A Nitsche method for incompressible fluids with general dynamic boundary conditions

Both Newtonian and non-Newtonian fluids may exhibit complex slip behaviour at the boundary. We examine a broad class of slip boundary conditions that generalises the commonly used Navier slip, perfect slip, stick-slip and Tresca friction boundary conditions. In particular, set-valued, non-monotone, noncoercive and dynamic relations may occur. For a unifying framework of such relations, we present a fully discrete numerical scheme for the time-dependent Navier-Stokes equations subject to impermeability and general slip type boundary conditions on polyhedral domains. Based on compactness arguments, we prove convergence of subsequences, finally ensuring the existence of a weak solution. The numerical scheme uses a general inf-sup stable pair of finite element spaces for the velocity and pressure, a regularisation approach for the implicit slip boundary condition and, most importantly, a Nitsche method to impose the impermeability and a backward Euler time stepping. One of the key tools in the convergence proof is an inhomogeneous Korn inequality that includes a normal trace term.