Variational inequality solutions and finite stopping time for a class of shear-thinning flows

The aim of this paper is to study the existence of a finite stopping time for solutions in the form of variational inequality to fluid flows following a power law (or Ostwald-DeWaele law) in dimension $N \in \{2,3\}$. We first establish the existence of solutions for generalized Newtonian flows, valid for viscous stress tensors associated with the usual laws such as Ostwald-DeWaele, Carreau-Yasuda, Herschel-Bulkley and Bingham, but also for cases where the viscosity coefficient satisfies a more atypical (logarithmic) form. To demonstrate the existence of such solutions, we proceed by applying a nonlinear Galerkin method with a double regularization on the viscosity coefficient. We then establish the existence of a finite stopping time for threshold fluids or shear-thinning power-law fluids, i.e. formally such that the viscous stress tensor is represented by a $p$-Laplacian for the symmetrized gradient for $p \in [1,2)$.