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

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 with thixotropic and shear-thinning flows. The existence of dissipative solutions for such flows is known since [1]. We submit here an alternative approach using variational inequality solutions as presented in [17] 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 a class of fluids with threshold flows in dimension N ∈ { 2 , 3 } . More exactly, we show that if the extra non-linear term is of the form F ( s ) = s − α with α ∈ (0 , 1] when s is large, then there exists a finite stopping time. This result extends the existing results concerning the existence of a finite stopping time for Bingham fluids in dimension two.

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