Non-isothermal non-Newtonian fluid flow implemented during filling a coaxial channel in the gravity field is investigated with regard to the dissipative heating, the temperature dependence of the effective viscosity, and the presence of the free surface. The rheological behavior of the medium is described by the Сross-WLF model. The problem is solved numerically using the original computational technology based on the control volume approach and the method of invariants for satisfying the boundary conditions on the free boundary. The flow characteristics in the process of filling are demonstrated for various values of the constitutive parameters.