ − 1

2

)P

(I)0 −1

2

∂P(I)0

∂η

]0

−∞(30)

The upper limit vanishes because of the no-flux condition astop of the spring, so it follows that the flux must also vanisthe limit asη → −∞. Thus, we obtain

P[I]0 (η, τ, T ) = P

[I] ,∞0 (η, T ) + {τ-timescale decay terms} (31)

with

P[I] ,∞0 (η, T ) = A(T )λ eλη (λ = 2γ − 1) (32)

The prefactorA(T ) represents precisely the total probability ctained within the leading-order peak, and so it could be obtasimply from an overall normalization: whatever area extendyond the end from the Gaussian outer solution goes intoA(T )as given by Eq.(18). More rigorously, one can integratefirst-order Eq.(29) in η to conclude that

∂

∂τ

∫ 0

−∞P

[I]1 (η, τ, T ) dη

= − ∂

∂T

∫ 0

−∞P

[I]0 (η, τ, T ) dη

−[(

γ − 1

2

)P

[I]1 − 1

2

∂P[I]1

∂η

+(

γ − 1

2

)ηP

[I]0

]0

−∞(33)

-

λ

ote that the bracketed term gives zero net probability wheegrated inη. Matching Eq.(35)to the outer solution determinhe free coefficientB(T ). From Eq.(22)we see that

lim→−∞ εP[I] ,∞1 (η, T ) = εB(T ) = ε

{2A′(T )

λ

}(36)

.6. Uniformly valid asymptotic solution

Constructing the uniformly valid solution is relatively sile in this case, because the common limiting behavior (i

imit of outer solution and outer limit of inner solution) is jusonstant, given by Eq.(22). Combining Eqs.(8), (32)and(35),e find

(x, t) ∼ 1

�(t)

√2

πexp

{−1

2

[x

�(t)

]2}

+ ε−1 erfc

{1

�(t)ε√

2

}

× (2γ − 1) exp

{(2γ − 1)

(x − ε−1

ε

)}

+ ε

{[(2γ − 1)2

2A(t)

](x − ε−1

ε

)2

+ [2A′(t)](

x − ε−1

ε

)

+[

2

2γ − 1A′(t) − A(t)

]}

× exp

{(2γ − 1)

(x − ε−1

ε

)}+ O(ε2) (37)

ith A(t) andA(t) given by Eqs.(18)and(19).

20 L.C. Nitsche et al. / J. Non-Newtonian Fluid Mech. 133 (2006) 14–27

Fig. 1. PDF’s for a linear-locked dumbbell with fixed drag (Eq.(5)). Comparisonof the asymptotic solution(37) with numerical PDFs calculated using ASPH.Here,γ = 1 andε = 0.25.

For elongation rateγ = 1 and a fairly short spring (L = 4,ε = 0.25), Fig. 1 presents a comparison of the asymptotic so-lution (37) with numerical PDF’s obtained by applying a par-ticle method called atomistic smoothed particle hydrodynam-ics (ASPH) to the Smoluchowski equation(5) (see Nitsche andZhang[10] for a general description of the method for two-dimensional advection–diffusion problems). In brief, ASPHtracks a swarm of particles that move deterministically accordingto the local advective flux and also a relative diffusional velocitydue to a suitably formulated mutual repulsion among a localizedsubset of particles. A weight-function average extracts the localdensity of particles to yield the PDF. The calculations reportedhere are essentially the same as those presented in the Ph.D.thesis of Zhang[29]. We used 250 particles in all, with roughly20 particles in each local sum. (For the FENE with variable fric-tion (Fig. 5), the corresponding parameters were 500 and 30.)Even though the spring is not very long, the asymptotic PDF’sare very accurate, and illustrate how probability drains from thecentral Gaussian peak into a boundary layer of approximatelyexponential shape at the limit of extension.

3.7. Second moment

In order to approximate integral properties of the PDF con-sistently at leading order, we need only the first two lines in theacl hici ionst nce[

〈

Tr t the

Fig. 2. Time evolution of the second moment for a linear-locked dumbbell withfixed drag. The asymptotic formula(38)is compared with the numerical solution.Results are also shown for the consistently averaged second-moment Eq.(41).

limit of extension of the spring—as though the exponential pro-file of ord(ε) width were replaced by a delta function centeredatx = ε−1.

P(x, t) ≈ 1

�(t)

√2

πexp

{−1

2

[x

�(t)

]2}

+ erfc

{1

�(t)ε√

2

}δ(x − ε−1) (39)

With ε held fixed, the asymptotic behavior as� → ∞ is

〈x2〉 ∼ ε−2

{1 −

(2

3

√2

π

)1

ε�

}(40)

Fig. 2shows the second moment as a function of time, calcu-lated by substituting Eq.(11) for �(t) in the asymptotic formula(38). This asymptotic curve is in excellent agreement with re-sults calculated from the full numerical solution of the PDE(5)(see Zhang[29]).

The consistently averaged second-moment Eq.(15) suffersfrom a defect when it is applied to the Tanner model: there is noway to incorporate the limit of extension, aside from artificiallystopping the second moment when it runs into the hard stop ofthe spring:

〈x2〉 =(1 + 1

2γ−1

)e(2γ−1)t − 1

2γ−1, t < T

T y, andl rpo-r o theb nitet lead-i g-iw lei arge( tlyl con-

symptotic solution(37), because the ord(ε) inner correctionontributes negligibly when integrated over the ord(ε) boundaryayer. Here, we consider the second moment of the PDF, wndicates the degree of stretching. In two or three dimenshe second moment would yield information on birefringe16,8].

x2〉 ∼{

�2 erf(

1ε�

√2

)− �

ε

√2π

exp(

−12ε2�2

)}central Gaussian

+{(

1ε2 − 2

2γ−1

)erfc

(1

ε�√

2

)}boundary layer

(38)

he leading boundary-layer contribution in Eq.(38)comes fromegarding all of the associated probability concentrated a

h,

ε−2, t ≥ T

T = 1

2γ − 1ln

[2γ − 1 + ε2

2γε2

](41)

he consistently averaged second moment rises too quicklevels off abruptly with a corner—because it does not incoate the continuous drainage of the central Gaussian intoundary layer. The terminal value does not account for fi

hickness of the boundary layer, and represents only theng ε−2 term in Eq.(38). This explains why consistent averang overpredicts the long-time asymptote by 2/(2γ − 1) = 2,hich can be read off fromFig. 2. The discrepancy is noticeab

n this example because the limit of extension is not very lL = ε−1 = 4); it would become unimportant for significanonger dumbbells. Thus, at steady state the deficiency of

L.C. Nitsche et al. / J. Non-Newtonian Fluid Mech. 133 (2006) 14–27 21

sistent averaging (which is equivalent to the Gaussian closurein this simple case) remains largely hidden: only the transientreveals it.

3.8. The L closure: box-spike approximation of the PDF

The width parameter�(t) that specifies the Gaussian peakin Eq. (39) at each instant can be replaced by an equivalentparameter: the (suitably scaled) probability density atx = 0:

Q(t)def= P(0, t)

ε= 1

ε�(t)

√2

π(42)

Eq.(38)can then be converted into a relation between〈x2〉 andQ.

ε2〈x2〉 ∼(

2

πQ2

)erf

(Q

√π

2

)−(

2

πQ

)exp

(−πQ2

4

)

+ erfc

(Q

√π

2

)(43)

The L closure of Lielens and coworkers[8,9,16] involves avery simple but useful model for the time-evolving PDF, whichis suggested by the progression of curves inFig. 1. By the timeP(0, t) has decayed to ord(ε), which meansQ(t) = ord(1), theGaussian core is nearly constant over the domain 0< x < ε−1.Thus, we represent the PDF with a completely flat outer pro-

q.otalyet

≥ 1

≤ Q < 1(44)

p-d

e-bilit

seen

Fig. 3. A simplified box-spike representation of the PDF (Eq.(44)), accordingto the L closure of Lielens et al.[8]. The central peak is modeled as a rectangularpulse, and the boundary layer is modeled as a delta function located at the limit ofextension. Probability starts accumulating in the delta function when the widthof the rectangle reaches the length of the spring.

4. Asymptotics for the FENE dumbbell with variablefriction

New elements in the Smoluchowski equation are: (i) theFENE factor in the spring law, which causes the restoring force toblow up as the spring stretches toward its limit of extension, and(ii) the empirical, configuration-dependent friction factor, whichmodifies the diffusive flux and that part of the advective fluxthat comes from the spring pulling back against the stretchingflow.

∂P

∂t+ ∂

∂x

{(γ − 1

2

1

1 + kx

1

1 − ε2x2

)xP−1

2

1

1 + kx

∂P

∂x

}=0,

0 < x < ε−1 (48)

F hes rousasymptotics (Eqs.(43) and(47)), are compared with the simple approximate

file, which is augmented with the inner delta function from E(39)—the latter being suitably weighted to preserve the tprobability. At earlier times, when the outer solution has notreached the end, one can model the PDF with a rectangle.

PL(x, t) ={

εQ[H(x) − H(x − [εQ]−1)], Q

εQ[H(x) − H(x − ε−1)] + (1 − Q)δ(x − ε−1), 0

with H(x) the Heavyside unit step function. This L closure reresentation of the PDF is shown inFig. 3, and the associatedependence of the second moment uponQ is as follows:

ε2〈x2〉L ={ 1

3Q2 , Q ≥ 1

1 − 2Q3 , Q < 1

(45)

The linear dependence ofε2〈x2〉 uponQ whenQ < 1 matchesthe first two terms in the Taylor expansion of Eq.(43) aboutQ = 0.

Based upon Eqs.(44)and(45), there is a piecewise linear rlationship between the second moment and the total probaA accumulated in the delta function (seeFig. 4).

A ={

0, 0 ≤ ε2〈x2〉 ≤ 13

−12 + 3

2ε2〈x2〉, 13 ≤ ε2〈x2〉 ≤ 1

(46)

LettingQ run from 0 toε−1(2/π)1/2 in Eq.(43)and the suitableversion of Eq.(18),

A = erfc

{Q

√π

2

}(47)

we can obtain a corresponding asymptotic curve, which is

in Fig. 4to validate the simple approximation(46)quite well.

formula (46) resulting from the family of simplified box-spike PDF’s (Fig. 3)from the L closure[8].

y

ig. 4. Amount of probabilityA accumulated in the boundary layer vs. tcaled second moment for the linear-locked dumbbell with fixed drag. Rigo

22 L.C. Nitsche et al. / J. Non-Newtonian Fluid Mech. 133 (2006) 14–27

The FENE factor brings the limit of extension explicitly into thePDE, and nonlinear terms will require closure assumptions ofsome sort.

The initial equilibrium PDF is

P(x, 0) ={∫ ε−1

0(1 − ε2x2)1/2ε2

dx

}−1

(1 − ε2x2)1/2ε2(49)

and the final steady-state PDF is

P(x, ∞) ={∫ ε−1

0(1 − ε2x2)1/2ε2

eγx2(3+2kx)/3 dx

}−1

× (1 − ε2x2)1/2ε2eγx2(3+2kx)/3 (50)

Here, we will proceed less systematically than with dumbbellModel A, and develop first only as much of the boundary-layerstructure as is required to motivate the L closure.

4.1. Stretched boundary-layer coordinate

The FENE factor in Eq.(48)causes probability to accumulatenear that pointx = λ where the advective velocity vanishes:

V (λ; ε)

λ= γ − 1

2

(1

1 + kλ

)(1

1 − ε2λ2

)= 0 (51)

s

V1(η) = [4γ (1 − γ)]η − [16k2γ3]η2 (60)

D0(η) =[

1

2k

](61)

D1(η) =[

1 − 4γ

8k2γ

](62)

At leading order we have a linear velocity (equivalently, a har-monic binding potential) and a constant diffusivity. On this basisone would expect the inner solution to be a Gaussian peak.

4.2. Two-timescale expansion for the inner solution

As suggested by theε factor multiplying the time derivativein Eq.(48), we use the two-timescale expansion

P [I] (η, t; ε) ∼ ε−1P[I]0 (η, τ, T ) + P

[I]1 (η, τ, T )

+ εP[I]2 (η, τ, T ) + · · · (63)

with the fast and slow time variables

τ = ε−1t, T = t (64)

Note that the disparity between timescales is one order lowerthan for dumbbell Model A (cf. Eq.(26)).

Substituting the expansion(63) into the inner PDE written inboth time variables,{

a r-ca

Ui d tob

P

w thel

A

An algebraic perturbation yields

λ(ε) ∼ 1

ε−(

1

4kγ

)+(

8γ − 3

32k2γ2

)ε −

((1−2γ)2

16k3γ3

)ε2 + O(ε3)

(52)

The width of the final steady-state peak(50) scales withε,which suggests the inner-variable scaling

x = λ + ηε (53)

Written in terms ofη, the Smoluchowski equation(48)become

ε∂P

∂t+ ∂

∂η

[V (η; ε)P − D(η; ε)

∂P

∂η

]= 0 (54)

with the advective and diffusive coefficients

V (η; ε) ={

γ − 1

2

[1

1 + k(λ + ηε)

] [1

1 − ε2(λ + ηε)2

]}× (λ + ηε) (55)

D(η; ε) = 1

2ε

[1

1 + k(λ + ηε)

](56)

Local expansions of these functions aboutη = 0 take the form

V (η; ε) = V0(η) + εV1(η) + O(ε2) (57)

D(η; ε) = D0(η) + εD1(η) + O(ε2) (58)

with

V0(η) = [−4kγ2]η (59)

∂P [I]

∂τ+ ∂

∂η

[V0(η)P [I] − D0

∂P [I]

∂η

]}

+ ε

{∂P [I]

∂T+ ∂

∂η

[V1(η)P [I] − D1

∂P [I]

∂η

]}+ · · · = 0

(65)

nd collecting terms of like powers ofε, we obtain again a hierahy of PDE’s inη andτ, with the slow time variableT appearings a parameter.

∂P[I]0

∂τ+ ∂

∂η

[V0(η)P [I]

0 − D0(η)∂P

[I]0

∂η

]= 0 (66)

∂P[I]1

∂τ+ ∂

∂η

[V0(η)P [I]

1 − D0(η)∂P

[I]1

∂η

]

= −∂P[I]0

∂T− ∂

∂η

[V1(η)P [I]

0 − D1∂P

[I]0

∂η

](67)

sing the coefficientsV0(η) andD0(η) from Eqs.(59)and(61)n Eq. (66), the leading-order quasi-steady solution is foune a symmetric Gaussian peak.

[I] ,∞0 (η, T ) = A(T )

(2kγ√

π

)e−(2kγη)2 (68)

hereA(T ) represents the total probability contained withineading-order peak, as before. The analog of Eq.(34) is

′(T ) =[V0(η)P [I]

1 − D0∂P

[I]1

∂η

]−∞

(69)

L.C. Nitsche et al. / J. Non-Newtonian Fluid Mech. 133 (2006) 14–27 23

Fig. 5. Numerical PDF’s for a FENE dumbbell with variable drag (Eq.(48)).Here,γ = 1, ε = 0.25 andk = √

2/2.

From this we find the asymptotic decay of the inner solution:

P[I] ,∞1 (η, T ) ∼ −

[A′(T )

4kγ2

]1

ηasη → ∞ (70)

That P [I] ,∞1 should vanish in the outer limit could have been

anticipated from the fact that there is no ord(1) contributionin the inner limit of the outer solution. The zeroth-order innersolutionP

[I] ,∞0 (η, T ) suffices for calculating integral properties

at leading order.

4.3. Summary of the inner solution

Thus far, we have uncovered a boundary-layer structure thatis similar to that for dumbbell Model A, in that it is a peak oford(ε) width. The main differences are:

• Its Gaussian shape(68), as opposed to the asymmetric expo-nential profile(32).

• Its location centered atx = λ (Eq.(52)), slightly inboard fromthe limit of extensionx = ε−1.

• A smaller disparity between the inner and outer timescales(cf. Eqs.(26)and(64)).

These features (confirmed by the numerical PDF’s shown inFig. 5) will not hinder applicability of the L closure.

4

as

b ngeb olvef liedi ,a emb brai

Fig. 6. Time evolution of the second moment for a FENE dumbbell with variabledrag, corresponding toFig. 5. Numerical results are compared with the Peterlin-type closure(74)and the L closure[8] (Eqs.(71)and(81)).

operations:

〈1〉 def=⟨

x2

1 + kx

⟩≈ 〈x2〉

1 + k√

〈x2〉(72)

〈2〉 def=⟨

x2

(1 + kx)(1 − ε2x2)

⟩≈ 〈x2〉

(1 + k√

〈x2〉)(1 − ε2〈x2〉)(73)

More expedient than mathematically justifiable, this approachhas been widely used in polymer kinetic theory. In any case, theODE for the second moment is now closed:

d〈x2〉dt

= 2γ〈x2〉 − 〈x2〉(1 + k

√〈x2〉)(1 − ε2〈x2〉)

+ 1

1 + k√

〈x2〉(74)

Fig. 6compares the time dependence of the second moment ascalculated from this ODE with the numerical results. Althoughthe FENE factor causes〈x2〉 to level off of its own accord (incontrast to the consistently averaged curve inFig. 2), the inter-mediate rise is still much too rapid, and the curve again bendstoo abruptly toward the final steady asymptote. The obvious cul-prits are the closure assumptions(72) and(73). Their validityis not a matter of speculation or convenience: it depends on theactual PDF’s, and can be checked directly against the numericals ni esca s theai ywt ls bbell.T deledw oper-t ctuala nexts

.4. A Peterlin-type closure

Compared with the exactly solvable case of Eq.(5), the FENEnd configuration-dependent drag factors in Eq.(48) bring aignificant complication. The ODE(14) is no longer closed

d〈x2〉dt

= 2γ〈x2〉 −⟨

x2

(1 + kx)(1 − ε2x2)

⟩+ 1

〈x2〉⟨

x2

1 + kx

⟩(71)

ecause the averages of the velocity and diffusivity can no loe evaluated directly in terms of the second moment being s

or. Additional information is required, and this is often suppn the form of the Peterlin approximation[1,7,15]—an assumedpproximate interchangeability of the order of taking the ensle average (integral against the PDF) and nonlinear alge

rd

-c

olution of the Smoluchowski equation(48). This comparisos made inFig. 7andFig. 8. Each point on the numerical curvorresponds to a particular time in the family of PDF’s (Fig. 5),t which the second moment was calculated and plotted abscissa. While the first Peterlin-type closure assumption(72)

s surprisingly accurate, the second one(73) gives an entirelrong shape at large extensions. Nevertheless, Eq.(73) is seen

o be accurate at one special point:〈x2〉=λ2, which is the finateady-state (at leading order) for the stretched FENE dumhat is why steady-state properties can be accurately moith the Peterlin-type closure, even though the transient pr

ies are poorly predicted. This fortunate intersection of the and Peterlin curves is not coincidental, as will be seen in theection.

24 L.C. Nitsche et al. / J. Non-Newtonian Fluid Mech. 133 (2006) 14–27

4.5. The L closure

As with the linear-locked dumbbell, a single Gaussian peakcannot sufficiently capture the transient structure of the actualPDF: a spreading central core that drains into an accumulatingboundary layer near the limit of extension. Thus, we are led toconsider a slightly modified version of Eq.(44), in which thelocation of the inner spike is suitably modified according to Eq.(52).

PL(x, t) ={

λ−1Q[H(x) − H(x − λQ−1)], Q ≥ 1

λ−1Q[H(x) − H(x − λ)] + (1 − Q)δ(x − λ), 0 ≤ Q < 1(75)

This simple form allows the integrals on the left-hand sides ofEqs.(72)and(73)to be evaluated analytically, and gives a lineardependence uponQ whenQ < 1.

〈j〉L ={

αj(λQ−1), Q ≥ 1

βj(λ) + [αj(λ) − βj(λ)]Q, 0 ≤ Q < 1(76)

with

α1(x) = 1

xk3

{ln(1 + kx) − kx + k2x2

2

}(77)

β1(x) = x2

1 + kx(78)

α

β

U oo

〈λ3

N thc

〈

〈Tv hd lsmc6

c ta intp nlf a

the Peterlin curve will not level offexactly at〈x2〉 = λ2, becausethe steady-state version of the ODE(74)

γ − 1

2

(1

1 + k√

〈x2〉

)(1

1 − ε2〈x2〉)

(83)

+ 1

2〈x2〉 ×(

1

1 + k√

〈x2〉

)= 0

adds to the two convective terms from Eq.(51)a third, diffusiveterm of orderε2. As the final steady state is slightly displacedfrom intersection between the Peterlin and L closure curves inFigs. 7 and 8, the two approximations will have diverged fromeach other. This is why the long-time asymptotes inFig. 6differmarginally.

For the boundary-layer accumulation we note the followinganalog of Eq.(46), which can be derived from the box-spikeshape of the PDF:(75).

A ={

0, 0 ≤ 〈x2〉 ≤ λ2

3

−12 + 3

2〈x2〉λ2 , λ2

3 ≤ 〈x2〉 ≤ λ2(84)

Finally, in view of Eqs.(52), (77)–(80), the linear portion of thep-

tions

)

tri-

bell

)

)

2(x) = 1

x

{ln(1 + kx)

k(k2 − ε2)− ln(1 + εx)

2ε2(k − ε)− ln(1 − εx)

2ε2(k + ε)

}(79)

2(x) = x2

(1 + kx)(1 − ε2x2)(80)

pon relatingQ to the second moment by the applicable analf Eq.(45), we find the closure relation

j〉L =

αj(√

3〈x2〉), 0 ≤ 〈x2〉 ≤32

[1 − 〈x2〉

λ2

]αj(λ) +

[−1

2 + 32

〈x2〉λ2

]βj(λ), λ2

3 ≤ 〈x2〉 ≤

ote in particular the two endpoints of the linear segment ofurve:

j〉L = αj(λ) when〈x2〉 = 1

3λ2,

j〉L = βj(λ) when〈x2〉 = λ2 (82)

hese approximations are plotted inFigs. 7 and 8, and agreeery closely with the numerical results, especially where tumbbell is appreciably stretched out. Substituting the new cure relations(81)into the second-moment ODE(71), we obtainuch more accurate results for the transient behavior of〈x2〉, as

ompared with the Peterlin-type closures(72)and(73)(seeFig.).

Eq.(81)happens to coincideprecisely with the Peterlin-typelosures(72)and(73)when〈x2〉 = λ2, because at this one poinll of the probability has drained into the delta function spike

he simplified PDF(75) (cf. Eqs.(78), (80) and(82)). This ex-lains the fortuitous success of the Peterlin-type closures (o

or thesteady-state properties. It should be noted, however, th

g

2

λ2(81)

e

eo-

y)t

closure relation(81) yields the following leading-order asymtotic forms asε → 0:⟨

x2

1 + kx

⟩∼ 1

kε

[1

4+ 3

4ε2〈x2〉

](85)

⟨x2

(1 + kx)(1 − ε2x2)

⟩∼ 2γ

ε2

[−1

2+ 3

2ε2〈x2〉

](86)

for 1/3 ≤ ε2〈x2〉 ≤ 1.

5. Transient stresses

In order to compute the stress, the Smoluchowski equa(5) and(48)can be summarized in the encompassing form

∂P

∂t+ ∂

∂x

{P

[γx − 1

2

F{P}(x)

ζ(x)

]}= 0 (87

where the driving forceF{P}(x) consists of an enthalpic conbution (spring) and an entropic contribution (diffusion):

F{P}(x) = f (x) + ∂ ln P

∂x(88)

The spring laws and friction coefficients for the two dumbmodels are

f (x) = x, ζ(x) = 1 (Model A) (89

f (x) = x(1 − ε2x2)−1, ζ(x) = 1 + kx (Model B) (90

L.C. Nitsche et al. / J. Non-Newtonian Fluid Mech. 133 (2006) 14–27 25

Fig. 7. Averages of the diffusion term in the second-moment ODE(71). Integralsof the numerical PDF’s fromFig. 5are compared with the Peterlin-type closure(72)and the L closure(81).

The contribution of one dumbbell to the stress in the polymersolution is given by a weighted average of the driving force,

τ(t) =∫ ε−1

0xPF{P}(x) dx (91)

The factorxP in this integral is proportional to the number den-sity of dumbbells stretched to lengthx that cut across a givenplane.

5.1. Linear-locked dumbbell with fixed friction

In the case of the linear-locked dumbbell there is a contribu-tion to the stress from the hard stop of the spring atx = ε−1.This can be seen by adding to the driving force(88)a force termdue to an infinitely high, soft potential barrier operative on avery short lengthscale that is negligible compared to the ord(ε)boundary-layer thickness described in Section3.4.

F{P}(x) = x + ∂ ln P

∂x+ dΦ

dx(92)

For this vanishingly thin potential zone an inner/two-time anal-ysis similar to that presented in Sections3.5and4.2shows thatthe near-wall PDF rapidly equilibrates to the Boltzmann distri-bution

P(x, t) ∼ P(ε−1, t) e−Φ(x) (93)

I

ε

a

E aineb

ε

Fig. 8. Averages of the spring term in the second-moment ODE(71). Integralsof the numerical PDF’s fromFig. 5are compared with the Peterlin-type closure(73)and the L closure(81).

against the reduced second moment from Eq.(43). The corre-sponding (piecewise-linear) formula for the L closure resultsfrom inserting Eq.(46) into (94):

ε2τ ={

ε2〈x2〉, 0 ≤ ε2〈x2〉 ≤ 13

(1 − 2γ) + (6γ − 2)ε2〈x2〉, 13 ≤ ε2〈x2〉 ≤ 1

(96)

These two curves are compared inFig. 9, for γ = 1.

5.2. FENE dumbbell with variable friction

Since the probability density vanishes at the limit of extension(where the nonlinear Warner restoring force blows up), the one-dimensional version of the standard Kramers formula[1,7–9,16]applies for the stress.

τ(t) =∫ ε−1

0P(x, t)xf (x) dx − 1 (97)

The boundary-layer contribution to the stress dominates for theFENE dumbbell with variable friction. Nearx = λ the spring-force factor is

xf (x) = x2

1 − ε2x2 = 2kγ

ε3 + O(ε−2) (98)

F dragi

ntegratingxP against the new driving force gives2τ(t) = ε2(〈x2〉 − 1) + 2εP(ε−1, t) ∼ ε2〈x2〉 + (4γ − 2)A(t)

(94)

t leading order.Running through the intermediate parameterQ defined by

q.(42), the asymptotic stress–extension curve can be obty plotting the reduced stress,

2τ ∼(

2

πQ2

)erf

(Q

√π

2

)−(

2

πQ

)exp

(−πQ2

4

)

+ (4γ − 1) erfc

(Q

√π

2

)(95)

d

ig. 9. Stress–extension curves for the linear-locked dumbbell with fixedn suddenly started uniaxial extension.

26 L.C. Nitsche et al. / J. Non-Newtonian Fluid Mech. 133 (2006) 14–27

Fig. 10. Stress–extension curves for the FENE dumbbell with variable drag insuddenly started uniaxial extension. Here,γ = 1, ε = 0.25 andk = √

2/2.

The leading-order stress is then found to be

ε2τ(t) ∼ 2kγ

εA(t) (99)

with A given in terms of〈x2〉 by Eq.(84). Thus, the L closureyields the following stress–extension curve:

ε2τ ∼

0, 0 ≤ 〈x2〉 ≤ λ2

32kγε

[−1

2 + 32

〈x2〉λ2

], λ2

3 ≤ 〈x2〉 ≤ λ2(100)

with λ being given by Eq.(52). Fig. 10shows that this analyticalformula compares much better with the corresponding curvefrom the numerical solution than does the Peterlin closure.

Note the ord(ε−1) scaling in Eq.(100), whereby stresses canbuild up to much higher values than for the linear-locked dumb-bell with fixed friction, when the extensibility parameter is large(cf. Eq.(96)).

6. Concluding remarks

This paper has applied the L closure of Lielens and cowork-ers[8,9,16], which was originally developed for FENE dumb-bells with fixed friction, to two alternative dumbbell modelsof polymers in a dilute solution undergoing uniaxial extension:a linear-locked dumbbell with fixed friction (Model A) and aFENE dumbbell with variable friction (Model B). The two mainf clos stal hod[ en-s readi getss thes lityd inald ric-ti withv ido ectiod ility

near the end within a thin boundary layer. There is a separationof timescales, whereby the boundary layer equilibrates rapidlyto the relatively slow outflux of probability toward the end. Toaddress this behavior, singular perturbation analysis was com-bined with the method of multiple scales. The location of theboundary layer (terminal spike) remains fixed as the dumbbellgets stretched, which fixes one of the two state variables in theL closure, and thereby reduces it to first order.

For dumbbell Model A we have obtained a uniformly validasymptotic approximation for the full PDF, which compares wellwith the numerical solution even for a relatively short dumb-bell. The outer solution (spreading central core of probability)is exactly a Gaussian distribution, which is not constrained bythe hard stop in the spring and (in the absence of the boundarylayer) would allow a finite probability beyond the limit of exten-sion. A Taylor expansion of this Gaussian profile leads preciselyto the rectangular portion of the L closure PDF. The inner solu-tion takes whatever probabilitywould have accumulated beyondthe limit of extension, and piles it up in an asymptotically nar-row, exponential (Boltzmann) distribution against the stop in thespring. The finite probability density at the end contributes tothe stress.

For dumbbell Model B we have pursued the asymptotics farenough to establish the leading-order behavior of the inner solu-tion, which is an asymptotically narrow Gaussian peak centeredinboard of the limit of extension, precisely where the spring’sr nga-t tingt con-f mustb s, thes ble,a gimeo ainstn ppli-c tea ningt icht glys ions.

limito blya ls, tow el Bc inedb

theL rlinc bbellw tiont sients ee ura el—u nds ENEd an

eatures of the simplified PDF shape associated with the Lure have been examined with reference to the full PDF, as eished with: (i) numerical solution via an atomistic SPH met10,29]and (ii) asymptotic analysis for the limit of a large extibility parameter at fixed elongation rate. Flatness of the spng central core of probability density, when the dumbbellignificantly extended, justifies the rectangular portion ofimplified PDF. An asymptotically narrow spike in probabiensity at or near the limit of extension motivates the termelta function. It is noted that the FENE dumbbell with fixed f

ion doesnot possess the required asymptotic structure[29], ands therefore less amenable to the L closure than the FENEariable friction. Only with increased frictional grip of the flun the dumbbell as the latter becomes extended does advominate over diffusion to confine the accumulated probab

-b-

-

n

estoring force balances the stretching influence of the eloional flow, and the advective velocity vanishes. In contrache Smoluchowski equation to its second-moment form, theormational averages of two specific nonlinear expressionse closed in terms of the second moment. For these quantitieimplified, L closure shape of the PDF is analytically tractand yields closure functions that are actually linear in the ref significant stretching. These closure laws (validated agumerical results based upon the full PDF) show why the aable form of the Peterlin closure[13,15,7,8]should be accurat the final steady state, while seriously missing the interve

ransients. In the limit of a large extensibility parameter (in whhe L closure is asymptotically justified), we obtain appealinimple expressions for the coefficients in the closure funct

Example calculations for a moderate (dimensionless)f extension (L = 4) show the L closure to yield reasonaccurate stress–extension curves for both dumbbell modehich it had previously not been applied. Stresses for Modan reach much higher levels than for Model A, as is explay the asymptotic scaling.

Finally, as an effort toward completeness, we compareclosure to two simpler modifications of the original Pete

losure, which have thus far been applied to the FENE dumith fixed friction (as opposed to the case of variable fric

reated here) in an attempt to improve the predictions of trantresses and hysteresis. The FENE-P∗ closure[22] regarded thxtensibility parameter and relaxation time (equivalently, oLndγ) as adjustable fitting parameters in the FENE-P modsing new valuesL′ and γ ′ to match the shear modulus ateady zero-shear-rate viscosity of the corresponding full Fescribed by givenL andγ. Thereby,L′ was reduced to less th

L.C. Nitsche et al. / J. Non-Newtonian Fluid Mech. 133 (2006) 14–27 27

66% ofL. Improvements in the predictions of initial build-up ofthe elongational viscosity were offset by terminal asymptotesthat were much too low. This trade-off highlights the limitationof lumping the outflowing probability together (effectively ina delta function spike), so that it must arrive all at once nearthe limit of extension—however that parameter is chosen. Thesecond-order FENE-P2 closure[8] distributed the total probabil-ity between two delta function spikes: (i) the movable peak fromthe FENE-P and (ii) an additional spike confined to zero exten-sion. This simple representation of the PDF was, in principle,able to represent the drainage of probability from a grossly styl-ized central core into an accumulating peak at larger extensions.Evolution equations for the second and fourth moments wereclosed in terms of the two independent shape parameters (loca-tion α of, and amountβ of probability in, the movable spike).In implementing this idea, however, an initially unexpected de-generacy surfaced, whereby the two moments (state variables)ended up being connected by an additional constraint. Thus,the FENE-P2 approach reduced to a first-order FENE-P closurewith a reduced extensibility parameter, which made it essentiallyequivalent to the FENE-P∗.

Throughout the start-up of elongational flow, our asymptoticanalysis shows the location of the boundary layer (stylized inthe L closure as a delta function spike) to remain fixed whilethe time-evolving PDF drains probability from the central core(stylized as a rectangle) into the boundary layer. Thus, the Lc mom for tF

A

ni-v er-s l fort

R

oly-New

nder

nger,

ows,

ly-315

c-319

mb-

ap-, J.

[9] G. Lielens, R. Keunings, V. Legat, The FENE-L and FENE-LS closureapproximations to the kinetic theory of finitely extensible dumbbells, J.Non-Newtonian Fluid Mech. 87 (1999) 179–196.

[10] L.C. Nitsche, W. Zhang, Atomistic SPH and a link between diffusion andinterfacial tension, AIChE J. 48 (2002) 201–211.

[11] H.C. Ottinger, A model of dilute polymer solutions with hydrodynamicinteraction and finite extensibility. I. Basic equations and series expansions,J. Non-Newtonian Fluid Mech. 26 (1987) 207–246.

[12] H.C.Ottinger, Stochastic Processes in Polymeric Fluids, first ed., Springer,Berlin, 1996.

[13] A. Peterlin, Hydrodynamics of macromolecules in a velocity field withlongitudinal gradient, J. Polym. Sci. Polym. Lett. 4B (1966) 287–291.

[14] J.M. Rallison, E.J. Hinch, Do we understand the physics in the constitutiveequation? J. Non-Newtonian Fluid Mech. 29 (1988) 37–55.

[15] P. Singh, L.G. Leal, Computational studies of the FENE dumbbell modelwith conformation-dependent friction in a co-rotating two-roll mill, J. Non-Newtonian Fluid Mech. 67 (1996) 137–178.

[16] R. Sizaire, G. Lielens, I. Jaumain, R. Keunings, V. Legat, On the hystereticbehaviour of dilute polymer solutions in relaxation following extensionalflow, J. Non-Newtonian Fluid Mech. 82 (1999) 233–253.

[17] A.J. Szeri, A deformation tensor model for nonlinear rheology of FENEpolymer solutions, J. Non-Newtonian Fluid Mech. 92 (2000) 1–25.

[18] R.I. Tanner, Stresses in dilute solutions of bead-nonlinear-spring macro-molecules. II. Unsteady flows and approximate constitutive relations,Trans. Soc. Rheol. 19 (1975) 37–65.

[19] R.I. Tanner, Stresses in dilute solutions of bead-nonlinear-spring macro-molecules. III. Friction coefficient varying with dumbbell extension, Trans.Soc. Rheol. 19 (1975) 557–582.

[20] R.I. Tanner, Engineering Rheology, second ed., Oxford University Press,New York, 2000.

[21] R.I. Tanner, W. Stehrenberger, Stresses in dilute solutions of bead-ws, J.

bead-ws, J.

[ tionh. 75

[ jec-.N.vol.

[ s of379–

[ on fornian

[ n of

[ thon-

[ s in

[ nom-niver-

[ od-004)

[ las-956)

[ iousn, J.

losure also reduces to first order. But the resulting secondents and stresses are substantially more accurate thanENE-P2 [8].

cknowledgements

LCN would like to thank Professor E.J. Hinch, DAMTP, Uersity of Cambridge, UK, for a series of very helpful convations during the early stages of this work. LEW is gratefuhe support of the 3M Company.

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