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

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 $L^p\left(0,T;\left(W^{1,p}{\left(\Omega \right)}^3\right)\right)$, where $\Omega$ 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 $L^{p^{\prime }}\left(0,T;\left(W^{1,p^{\prime }}{\left(\Omega \right)}^3\right)\right)$ with $p^{\prime }>2$ 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.