Existence of global weak solutions to some regularized kinetic models for dilute polymers
We study the existence of global-in-time weak solutions to a coupled microscopic-macroscopic bead-spring model which 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 i...
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| Format: | Journal article |
| Language: | English |
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2007
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| author | Barrett, J Sueli, E |
| author_facet | Barrett, J Sueli, E |
| author_sort | Barrett, J |
| collection | OXFORD |
| description | We study the existence of global-in-time weak solutions to a coupled microscopic-macroscopic bead-spring model which 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 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 as 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 which satisfies a Fokker-Planck-type parabolic equation, a crucial feature of which is the presence of a center-of-mass diffusion term. The anisotropic Friedrichs mollifiers, which naturally arise in the course of the derivation of the model in the Kramers expression for the extra-stress tensor and in the drag term in the Fokker-Planck equation, are replaced by isotropic Friedrichs mollifiers. We establish the existence of global-in-time weak solutions to the model for a general class of spring-force-potentials including, in particular, the widely used finitely extensible nonlinear elastic (FENE) potential. We justify also, through a rigorous limiting process, certain classical reductions of this model appearing in the literature which exclude the center-of-mass diffusion term from the Fokker-Planck equation on the grounds that the diffusion coefficient is small relative to other coefficients featuring in the equation. In the case of a corotational drag term we perform a rigorous passage to the limit as the Friedrichs mollifiers in the Kramers expression and the drag term converge to identity operators. © 2007 Society for Industrial and Applied Mathematics. |
| first_indexed | 2024-03-07T04:52:36Z |
| format | Journal article |
| id | oxford-uuid:d57b3bee-c105-4227-bd16-bd6ae519ee4c |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T04:52:36Z |
| publishDate | 2007 |
| record_format | dspace |
| spelling | oxford-uuid:d57b3bee-c105-4227-bd16-bd6ae519ee4c2022-03-27T08:26:12ZExistence of global weak solutions to some regularized kinetic models for dilute polymersJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:d57b3bee-c105-4227-bd16-bd6ae519ee4cEnglishSymplectic Elements at Oxford2007Barrett, JSueli, EWe study the existence of global-in-time weak solutions to a coupled microscopic-macroscopic bead-spring model which 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 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 as 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 which satisfies a Fokker-Planck-type parabolic equation, a crucial feature of which is the presence of a center-of-mass diffusion term. The anisotropic Friedrichs mollifiers, which naturally arise in the course of the derivation of the model in the Kramers expression for the extra-stress tensor and in the drag term in the Fokker-Planck equation, are replaced by isotropic Friedrichs mollifiers. We establish the existence of global-in-time weak solutions to the model for a general class of spring-force-potentials including, in particular, the widely used finitely extensible nonlinear elastic (FENE) potential. We justify also, through a rigorous limiting process, certain classical reductions of this model appearing in the literature which exclude the center-of-mass diffusion term from the Fokker-Planck equation on the grounds that the diffusion coefficient is small relative to other coefficients featuring in the equation. In the case of a corotational drag term we perform a rigorous passage to the limit as the Friedrichs mollifiers in the Kramers expression and the drag term converge to identity operators. © 2007 Society for Industrial and Applied Mathematics. |
| spellingShingle | Barrett, J Sueli, E Existence of global weak solutions to some regularized kinetic models for dilute polymers |
| title | Existence of global weak solutions to some regularized kinetic models for dilute polymers |
| title_full | Existence of global weak solutions to some regularized kinetic models for dilute polymers |
| title_fullStr | Existence of global weak solutions to some regularized kinetic models for dilute polymers |
| title_full_unstemmed | Existence of global weak solutions to some regularized kinetic models for dilute polymers |
| title_short | Existence of global weak solutions to some regularized kinetic models for dilute polymers |
| title_sort | existence of global weak solutions to some regularized kinetic models for dilute polymers |
| work_keys_str_mv | AT barrettj existenceofglobalweaksolutionstosomeregularizedkineticmodelsfordilutepolymers AT suelie existenceofglobalweaksolutionstosomeregularizedkineticmodelsfordilutepolymers |