Second grade fluids with boundery conditions

Abstract

It is well-known that non-Newtonian fluids such as polymers melts do not satisfy the usual adherence boundary condition. On the other hand, the available theory relies heavily on the no-slip assumption. The purpose of this work is to establish the well-posedness of the initial-boundary-value problem for flows of second grade fluids subject to general partial slip boundary conditions. It is assumed that the fluid satisfies the usual thermodynamical restrictions, that the domain of flow is bounded and simply connected, and that the slip yield stress is zero. The proof is based on a fixed-point formulation of the problem which decomposes it into three linear ones: a Stokes type problem and two transport problems. After proving the solvability of these auxiliary problems by the Faedo-Galerkin method, the existence of a unique classical solution, local in time, is established by means of a Schauder fixed point theorem. Then global a priori estimates are derived to obtain a unique global classical solution for sufficiently small data and large viscosity. The solution is found to be stable under mild restrictions on the slip operator.