A nonlinear model of the non-isothermal slip flow between two parallel plates

Abstract

We study a mathematical model describing the steady-state non-isothermal flow of a viscous fluid between two parallel plates under the Navier slip boundary condition. The flow is driven by an applied pressure gradient. The dependence of the viscosity, thermal conductivity and slip coefficients on the temperature is taking into account. The model under consideration is a boundary value problem for a system of nonlinear coupled ordinary differential equations. We give the weak formulation of this problem and establish sufficient conditions for the existence and uniqueness of a weak solution. To construct weak solutions, we propose an algorithm based on the Galerkin procedure, methods of the topological degree theory, and compactness arguments. Moreover, explicit expressions for the velocity and the temperature on the plates are obtained.