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Finite element approximation of finitely extensible nonlinear elastic 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 weak solutions to a general class of coupled FENE-type finitely extensible nonlinear elastic dumbbell models that arise from the kinetic theory of dilute solutions of polymeric liquids with noninteracting polymer chains. The class of models involves the unsteady incompressible Navier–Stokes equations in a bounded domain Ω ⊂ ℝd, d = 2 or 3, for the velocity and the pressure of the fluid, with an elastic extra-stress tensor appearing on the right-hand side in the momentum equation. The extra-stress tensor stems from the random movement of the polymer chains and is defined through the associated probability density function that satisfies a Fokker–Planck type parabolic equation, a crucial feature of which is the presence of a centre-of-mass diffusion term. We require no structural assumptions on the drag term in the Fokker–Planck equation; in particular, the drag term need not be corotational. We perform a rigorous passage to the limit as first the spatial discretization parameter, and then the temporal discretization parameter tend to zero, and show that a (sub)sequence of these finite element approximations converges to a weak solution of this coupled Navier–Stokes–Fokker–Planck system. The passage to the limit is performed under minimal regularity assumptions on the data: a square-integrable and divergence-free initial velocity datum \hbox{$\absundertilde$}u0~ for the Navier–Stokes equation and a nonnegative initial probability density function ψ0 for the Fokker–Planck equation, which has finite 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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