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arXiv · 2112.02871

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

Laurent Chupin·Nicolae Cîndea·Geoffrey Lacour
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Abstract

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 $p \in [1,2)$.

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Laurent Chupin, Nicolae Cîndea, Geoffrey Lacour. 2021-12-06. Variational inequality solutions and finite stopping time for a class of shear-thinning flows. https://doi.org/10.1007/s10231-024-01457-9

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