Long time existence of solutions to 2D Navier-Stokes equations with inflow-outflow and heat convection (Q648540)

An incompressible, viscous flow in a cylindrical pipe is considered. The flow is described by the non-stationary Navier-Stokes equations coupled with the heat equation. On some parts of the boundary, inflow-outflow conditions are considered; on the other parts, slip boundary conditions are assumed. For the temperature, Neumann boundary conditions are imposed. After establishing some a priori estimates for the velocity and for the temperature, the main result of this article is obtained. By means of Leray-Schauder's fixed point theorem, the global existence of the solutions for the considered coupled system with inflow-outflow conditions is proved.