  The aim of this paper is to study the existence of variational inequality
weak solutions and of a finite stopping time for a large class of generalized
Newtonian fluids shear-thinning flows. The existence of dissipative solutions
for such flows is known since \cite{abbatiello-feireisl-20}. We submit here an
alternative approach using variational inequality solutions as presented in
\cite{duvaut-lions} in the two-dimensional Bingham flow. In order to prove the
existence of such solutions we regularize the non-linear term and then we apply
a Galerkin method for finally passing to the limit with respect to both
regularization and Galerkin discretization parameters. In a second time, we
prove the existence of a finite stopping time for Ostwald-De Waele and Bingham
flows in dimension \(N \in \{2, 3\}\).
