Variational and Quasi-variational solutions to thick flows

TL;DR

This paper develops variational and quasi-variational inequality models for thick fluid flows, establishing existence and uniqueness of solutions, and extends to variable thresholds linked to the flow itself.

Contribution

It introduces a novel variational framework for thick fluid flows with variable deformation thresholds, including existence and uniqueness results for both viscous and inviscid cases.

Findings

Proved existence and uniqueness of solutions for variational inequalities in viscous flows.

Extended the model to quasi-variational inequalities with solution-dependent thresholds.

Established continuous dependence of solutions on parameters and thresholds.

Abstract

We formulate the flow of thick fluids as evolution variational and quasi-variational inequalities, with a variable threshold on the absolute value of the deformation rate tensor. In the variational case, we show the existence and uniqueness of strong and weak solutions in the viscous case and also the existence of strong and weak solutions in the inviscid case. These problems correspond to solve, respectively, the Navier-Stokes and the Euler equations with an additional generalised Lagrange multiplier associated with the threshold on the deformation rate tensor. Applying the continuous dependence of strong and weak solutions to the variational inequalities for the Navier-Stokes with constraints on the derivatives, and on their respective generalised Lagrange multipliers, we can solve the case of the variable threshold depending on the solution itself that correspond to quasi-variational problems. \vspace{2mm} $$ \text{Dedicated to Vsevolod Alekseevich Solonnikov, {\em in memoriam}}$$