Abstract
In this chapter we are concerned with the equation for steady flow of heat-conducting incompressible Newtonian fluids with dissipative heating under mixed boundary conditions. The boundary conditions for fluid may include Tresca slip, leak condition, one-sided leak conditions, velocity, pressure, vorticity, stress together and the conditions for temperature may include Dirichlet, Neumann and Robin conditions together. On the basis of results of Sect. 3.1, we get variational formulations consisting of a variational inequality for velocity and a variational equation for temperature, which are equivalent to the original PDE problems for smooth solutions.
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