Partial regularity of weak solutions of the stationary $3D$-Boussinesq system

In this paper we study smoothness of weak solutions of the three-dimensional stationary Boussinesq system describing steady state motion of viscous heat-convergent Newtonian fluid whose viscosity may depend on the temperature of the fluid. The principal feature of the system under consideration is the dissipative term involved into the energy balance equation. This term is equal to the product of the stress and the strain velocity tensors and has the quadratic growth with respect to the gradient of the velocity field. We prove the partial regularity of solutions to this system and give an estimate of the Hausdorff measure of the singular set.