Stokes system with direction dependent slip boundary condition coupled with the heat equation: finite element analysis

Abstract Direction-dependent Navier type slip boundary condition occurs in hydraulic networks (water supply, biological flows, oil ducts, etc...). The heat equation coupled with Stokes or Navier–Stokes problems with viscosity dependent temperature are most often studied in combination with Dirichlet type boundary condition on the velocity field and the temperature. In this work, we propose a formulation of the Stokes system with anisotropic slip boundary condition on the velocity coupled with the heat. More precisely, we carry out (i) a complete analysis of the continuous and discrete formulations in two and three dimensions (ii) formulate and analyse an iterative scheme, (iii) verification of the convergence properties and finally (iv) validation of the model with two dimensional simulation of Lid-driven cavity problem.