ON THE EQUATIONS GOVERNING THE FLOW OF MECHANICALLY INCOMPRESSIBLE, BUT THERMALLY EXPANSIBLE, VISCOUS FLUIDS

Abstract

In this paper, the stationary flow of a heat conducting viscous fluid, which is mechanically incompressible but thermally expansible is studied. The flow takes place in a bounded domain and the discharge is prescribed. The thermodynamical modeling of this situation is discussed first. Then the stationary model with zero Eckert number and prove existence of a solution is studied. Using these results, the Oberbeck–Boussinesq system obtained in the zero expansivity limit is proved. Next, uniqueness for small data and the regularity of the weak solutions is proved. For the unique regular solution the higher order corrections for Boussinesq' approximation and we constructed the error estimate with respect to the thermal expansivity coefficient is proved. The next order correction in this limit is an Oseen type momentum equation coupled with a linear advection/diffusion equation for the temperature. Such higher order correction seems to be new in the literature.