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Existence of global weak solutions for some polymeric flow models

by J. W. Barrett and Ch. Schwab and E. Süli

(Report number 2005-02)

Abstract
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 diluted solutions of polymeric liquids with noninteracting polymer chains. The model consists of the unsteady incompressible Navier--Stokes equations in a bounded domain $\Omega \subset \mathbb{R}^d$, d=2,3, for the velocity and the pressure of the fluid, with an extra-stress tensor as 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 degenerate parabolic equation. Upon appropriate smoothing of the convective velocity field in the Fokker--Planck equation, and in some circumstances, of the extra-stress tensor, we establish the existence of global-in-time weak solutions to this regularised bead-spring model for a general class of spring-force potentials including in particular the widely used FENE (Finitely Extensible Nonlinear Elastic) model.

Keywords:

@Techreport{BSS05_341,
  author = {J. W. Barrett and Ch. Schwab and E. S\"uli},
  title = {Existence of global weak solutions for some polymeric flow models},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2005-02},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2005/2005-02.pdf },
  year = {2005}
}

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