9. The Steady Equations for Heat-Conducting Fluids

The chapter delves into the steady equations governing heat-conducting, incompressible Newtonian fluids with dissipative heating under various boundary conditions. It presents variational formulations involving a variational inequality for velocity and a variational equation for temperature, equivalent to the original PDE problems for smooth solutions. The existence of solutions is rigorously proven using auxiliary problems and parameter approximations. The chapter also highlights the conditions under which solutions exist for both static and total pressure cases, offering a detailed analysis of aerospace and engineering applications. The content is enriched with bibliographical remarks, situating the work within the broader context of existing research.