Unsteady 3D-Navier–Stokes system with Tresca’s friction law

Abstract

We consider unsteady incompressible 3D fluid flow with nonconstant viscosity subjected to nonlinear boundary conditions of Tresca’s type: when the modulus of the shear stress at the boundary is smaller than a given threshold, the classical no-slip condition holds; otherwise, the modulus of the shear stress at the boundary equals the given threshold and the fluid is allowed to slip. The problem is thus described by a nonlinear parabolic variational inequality. We construct a sequence of approximate solutions by using both a regularization of the free boundary condition due to friction and a penalty method, reminiscent of the “incompressibility limit” of compressible fluids, allowing us to get a better insight 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.