Existence and equilibration of global weak solutions to finitely extensible nonlinear bead-spring chain models for dilute polymers
We show the existence of global-in-time weak solutions to a general class of coupled FENE-type bead-spring chain 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-Stok...
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| Format: | Journal article |
| Language: | English |
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2010
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| author | Barrett, J Süli, E |
| author_facet | Barrett, J Süli, E |
| author_sort | Barrett, J |
| collection | OXFORD |
| description | We show the existence of global-in-time weak solutions to a general class of coupled FENE-type bead-spring chain 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 in two or three space dimensions 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 by the Kramers expression through the associated probability density function that satisfies a Fokker-Planck-type parabolic equation, a crucial feature of which is the presence of a center-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. With a square-integrable and divergence-free initial velocity datum for the Navier-Stokes equation and a nonnegative initial probability density function for the Fokker-Planck equation, which has finite relative entropy with respect to the Maxwellian of the model, we prove the existence of a global-in-time weak solution to the coupled Navier-Stokes-Fokker-Planck system. It is also shown that in the absence of a body force, the weak solution decays exponentially in time to the equilibrium solution, at a rate that is independent of the choice of the initial datum and of the centre-of-mass diffusion coefficient. |
| first_indexed | 2024-03-06T22:55:08Z |
| format | Journal article |
| id | oxford-uuid:601bb21e-c77d-4e00-9957-edc124fad59c |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-06T22:55:08Z |
| publishDate | 2010 |
| record_format | dspace |
| spelling | oxford-uuid:601bb21e-c77d-4e00-9957-edc124fad59c2022-03-26T17:51:16ZExistence and equilibration of global weak solutions to finitely extensible nonlinear bead-spring chain models for dilute polymersJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:601bb21e-c77d-4e00-9957-edc124fad59cEnglishSymplectic Elements at Oxford2010Barrett, JSüli, EWe show the existence of global-in-time weak solutions to a general class of coupled FENE-type bead-spring chain 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 in two or three space dimensions 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 by the Kramers expression through the associated probability density function that satisfies a Fokker-Planck-type parabolic equation, a crucial feature of which is the presence of a center-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. With a square-integrable and divergence-free initial velocity datum for the Navier-Stokes equation and a nonnegative initial probability density function for the Fokker-Planck equation, which has finite relative entropy with respect to the Maxwellian of the model, we prove the existence of a global-in-time weak solution to the coupled Navier-Stokes-Fokker-Planck system. It is also shown that in the absence of a body force, the weak solution decays exponentially in time to the equilibrium solution, at a rate that is independent of the choice of the initial datum and of the centre-of-mass diffusion coefficient. |
| spellingShingle | Barrett, J Süli, E Existence and equilibration of global weak solutions to finitely extensible nonlinear bead-spring chain models for dilute polymers |
| title | Existence and equilibration of global weak solutions to finitely
extensible nonlinear bead-spring chain models for dilute polymers |
| title_full | Existence and equilibration of global weak solutions to finitely
extensible nonlinear bead-spring chain models for dilute polymers |
| title_fullStr | Existence and equilibration of global weak solutions to finitely
extensible nonlinear bead-spring chain models for dilute polymers |
| title_full_unstemmed | Existence and equilibration of global weak solutions to finitely
extensible nonlinear bead-spring chain models for dilute polymers |
| title_short | Existence and equilibration of global weak solutions to finitely
extensible nonlinear bead-spring chain models for dilute polymers |
| title_sort | existence and equilibration of global weak solutions to finitely extensible nonlinear bead spring chain models for dilute polymers |
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