On the boundary value problem for a model of nonisothermal flows of a non-Newtonian fluid

Under study is a stationary model describing non-Newtonian fluid flows with the viscosity dependent on the strain rate and the heat transfer in a bounded 3D domain. This model is a strongly nonlinear system of coupled partial differential equations for the velocity field, temperature, and pressure. On the boundary of the flow domain, the system is supplemented with a no-slip condition and a linear Robin-type boundary condition for the temperature. An operator formulation of this boundary-value problem is proposed. Using the properties of $d$-monotone operators and the Leray–Schauder Fixed Point Theorem, we prove the existence of weak solutions under natural conditions for the data of the model. It is also shown that the solutions set is bounded and closed.