<p>An investigation is made into the processes of hydrodynamics and heat transfer in a generalized Couette flow of a non-Newtonian polymer compound in a convergent channel under boundary conditions of the first-kind. Consideration is given to a steady nonisothermal process. The low values of the Reynolds criterion and the high values of the Peclet criterion make it possible to disregard the forces of gravity and inertia, and parallel thermal conductivity. As a rheological model, use is made of an Ellis model with a viscosity depending on the temperature and dispersion degree of the filler. An analysis is made of high-viscosity media, so the energy equation accounts for a dissipative term. From the equation of motion using this rheological model, the velocity profile is obtained in an explicitly expressed form. The variable viscosity has a significant effect on temperature, stress, and velocity fields along the channel length. As a result, this has a substantial effect on the velocity of the process of filler dispersion. The problem has been solved numerically using the method of finite differences.</p>