Unsteady 3D-Navier-Stokes System with Tresca's Friction Law

TL;DR

This paper models unsteady 3D incompressible fluid flow with temperature-dependent viscosity under mixed boundary conditions, including Tresca's friction law, and establishes existence of solutions through regularization and limit processes.

Contribution

It introduces a novel regularization and penalty approach to handle Tresca's friction law in unsteady 3D Navier-Stokes equations with variable viscosity, providing new existence results.

Findings

Established existence of solutions for the unsteady 3D Navier-Stokes system with Tresca's friction law.

Developed a regularization and penalty method to analyze free boundary conditions.

Demonstrated the link between velocity and pressure fields through compactness arguments.

Abstract

Motivated by extrusion problems, we consider a non-stationary incompress-ible 3D fluid flow with a non-constant (temperature dependent) viscosity, subjected to mixed boundary conditions with a given time dependent velocity on a part of the boundary and Tresca's friction law on the other part. We construct a sequence of approximate solutions by using a regularization of the free boundary condition due to friction combined with a particular penalty method, reminiscent of the " incompressibility limit " of compressible fluids, allowing to get better insights into the links between the fluid velocity and pressure fields. Then we pass to the limit with compactness arguments to obtain a solution to our original problem.