Existence for non-isothermal fluid flows with Tresca's friction and Cattaneo's heat law

We consider a two dimensional non-isothermal incompressible fluid flow involving heat transfer and friction. The problem is described by the non-stationary Navier–Stokes system and by the hyperbolic heat equation derived from Cattaneo's heat flux law, which allows to overcome the paradox of infinite propagation speed. We assume a strong coupling due to temperature dependent viscosity and velocity dependent heat capacity. Moreover we consider Tresca's friction boundary conditions for the flow. We prove an existence result for this problem by using a fixed point technique.