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Pré-Publication, Document De Travail Année : 2022

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

Résumé

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 \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 a class of fluids with threshold flows in dimension \(N \in \{2, 3\}\). More exactly, we show that if the extra non-linear term is of the form \(F(s) = s^{-\alpha}\) with $\alpha \in (0, 1]$ 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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Dates et versions

hal-03461241 , version 1 (04-12-2021)
hal-03461241 , version 2 (05-12-2022)
hal-03461241 , version 3 (30-05-2023)

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Laurent Chupin, Nicolae Cîndea, Geoffrey Lacour. Variational inequality solutions and finite stopping time for a class of thixotropic or shear-thinning flows. 2022. ⟨hal-03461241v2⟩
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