Flows of weakly compressible viscoelastic fluids through a regular bounded domain with inflow-outflow boundary conditions

Pages: 625 - 642, Volume 9, Issue 3, May 2010

doi:10.3934/cpaa.2010.9.625       Abstract        Full Text (255.5K)       Related Articles

Zaynab Salloum - Université Paris-Est, Laboratoire d'Analyse et de Mathématiques Appliquées, UMR CNRS 8050, 61 avenue du Général de Gaulle, 94010 Créteil Cedex, France (email)

Abstract: We study steady isothermal motions of a nonlinear weakly compressible viscoelastic fluids of Oldroyd type in a bounded domain $\Omega\subset\mathbb{R}^2$, with given non-zero velocities on the boundary of $\Omega$. We suppose that the pressure $p$ and the extra-stress tensor $\tau$ are prescribed on the part of the boundary corresponding to entering velocities. A uniqueness and existence result for the solution $(\mathbf u,p,\tau)$ is established in $W^{2,q}(\Omega)\times W^{1,q}(\Omega)\times W^{1,q}(\Omega)$ with $ 2 < q < 3$. The proof follows from an analysis of a linearized problem. The fixed point theorem is used to establish the existence of a solution. The solutions of two transport equations for $p$ and $\tau$ are obtained by integration along the streamlines.

Keywords:  Viscoelastic fluids, singularities, weakly compressible, viscous flows, fixed-point arguments.
Mathematics Subject Classification:  Primary: 76A05, 76A10, 76N10; Secondary: 35B30.

Received: May 2009;      Revised: September 2009;      Published: January 2010.