10. The Non-steady Equations for Heat-Conducting Fluids

This chapter delves into the complexities of non-steady equations for heat-conducting fluids, focusing on the motion of incompressible Newtonian fluids under mixed boundary conditions. It begins by establishing a variational formulation for the problem, which includes a time-dependent variational inequality for velocity and a variational equation for temperature. The chapter then proves the existence of a solution to the problem, demonstrating that if the buoyancy effect and energy dissipation due to expansion are small enough, a solution with a 'defect measure' can be found. The main result is supported by a thorough analysis and estimates of solutions to an approximate problem, showcasing the intricate nature of the problem and the innovative approaches needed to solve it. The chapter concludes with a discussion of the open challenges and related works in the field, highlighting the significance of this research in advancing our understanding of heat-conducting fluids.