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IMA Journal of Numerical Analysis Advance Access originally published online on September 18, 2008
IMA Journal of Numerical Analysis 2009 29(4):937-959; doi:10.1093/imanum/drn022
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© The author 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Numerical approximation of corotational dumbbell models for dilute polymers

John W. Barrett

Department of Mathematics, Imperial College London, London SW7 2AZ, UK

Endre Süli{dagger}

Numerical Analysis Group, Mathematical Institute, University of Oxford, 24–29 St Giles’, Oxford, OX1 3LB, UK

{dagger} Email: endre.suli{at}maths.ox.ac.uk

Received on 28 July 2007. Revised on 4 January 2008.


  Abstract

We construct a general family of Galerkin methods for the numerical approximation of weak solutions to a bead-spring model that arises from the kinetic theory of dilute solutions of polymeric liquids with noninteracting polymer chains. The model consists of the unsteady incompressible Navier–Stokes equations, for the velocity and the pressure of the fluid, with an elastic extra-stress tensor as 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 satisfying a Fokker–Planck-type parabolic equation. We focus on finitely extensible nonlinear elastic-type dumbbell models. In the case of a corotational drag term, we perform a rigorous passage to the limit as the spatial and temporal discretization parameters tend to zero and show that a (sub)sequence of numerical solutions converges to a weak solution of this coupled Navier–Stokes–Fokker–Planck system.

Key Words: Galerkin methods; polymeric flow models; existence of weak solutions; Navier–Stokes equations; Fokker–Planck equations; FENE


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