Newtonian Limit for Weakly Viscoelastic Fluid Flows of Oldroyd Type

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 IR 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 SobolevH theory and the Besov-B 2 theory to reach the critical case s = N/2) to the corresponding newtonian quantities. This 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.