Newtonian limit for weakly viscoelastic fluid flows of Olroyds' type

TL;DR

This paper investigates the Newtonian limit of weakly viscoelastic fluids of Oldroyd type, proving convergence of flow variables as elasticity effects diminish, and establishing global existence results in 2D.

Contribution

It provides rigorous convergence results for weakly viscoelastic flows to Newtonian flows in various function spaces and introduces new global existence results in two dimensions.

Findings

Velocity and stress tensor converge to Newtonian quantities as Weissenberg number tends to zero.

Established global existence of weakly viscoelastic flows in 2D.

Proved convergence in Sobolev and Besov spaces for ill-prepared data.

Abstract

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.