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

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May 2010, 9(3): 625-642. doi: 10.3934/cpaa.2010.9.625

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

1.	Université Paris-Est, Laboratoire d'Analyse et de Mathématiques Appliquées, UMR CNRS 8050, 61 avenue du Général de Gaulle, 94010 Créteil Cedex, France

Received May 2009 Revised September 2009 Published January 2010

Abstract
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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: fixed-point arguments., singularities, weakly compressible, viscous flows, Viscoelastic fluids.

Mathematics Subject Classification: Primary: 76A05, 76A10, 76N10; Secondary: 35B3.

Citation: Zaynab Salloum. Flows of weakly compressible viscoelastic fluids through a regular bounded domain with inflow-outflow boundary conditions. Communications on Pure & Applied Analysis, 2010, 9 (3) : 625-642. doi: 10.3934/cpaa.2010.9.625

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