Unsteady non-Newtonian fluid flows with boundary conditions of friction type: The case of shear thinning fluids

Abstract

Following the previous part of our study on unsteady non-Newtonian fluid flows with boundary conditions of friction type we consider in this paper the case of pseudo-plastic (shear thinning) fluids. The problem is described by a p-Laplacian non-stationary Stokes system with p<2 and we assume that the fluid is subjected to mixed boundary conditions, namely non-homogeneous Dirichlet boundary conditions on a part of the boundary and a slip fluid-solid interface law of friction type on another part of the boundary. Hence the fluid velocity should belong to a subspace of Lp(0,T;(W1,p(Ω)3)), where Ω is the flow domain and T>0, and satisfy a non-linear parabolic variational inequality. In order to solve this problem we introduce first a vanishing viscosity technique which allows us to consider an auxiliary problem formulated in Lp′(0,T;(W1,p′(Ω)3)) with p′>2 the conjugate number of p and to use the existence results already established in Boukrouche et al. (2020). Then we apply both compactness arguments and a fixed point method to prove the existence of a solution to our original fluid flow problem.