Finite element approximation of finitely extensible nonlinear elastic dumbbell models for dilute polymers
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...
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| Format: | Report |
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EDP Sciences, SMAI
2011
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| _version_ | 1826295994183057408 |
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| author | Barrett, J Suli, E |
| author_facet | Barrett, J Suli, E |
| author_sort | Barrett, J |
| collection | OXFORD |
| description | 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 $\Omega \subset R^d$, 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 $u_0$ for the Navier-Stokes equation and a nonnegative initial probability density function $\psi_0$ for the Fokker-Planck equation, which has finite relative entropy with respect to the Maxwellian M. |
| first_indexed | 2024-03-07T04:09:31Z |
| format | Report |
| id | oxford-uuid:c759acd0-b132-407d-abd9-06f90dcbdf5d |
| institution | University of Oxford |
| last_indexed | 2024-03-07T04:09:31Z |
| publishDate | 2011 |
| publisher | EDP Sciences, SMAI |
| record_format | dspace |
| spelling | oxford-uuid:c759acd0-b132-407d-abd9-06f90dcbdf5d2022-03-27T06:44:26ZFinite element approximation of finitely extensible nonlinear elastic dumbbell models for dilute polymersReporthttp://purl.org/coar/resource_type/c_93fcuuid:c759acd0-b132-407d-abd9-06f90dcbdf5dMathematical Institute - ePrintsEDP Sciences, SMAI2011Barrett, JSuli, EWe 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 $\Omega \subset R^d$, 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 $u_0$ for the Navier-Stokes equation and a nonnegative initial probability density function $\psi_0$ for the Fokker-Planck equation, which has finite relative entropy with respect to the Maxwellian M. |
| spellingShingle | Barrett, J Suli, E Finite element approximation of finitely extensible nonlinear elastic dumbbell models for dilute polymers |
| title | Finite element approximation of finitely extensible nonlinear elastic dumbbell models for dilute polymers |
| title_full | Finite element approximation of finitely extensible nonlinear elastic dumbbell models for dilute polymers |
| title_fullStr | Finite element approximation of finitely extensible nonlinear elastic dumbbell models for dilute polymers |
| title_full_unstemmed | Finite element approximation of finitely extensible nonlinear elastic dumbbell models for dilute polymers |
| title_short | Finite element approximation of finitely extensible nonlinear elastic dumbbell models for dilute polymers |
| title_sort | finite element approximation of finitely extensible nonlinear elastic dumbbell models for dilute polymers |
| work_keys_str_mv | AT barrettj finiteelementapproximationoffinitelyextensiblenonlinearelasticdumbbellmodelsfordilutepolymers AT sulie finiteelementapproximationoffinitelyextensiblenonlinearelasticdumbbellmodelsfordilutepolymers |