Mathematical analysis of steady non-isothermal flows of a micropolar fluid

This paper deals with a boundary value problem (BVP) describing the 3D steady non-isothermal flow of a micropolar fluid (with couple stresses) in a bounded vessel. The presence of couple shear stresses is a consequence of taking into account the rotational degrees of freedom for an elementary volume of a fluid. Since the governing equations of the couple stress fluid are of order 4, for a well-posed setting of a BVP modeling internal flows, it is not enough to prescribe the no-slip (stick) condition on solid walls on a vessel. Therefore, we come across the non-trivial issue of specifying extra boundary conditions for the velocity field that are reasonable from both physical and mathematical points of view. As one of approaches to solving this problem, we suggest introducing a vorticity-type boundary condition with a parameter, the choice of which determines one of two scenarios: either the no-slip regime together with the vanishing of the vorticity on the boundary (the “super-stick” regime) or the no-slip regime under the vanishing of the couple stresses on the boundary. The interpretation of both boundary conditions is proposed in the terms of the normal and tangential components of the couple stress vector. Another important feature of our work is that we take into account the viscous dissipation effect in the energy balance equation unlike conventional approaches that overlook this effect. We introduce both weak and strong formulations of the considered BVP and study the relationship between the ones. Applying a generalized version of the Leray–Schauder fixed-point theorem, we prove the existence of a weak solution and, under additional assumptions for the model data, the uniqueness of this solution. Moreover, some qualitative and quantitative properties of solutions are established. In particular, we analyze the convergence of the constructed solutions to the solutions of the stationary Navier–Stokes system as the couple stress viscosity coefficient tends to zero.