A Deterministic Multiscale Approach for Simulating Dilute Polymeric Fluids

Knezevic, David J.; Süli, Endre

doi:10.1007/978-3-642-00605-0_2

Cited by 1 publication

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“…A possible practical approach to its implementation might be based on operator-splitting, following the computational strategy advocated in the papers [19,20], for example; see also [21]. We note however that the schemes that were analyzed in [19][20][21] were shown to converge for the Fokker-Planck equation considered in isolation, i.e.…”

Section: Discussionmentioning

confidence: 99%

Finite element approximation of finitely extensible nonlinear elastic dumbbell models for dilute polymers

Barrett

Süli

2012

ESAIM: M2AN

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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 ⊂ R d , 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 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 .Mathematics Subject Classification. 35Q30, 35K65, 65M12, 65M60, 76A05, 82D60.

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