Strong Solutions to Buoyancy-Driven Flows in Channel-Like Bounded Domains

Abstract

We consider a boundary-value problem for steady flows of viscous incompressible heat-conducting fluids in channel-like bounded domains. The fluid flow is governed by balance equations for linear momentum, mass, and internal energy. The internal energy balance equation of this system takes into account the phenomena of the viscous energy dissipation and includes the adiabatic heat effects. The system of governing equations is provided by suitable mixed boundary conditions modeling the behavior of the fluid on fixed walls and open parts of the channel. Due to the fact that some uncontrolled “backward flow” can take place at the outlets of the channel, there is no control of the convective terms in balance equations for linear momentum and internal energy, and consequently, one is not able to prove energy type estimates. This makes the qualitative analysis of this problem more difficult. In this paper, the existence of the strong solution is proven by a fixed-point technique for sufficiently small external forces.