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Amine AMMAR - Effect of the inverse Langevin approximation on the solution of the
Fokker-Planck equation of non-linear dilute polymer - Journal of Non-Newtonian Fluid Mechanics n°231,
p.1-5 - 2016
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ContentslistsavailableatScienceDirect
Journal
of
Non-Newtonian
Fluid
Mechanics
journalhomepage:www.elsevier.com/locate/jnnfm
Effect
of
the
inverse
Langevin
approximation
on
the
solution
of
the
Fokker–Planck
equation
of
non-linear
dilute
polymer
Amine
Ammar
a,b,∗a LAMPA, Arts et Métiers ParisTech, 2 Boulevard du Ronceray, BP 93525, F-49035 Angers Cedex 01, France b UMSSDT, ENSIT, Université de Tunis, 5 Avenue Taha Hussien, Monteury 1008, Tunis, Tunisia
a
r
t
i
c
l
e
i
n
f
o
Article history:
Received 11 November 2015 Revised 11 February 2016 Accepted 12 February 2016 Available online 23 February 2016
Keywords:
Polymer kinetic theory Langevin function Inverse approximation
a
b
s
t
r
a
c
t
TheLangevinfunctionisdefinedbyL(x)=coth(x)− 1/x.Itsinverseisusefulformanyapplicationsand especiallyforpolymerscience.Astheinverseexactexpressionhasnoanalyticrepresentation,many ap-proximationshavebeen established.Themostfamous approximationis theone traditionallyused for thefinitelyextensiblenon-linearelastic(FENE)dumbbellmodel inwhichthe inverseisapproximated byL−1(y)=3y/(1− y2).Recently MartinKrögerhaspublishedapaperentitled‘Simple,admissibleand accurateapproximationsoftheinverseLangevinandBrillouinfunctions,relevantforstrongpolymer de-formation andflows’ (Kröger,2015)inwhichheproposed approximationswith veryreducederrorin relationtothenumericinverseoftheLangevinfunction.Thequestionweaimtoanalyzeinthisshort communicationis:whenoneusesthetraditionalapproximationratherthanthemoreaccurateone pro-posedbyKrögeristhatreallysignificantregardingthevalueoftheprobabilitydistributionfunction(PDF) intheframeworkofakinetictheorysimulation?Ifyeswhenwemovetotheupperscalebyevaluating thevalueofthestress,canweobserveasignificantdifference?
Bymakingsomesimple1Dsimulationsinhomogeneousextensionalflowitisdemonstratedinthis shortcommunicationthatthePDFpredictionwithinkinetictheoryframeworkaswellasthemacroscopic stressvaluearebothaffectedbythequalityoftheapproximation.
© 2016ElsevierB.V.Allrightsreserved.
1. Introduction
TheLangevinfunctionisdefinedas
L
(
x)
=coth(
x)
− 1/x (1)The inverse Langevin function is usedin rheology of polymer suspension andin themolecular stress function theory.Itresults fromthenon-Gaussianstatisticaltheoryofrubberelasticityasthe entropic force developed by polymer chains. When chain length approaches its maximal value corresponding to a fully stretched state, the chain force tends to infinity which impliesasymptotic behavioroftheinversefunctionnearthevaluey=1.The inverse Langevin function cannot be represented inan explicit form and necessitates an approximation using some series that uses non-rationalorrationalfunctions. Forthat, anaccurate approximation ofthisfunctioncan bebasedona highorderseriesexpansion.A highorderseriesexpansionisaneasywaytocharacterizea func-tion that cannot beexpressed ina closedform. There isa corre-lation betweenthenumber ofexpansion termsand theaccuracy
∗ Correspondence address: LAMPA, Arts et Métiers ParisTech, 2 Boulevard du Ron-
ceray, BP 93525, F-49035 Angers Cedex 01, France. Tel.: +33 621011302.
E-mail address: Amine.Ammar@ensam.eu
relatedtotheconvergencerate.Theexactvaluecanbesometimes difficulttoattain.Anexampleofsuchapproachescanbefoundin
[5]wherethe inverseLangevin function isrepresented by a Tay-lorseriesexpansionaround y=0basedonthe firstfournonzero coefficients L−1Kuh
(
y)
=3y+9 5y 3+297 175y 5+1539 875y 7+O(
y9)
(see[5]) (2)Near the singularity point y=1 the Taylor series cannot still accuratelydescribe thebehavioroftheinverseLangevin function. In this case, an approximation by a rational function defined as a fractionofpolynomials can be advantageous since itis able to reproducetheasymptoticbehavior.Some examplesof approxima-tionsthatcanbefoundintheliteraturearelistedbelow[6,7]:
L−1Tre
(
y)
= 3y 1+0.2y6− 0.6y2− 0.2y(see[7]) (3) L−1Puso(
y)
= 3y 1− y3 (see[6]) (4) L−1Coh(
y)
=y3− y 2 1− y2 (see[2]) (5) http://dx.doi.org/10.1016/j.jnnfm.2016.02.008 0377-0257/© 2016 Elsevier B.V. All rights reserved.2 A. Ammar / Journal of Non-Newtonian Fluid Mechanics 231 (2016) 1–5
The most famous approximation is the one introduced by Warnerin1972[8]traditionallycalledtheFENEapproximation.
L−1FENE
(
y)
=(
3y1− y2
)
(6)In thisworkwe aregoing tofocusourattentiononthe tradi-tionalFENEapproximationwhichwillbereferredtointheresults as‘FENE’.Ourattention willalso be focused on thetwo approx-imations proposed by Körger in [4] that will be respectively re-ferredtoas‘Kr1’and‘Kr2’ L−1Kr1
(
y)
=(
3y 1− y2)(
1+y2/2)
(7) L−1Kr2(
y)
=3y− y 5(
6y2+y4− 2y6)
(
1− y2)
(8)Cohenapproximation[2]hadbeenchosenfromthementioned fourapproximationstobeaddedtoourillustrationsbecauseithas thebestperformancefromthissetofEqs.(2)–(5)(asdiscussedin thementionedRef.[4]).Thispaperisorganizedasfollows:firstthe mainequationsofthekinetic theoryframeworkarerecalled,then theresolutiontechniqueisbrieflydescribedandfinallyresultsare discussed.
2. Polymerkinetictheoryequations
Thenon-rigiddumbbellmodelconsistsoftwobeadsconnected byaspringconnector.Thebeadservesasaninteractionpointwith thesolvent,andthespringcontainsthe localstiffnessdepending onlocalstretching(see[1]formoredetails).
Thedynamicofthechainisgovernedbyhydrostatic,Brownian, andconnectorforces.If wedenote by r˙1 andr˙2 thevelocities of thetwo beads located atpositions r1 andr2, thesethree contri-butionscan beeasily identified inthethree termsofeachof the followingequation: −
ζ
(
r˙2−v
0−κ
· r2)
− kBT∂
∂
r2(
ln)
− Fc=0 (9) −ζ
(
r˙1−v
0−κ
· r1)
− kBT∂
∂
r1(
ln)
+Fc=0 (10)where
ζ
isthedragcoefficient,v
isthevelocityfield,v
0isan aver-agevelocity,κ
isthevelocitygradienttensor(κ
i j=∂
v
i/∂
xj),kB is theBoltzmannconstant,Tistheabsolutetemperatureand(x)is theprobabilitydistributionfunctionforadumbbellconnector vec-torx=r2− r1.FromEqs.(9)and(10)wecanderivethefollowing equation: ˙ x=
κ
· x−2ζ
kBT∂
∂
x(
ln)
+F c(
x)
(11) Theconnectorforcecantakedifferentformsleadingtodifferent kineticmodels.Theconnectorforceisgivenby:Fc
(
x)
=h 3L−1 x x0 x (12)wherex=
|
x|
,his thespringcoefficient andx0 isthe maximum springlength.Aparticularityofthismodelisthatthereisno clo-sureapproximation abletosubstitute themicroscopicdescription byanequivalentconstitutivemacroscopicequation[3].The associ-atedevolutionofthedistributionfunctioncanbewrittenas:∂
∂
t =−∂
∂
x·κ
· x−2ζ
Fc(
x)
+2kBT
ζ
∂
2∂
x2 (13)The problemdefinedbyEq.(13)hasacharacteristicrelaxation time
θ
=ζ
/4h and a dimensionless finite extensibility parame-ter b=hx20/kBT. Thus vector x can be made dimensionless with
kBT/h,
κ
with1/θ
(soitcanbeviewedasaWeissenbergnumberWe),timewith
θ
andthepolymerstresstensorwithnckBT wherencisthenumberofchainsinaunitvolume.Consequently,the
di-mensionlessformofproblem(13)writes:
∂
∂
t =−∂
∂
x·κ
· x−1 2H(
x)
x+1 2
∂
2∂
x2 (14)whereH(x)becomesthedimensionlessconnectorforce,thatinthe FENEmodelresults:
HFENE
(
x)
=1
1− x2/b (15)
In order to take into account the Kröger approximations this expressionbecomes HKr1
(
x)
= 1(
1− x2/b)(
1+x2/(
2b))
(16) or HKr2(
x)
= 1− 1 15(
6x2/b+x4/b2− 2x6/b3)
1− x2/b (17)AndinthesamewaytheCohenapproximationgivesthefollowing function:
HCoh
(
x)
=1− x2/
(
3b)
1− x2/b (18)
To be able to evaluate exactly the accuracy of these approxima-tionswearegoingtoestablishareferencesolutionusingtheexact inverseoftheLangevinfunction.InthiscasethefunctionHExact(x)
will be calculated using a Newton’s method. Another alternative consiststousethe analytic(but longexpression)foran excellent approximationtotheexactinverseLangevinwhichisin[4].These twoapproachesgivesexactlythesameresults.
Moreover, a normalization condition is associated with the probabilitydistribution:
(
x)
dx=1 (19)Finally,therelationbetweenstatisticaldistributionofdumbbell configurations andthe polymerstress
τ
p is provided by Kramersexpression[1]
τ
p=H(
x)
xx− I=(
x)(
H(
x)
xx)
dx− I. (20)Ibeingtheunittensor(takesthevalue1inthe1Dcase). WemustnoticewithrespecttoEq.(14)that thisequation de-fines thetime evolution ofthe distribution function,whose inte-gration requiresto specifytheinitial distributiondenoted by
0. Thus, a reasonablechoice liesintakingas initialdistribution the equilibrium steady state related to a null velocity gradient. That distributioncanbeobtainedbysolvingthefollowingequation:
∂
∂
x· 1 2H(
x)
x0 +1 2
∂
20
∂
x2 =0 (21)The resultingdistribution
0 will be considered asthe initial condition.
3. FiniteElementresolutiontechnique
Weconsidersimpleflowscharacterizedbyhomogeneous veloc-itygradients, which impliesthat the previous materialderivative reducestothepartialderivative.Takingintoaccountthe homoge-nousgradientofvelocity,nodiscretizationinphysicalvariablesis required,andtherefore,weproceedbydiscretizingwithrespectto theconformationcoordinates.
∂
∂
t +E0(
x)
+E1(
x)
∂
∂
x − 1 2∂
2∂
x2 =0 (22)0.5 1 1.5 2 2.5 3 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
Steady state PDF for We=1 and b=10
x
Ψ
FENE Kr1 Kr2 Coh Exact 0 5 10 15 20 −1 0 1 2 3 4 5 6 7 8 9Evolution of stress for We=1 and b=10
time
σ
FENE Kr1 Kr2 Coh ExactFig. 1. Steady state PDF and transient stress for We = 1 and b = 10 .
From a practical pointof view,theconfiguration domain
is bounded. Thisdomain is chosen such that the distribution func-tion canbe assumedvanishingon itsboundary. Firstly,the prob-lemisformulatedintheFiniteElementframeworkusinga weight-ingfunction
∗
∗d
dtd
+
∗E0
(
x)
d+
∗E 1
(
x)
∂
∂
xd
− 1 2
∗
∂
2∂
x2d=0 where E0
(
x)
=∂
∂
x·κ
· x−1 2xH(
x)
E1(
x)
=κ
· x− 1 2xH(
x)
TheGalerkinFiniteElementdiscretizationwrites
(
x)
= ni=1
Ni
(
x)
i (24)
where n is the total number of nodes, Ni are compact support
shape functions for each node i. They take the value 1 for the nodeiandvanishforallothernodes(seeforexample[9]formore details). Due to the advection–diffusion character of Eq. (23) an appropriate stabilizationof the Finite Element scheme is needed to avoid numerical instabilities induced by the convection term. Anupwindingformulationisconsideredhere, whichmodifiesthe weightingfunction
∗
(
x)
= n i=1 Ni(
x)
i (25) 2.9 2.95 3 3.05 3.1 3.15 0 1 2 3 4 5 6
Steady state PDF for We=10 and b=10
x
Ψ
FENE Kr1 Kr2 Coh Exact 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 50 100 150 200Evolution of stress for We=10 and b=10
time
σ
FENE Kr1 Kr2 Coh ExactFig. 2. Steady state PDF and transient stress for We = 10 and b = 10 .
with Ni
(
x)
=Ni(
x)
+β
x 2 E1
(
x)
|
E1(
x)
|
·∂
Ni∂
x(
x)
(26)β
isrelatedtothelocalPecletnumberandgivenbyβ
=coth(
Pe)
− 1Pe (27)
wherethelocalPecletnumberiscalculatedby
Pe=
|
E1(
x)
|
x (28)
xbeingthelocaldistancebetweentwo nodesofthe mesh.The integrationofEq.(23)allowstoobtainalinearsystemthathasto besolvedateachtimestep.
∗TM
˙ +
∗TG
=0 (29) where Mi j= Ni
(
x)
Nj(
x)
dGi j= E0
(
x)
Ni(
x)
Nj(
x)
+E1(
x)
Ni(
x)
∂
Nj(
x)
∂
x +1 2∂
Ni(
x)
∂
x∂
Nj(
x)
∂
x dIntheframeworkofan Eulerimplicittime integrationscheme withatimestep
t,theupdatingofthePDFcanbedoneusing:
4 A. Ammar / Journal of Non-Newtonian Fluid Mechanics 231 (2016) 1–5 3 3.5 4 4.5 5 5.5 6 6.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
Steady state PDF for We=1 and b=50
x
Ψ
FENE Kr1 Kr2 Coh Exact 0 5 10 15 20 −10 0 10 20 30 40 50 60Evolution of stress for We=1 and b=50
time
σ
FENE Kr1 Kr2 Coh ExactFig. 3. Steady state PDF and transient stress for We = 1 and b = 50 .
4. Resultsanddiscussion
ThetwoparametersusedinthesimulationaretheWeissenberg numberWeandthe parameter characterizingthe chainlength b. Twovalues are considered forthe Weissenberg number
(
We=1 and 10) and also for the parameter b(
b=10 and 50). Hence, four parameter combinations are considered. In all of the re-sultspresentedbelowthe twoapproximations‘Kr1’ and‘Kr2’are foundto be similar andvery close tothe exact solution.For the PDF steady state and the transientstress the representations re-latedto both Kröger’s models are practically identical to the ex-act model.The Cohen model exhibitssmall differences. However thebehavioroftheFENEmodel(thedashedline)isfundamentally different.Inthefollowingfigures itisimpossibletodistinguishbetween the ‘Exact’, the ‘Kr1’ and the ‘Kr2’ curves asthey are practically superposed.Fig.4showsazoomedviewinordertohighlightthe differencesbetweenthecurves.
Fig. 1 showsthat forWe=1 andb=10 the PDF steadystate solutionexhibitsanotableerrorwiththeFENEapproximation.The extremevalue ofthe FENE curve is lower andshifted tothe left inrelationtotheother models.Alsothesteadystatestress value showsa relative difference of the order of20% compared to the exactvalue.
When the Weissenbergnumberisincreased
(
We=10,b=10)
as shown in Fig. 2, the same tendency is observed in the PDF steadystatesolution.Butafundamentaldifferenceinthetransient stresscan be seen. Infact when looking atthetime interval be-tween0.1and0.3a notabledifference inthe slopeofthe curves fortheFENE modelandtheothers canbe seen.This showshow6.8 6.85 6.9 6.95 7 0 1 2 3 4 5 6
Steady state PDF for We=10 and b=50
x
Ψ
FENE Kr1 Kr2 Coh Exact 6.912 6.914 6.916 6.918 6.92 6.922 6.924 6.926 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 5 5.2Zoom of the steady state PDF for We=10 and b=50
x
Ψ
FENE Kr1 Kr2 Coh Exact 0 0.5 1 1.5 2 −200 0 200 400 600 800 1000Evolution of stress for We=10 and b=50
time
σ
FENE Kr1 Kr2 Coh ExactFig. 4. Steady state PDF, zoom of the PDF and transient stress for We = 10 and
b = 50 .
the quality ofthe inverseapproximation can significantly modify theapparent relaxationtime andgives awarningconcerning the useoftheFENEapproximationinsimilarconditions.
ForFigs.3and4simulationshavebeenperformedwithWe= 1,10 andb=50. Itcan be observed that when the value ofb is increased, the FENE shift in the PDF curves becomes more pro-nounced.Althoughthe tendencyinthe stress evolutionseems to besimilar,thevalueofthesteadystaterelativeerrorremainslarge (of theorder of20% forWe=1). Assome of thesimulations re-sults are similar, an accurate estimation ofthe error forthe dif-ferentmodels ismadeusingtwo numericalcriteria:thefirst one isrelated tothe difference inthe steadystate PDFsandthe sec-ond one is related to the extra stress value (characterizing the
Table 1
Comparison of errors on the PDF for the different approximations.
Err We = 1 , b = 10 We = 10 , b = 10 We = 1 , b = 50 We = 10 , b = 50 We = 100 , b = 50 We = 0 . 1 , b = 50 FENE 34 .38 59 .16 75 .95 100 .0 99 .29 1 .013 Kr1 1 .027 1 .003 2 .894 0 .803 0 .103 0 .146 Kr2 0 .247 0 .439 0 .807 0 .394 0 .050 0 .089 Cohen 6 .382 1 .805 15 .76 2 .433 0 .265 0 .239 Table 2
Comparison of errors on the stress value for the different approximations.
Errτ We = 1 , b = 10 We = 10 , b = 10 We = 1 , b = 50 We = 10 , b = 50 We = 100 , b = 50 We = 0 . 1 , b = 50 FENE 20 .27 11 .36 20 .56 1 .696 3 .160 3 .695 Kr1 0 .169 0 .674 0 .519 0 .010 0 .033 0 .039 Kr2 0 .119 0 .609 0 .166 0 .004 0 .027 0 .616 Cohen 3 .706 0 .763 3 .738 0 .030 0 .036 1 .198
macroscopicscale).Thetwoerrordefinitionsare:
Err=100
(
FENE,∞ Kr1,Kr2,Coh−Exact∞
)
2d(
Exact∞)
2d(31)
Errτ=100
|
τ
FENE,Kr1,Kr2,Coh−τ
Exact|
τ
Exact(32) ThecalculationoftheseerrorsissummarizedinTables1and2. The twoerrorestimations,in termsofstressandPDF, show gen-erally the same tendencies and confirm the previously discussed conclusions.
In order to observe theeffect of the We numberlimit values two columns havebeenadded toTables 1 and2.Thesecolumns show the results of simulations with b=50 and We=100,0.1.
Thislimitanalysisconfirmstheprevioussimulationsandconfirms therobustness ofKröger’s approximation inrelation to theFENE one.The Cohen approximation inthisdifferent caseseems tobe acceptablealthoughitdoesnotexceedKröger’sapproximationsin termsofaccuracy.
5. Conclusion
Theanalysisofthe FENEapproximation fordilutepolymer ki-netictheoryconfirmsitspoorqualityintermsofthePDF predic-tionaswellasintermsofstresscalculation.Thiscouldbeharmful whenthismodelisusedinamicro-macrosimulation.Forallthose whoareinterestedincomputationalrheologyitishenceforth rec-ommendedtouseoneofthetwoapproximationsprovidedbyEq. (7)or(8)that are not muchmore expensive to write or to imple-mentthanEq.(6).
References
[1] R.B. Bird , C.F. Curtiss , R.C. Armstrong , O. Hassager , Dynamics of polymeric liq- uids, Kinetic Theory, vol. 2, John Wiley & Sons, 1987 .
[2] A. Cohen , A Padé approximant to the inverse Langevin function, Rheol. Acta 30 (1991) 270–273 .
[3] R. Keunings , On the Peterlin approximation for finitely extensible dumbells, J. Non-Newton. Fluid Mech. 68 (1997) 85–100 .
[4] M. Kröger , Simple, admissible, and accurate approximants of the inverse Langevin and Brillouin functions, relevant for strong polymer deformations and flows, J. Non-Newton. Fluid Mech. 223 (2015) 77–87 .
[5] W. Kuhn , F. Grun , Beziehungen zwischen elastischen Konstanten und Dehnungs- doppelbrechung hochelastischer Stoffe, Kolloid-Z. 101 (1942) 248–271 . [6] M. Puso , Mechanistic constitutive models for rubber elasticity and viscoelastic-
ity, University of California, Davis, 2003 (Ph.D. thesis) .
[7] L. Treloar , The Physics of Rubber Elasticity, Oxford University Press Inc., New York, 1975 .
[8] H.R. Warner , Kinetic theory and rheology of dilute suspensions of finitely ex- tendible dumbbells, Ind. Eng. Chem. Fundam. 11 (1972) 379–387 .
[9] O.C. Zienkiewicz , R.L. Taylor , J.Z. Zhu , The Finite Element Method: Its Basis And Fundamentals, Butterworth-Heinemann Ltd., 2005 .