The aim of this paper is to study the system consisting of Navier-Stokes equation and energy equation for an incompressible fluid, considered in a bounded domain in
, with homogeneous Dirichlet boundary conditions. In the case of bounded eddy viscosity, first, one proves that the system has weak solutions in the steady-state case and in the bidimensional evolution case. Then, existence of weak solutions is obtained when the energy equation is coupled to the three-dimensional evolution Stokes equation. Finally, the regularity of the solutions and the problem of passing to the limit in the equations are considered.