Steady-state flows of thermal viscous incompressible fluids with convective-radiation effects

The work deals with the coupled system constituted by the equations of motion and energy with nonlinear and nonlocal boundary conditions in order to describe the thermal flow motion of a class of non-Newtonian fluids and the convective-radiation balance, respectively. For the constitutive laws in an n-dimensional space (n = 2, 3), the stress tenser and the heat flux axe considered related with the (p, q) coercivity parameters for p > 2n/(n + 1) and q > np/(p(n + 1) - n), respectively. The radiation character on the boundary condition presents an additional difficulty to the problem which already includes energy-dependent viscosity and conductivity behaviors. In the framework of nonstandard Sobolev spaces we prove the existence of steady-state solutions applying a fixed point argument. We also study the existence of a solution to a coupled system motivated by the buoyancy driven flows from geophysical and astrophysical models.