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Finite element approximation of finitely extensible nonlinearelastic dumbbell models for dilute polymers

Published online by Cambridge University Press:  13 February 2012

John W. Barrett  and
Endre Süli
John W. Barrett
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK. j.barrett@ imperial.ac.uk
Endre Süli
Affiliation:
Mathematical Institute, University of Oxford, 24–29 St. Giles’, OX1 3LB Oxford, UK; endre.suli@ maths.ox.ac.uk
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Abstract

We construct a Galerkin finite element method for the numerical approximation of weaksolutions to a general class of coupled FENE-type finitely extensible nonlinear elasticdumbbell models that arise from the kinetic theory of dilute solutions of polymericliquids with noninteracting polymer chains. The class of models involves the unsteadyincompressible Navier–Stokes equations in a bounded domainΩ ⊂ ℝd, d = 2 or 3, forthe velocity and the pressure of the fluid, with an elastic extra-stress tensor appearingon the right-hand side in the momentum equation. The extra-stress tensor stems from therandom movement of the polymer chains and is defined through the associated probabilitydensity function that satisfies a Fokker–Planck type parabolic equation, a crucial featureof which is the presence of a centre-of-mass diffusion term. We require no structuralassumptions on the drag term in the Fokker–Planck equation; in particular, the drag termneed not be corotational. We perform a rigorous passage to the limit as first the spatialdiscretization parameter, and then the temporal discretization parameter tend to zero, andshow that a (sub)sequence of these finite element approximations converges to a weaksolution of this coupled Navier–Stokes–Fokker–Planck system. The passage to the limit isperformed under minimal regularity assumptions on the data: a square-integrable anddivergence-free initial velocity datum \hbox{$\absundertilde$}u0~ for the Navier–Stokes equation and a nonnegative initial probabilitydensity function ψ0 for the Fokker–Planck equation, which hasfinite relative entropy with respect to the Maxwellian M.

Keywords

Finite element methodconvergence analysisexistence of weak solutionskinetic polymer modelsFENE dumbbellNavier–Stokes equationsFokker–Planck equations
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 46 , Issue 4 , July 2012 , pp. 949 - 978
Copyright
© EDP Sciences, SMAI, 2012

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