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

John W. Barrett; Endre Süli

ESAIM: Mathematical Modelling and Numerical Analysis (2012)

  • Volume: 46, Issue: 4, page 949-978
  • ISSN: 0764-583X

Abstract

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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 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.

How to cite

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Barrett, John W., and Süli, Endre. "Finite element approximation of finitely extensible nonlinear elastic dumbbell models for dilute polymers." ESAIM: Mathematical Modelling and Numerical Analysis 46.4 (2012): 949-978. <http://eudml.org/doc/276384>.

@article{Barrett2012,
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\{$\absundertilde$\} 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.},
author = {Barrett, John W., Süli, Endre},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Finite element method; convergence analysis; existence of weak solutions; kinetic polymer models; FENE dumbbell; Navier–Stokes equations; Fokker–Planck equations; finite element method; Navier-Stokes equations; Fokker-Planck equations},
language = {eng},
month = {2},
number = {4},
pages = {949-978},
publisher = {EDP Sciences},
title = {Finite element approximation of finitely extensible nonlinear elastic dumbbell models for dilute polymers},
url = {http://eudml.org/doc/276384},
volume = {46},
year = {2012},
}

TY - JOUR
AU - Barrett, John W.
AU - Süli, Endre
TI - Finite element approximation of finitely extensible nonlinear elastic dumbbell models for dilute polymers
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2012/2//
PB - EDP Sciences
VL - 46
IS - 4
SP - 949
EP - 978
AB - 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{$\absundertilde$} 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.
LA - eng
KW - Finite element method; convergence analysis; existence of weak solutions; kinetic polymer models; FENE dumbbell; Navier–Stokes equations; Fokker–Planck equations; finite element method; Navier-Stokes equations; Fokker-Planck equations
UR - http://eudml.org/doc/276384
ER -

References

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