Study of the non-isothermal coupled problem with mixed boundary conditions in a thin domain with friction law

This paper deals with the asymptotic behavior of a coupled system involving of an incompressible Bingham fluid and the equation of the heat energy, in a three-dimensional bounded domain with Tresca free boundary friction conditions. First we prove the existence and uniqueness results for the weak solution. Second, we show the strong convergence of the velocity and the temperature. Then a specific Reynolds limit equation is obtained, and the uniqueness of the limit velocity and temperature are proved.