Newtonian limit for weakly viscoelastic fluid flows of Olroyds' type

Details Newtonian limit for weakly viscoelastic fluid flows of Olroyds' type Newtonian limit for weakly viscoelastic fluid flows of Olroyds' type

2008-03-03

arxiv.org/abs/0803.0228v1

Mathematics

Analysis of PDEs

Scientific paper

This paper is concerned with regular flows of incompressible weakly viscoelastic fluids which obey a differential constitutive law of Oldroyd type. We study the newtonian limit for weakly viscoelastic fluid flows in $\R^N$ or $\T^N$ for $N=2, 3$, when the Weissenberg number (relaxation time measuring the elasticity effect in the fluid) tends to zero. More precisely, we prove that the velocity field and the extra-stress tensor converge in their existence spaces (we examine the Sobolev-$H^s$ theory and the Besov-$B^{s,1}_2$ theory to reach the critical case $s= N/2$) to the corresponding newtonian quantities. These convergence results are established in the case of "ill-prepared"' data.We deduce, in the two-dimensional case, a new result concerning the global existence of weakly viscoelastic fluids flow. Our approach makes use of essentially two ingredients : the stability of the null solution of the viscoelastic fluids flow and the damping effect,on the difference between the extra-stress tensor and the tensor of rate of deformation, induced by the constitutive law of the fluid.

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