Hostname: page-component-5c6d5d7d68-sv6ng Total loading time: 0 Render date: 2024-08-15T18:52:04.770Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic Behavior of the Solution of the Distribution Diffusion Equation for FENE Dumbbell Polymer Model

Published online by Cambridge University Press:  10 August 2011

I. S. Ciuperca  and
L. I. Palade
I. S. Ciuperca
Affiliation:
Université de Lyon, Université Lyon 1, Institut Camille Jordan, UMR 5208 CNRS 69622 Villeurbanne, France
L. I. Palade*
Affiliation:
Université de Lyon, INSA de Lyon, Institut Camille Jordan UMR 5208 CNRS & Pôle de Mathématiques, 69621 Villeurbanne, France
*
⋆ Corresponding author. E-mail: liviu-iulian.palade@insa-lyon.fr
Get access

Abstract

This paper deals with the evolution Fokker-Planck-Smoluchowski configurational probability diffusion equation for the FENE dumbbell model in dilute polymer solutions. We prove the exponential convergence in time of the solution of this equation to a corresponding steady-state solution, for arbitrary velocity gradients.

Keywords

FENE dumbbell chainsFokker-Planck-Smoluchowski equationentropy functionsectorial operators
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 6 , Issue 5: Complex Fluids , 2011 , pp. 84 - 97
Copyright
© EDP Sciences, 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Barret, J. W., Schwab, C., Süli, E.. Existence of global weak solutions for some polymeric flow models. Math. Model. Meth. Appl. Sci., 15 (2005), No. 6, 939–983. CrossRefGoogle Scholar
R. B. Bird, R. C. Armstrong, O. Hassager. Dynamics of Polymeric Liquids, Vol. 1: Fluid Mechanics. J. Wiley & Sons, New York, 1987.
R. B. Bird, R. C. Armstrong, O. Hassager. Dynamics of Polymeric Liquids, Vol. 2: Kinetic Theory. J. Wiley & Sons, New York, 1987.
A. V. Busuioc, I. S. Ciuperca, D. Iftimie and L. I. Palade. The FENE dumbbell polymer model: existence and uniqueness of solutions for the momentum balance equation. Journal of Dynamics and Differential Equations, submitted, 2011.
J. A. Carillo, S. Cordier, S. Mancini. A decision-making Fokker-Planck model in computational neuroscience. To appear in Journal of Mathematical Biology, 2011.
Chupin, L.. The FENE model for viscoelastic thin film flow. Methods Appl. Anal., 16 (2009), No. 2, 217–261. Google Scholar
Ciuperca, I. S., Palade, L. I.. The steady state configurational distribution diffusion equation of the standard FENE dumbbell polymer model: existence and uniqueness of solutions for arbitrary velocity gradients. Mathematical Models & Methods in Applied Sciences, 19 (2009), 2039–2064. CrossRefGoogle Scholar
Ciupercă, I. S., Palade, L. I.. On the existence and uniqueness of solutions of the configurational probability diffusion equation for the generalized rigid dumbbell polymer model. Dynamics of Partial Differential Equations, 7 (2010), 245–263. CrossRefGoogle Scholar
Du, Q., Liu, C., Yu, P.. FENE dumbbell model and its several linear and nonlinear closure approximations. Multiscale Model. Simul., 4 (2005), No. 3, 709–731. CrossRefGoogle Scholar
D. Henry. Geometric Theory of semilinear parabolic equations. Lecture notes in mathematics, Vol. 840. Springer Verlag, New York, 1981.
R. R. Huilgol, N. Phan-Thien. Fluid Mechanics of Viscoelasticity. Elsevier, Amsterdam, 1997.
Jourdain, B., Le Bris, C., Lelièvre, T., Otto, F.. Long-time asymptotics of a multiscale model for a polymeric fluid flows. Arch. Rational Mech. Anal., 181 (2006), 97–148. CrossRefGoogle Scholar
J. G. Kirkwood. Macromolecules, edited by P. L. Auer. Gordon and Breach, 1968.
R. G. Larson. Constitutive Equations for Polymer Melts and Solutions. Butterworths, Boston, 1988.
Lin, F., Zhang, P., Zhang, Z.. On the global existence of smooth solution to the 2-D FENE Dumbell Model. Commun. Math. Phys., 277 (2008), 531–553. CrossRefGoogle Scholar
Masmoudi, N.. Well-Posedness for the FENE dumbbell model of polymeric flows. Comm. Pure Appl. Math., 61 (2008), No. 12, 1685–1714. CrossRefGoogle Scholar
F. A. Morrison. Understanding Rheology. Oxford University Press, Oxford, 2001.
J. Nečas. Les méthodes directes en théorie des équations elliptiques. Masson, Paris, 1967.
S. Cleja-Ţigoiu, V. Ţigoiu. Rheology and Thermodynamics, Part I - Rheology. Editura Universităţii din Bucureşti, Bucureşti, 1998.
Volpert, V. A., Volpert, A. I.. Location of spectrum and stability of solutions for monotone parabolic system. Advances in Differential Equations, 2 (1997), No. 5, 811–830. Google Scholar
Zhang, H., Zhang, P.. Local existence for the FENE-dumbbell model of polymeric fluids. Arch. Ratl. Mech. Anal., 181 (2006), 373–400. CrossRefGoogle Scholar