Solvability of the Boussinesq Approximation for Water Polymer Solutions

We consider nonlinear Boussinesq-type equations that model the heat transfer and steady viscous flows of weakly concentrated water solutions of polymers in a bounded three-dimensional domain with a heat source. On the boundary of the flow domain, the impermeability condition and a slip condition are provided. For the temperature field, we use a Robin boundary condition corresponding to the classical Newton law of cooling. By using the Galerkin method with special total sequences in suitable function spaces, we prove the existence of a weak solution to this boundary-value problem, assuming that the heat source intensity is bounded. Moreover, some estimates are established for weak solutions.