Thermal Analysis of the Downstream Spreading of a Planar Power-Law Liquid Jet with Convective Free-Surface Cooling

The two-dimensional thermal liquid jet of a non-Newtonian power-law fluid is investigated under shear-rate-dependent thermal diffusivity, resulting in a one-way coupled nonlinear system governing momentum and thermal transport. Two physically distinct free-surface thermal boundary conditions are examined: adiabatic insulation and convective heat loss. Conservation laws and conserved quantities for the governing system are derived systematically using the multiplier method. By coupling an appropriate conserved vector with an admitted Lie point symmetry, the governing partial differential equations are reduced to a coupled system of ordinary differential equations. Closed-form parametric families of solutions are then obtained for the thermal field. The analysis reveals fundamentally different thermal transport mechanisms across rheological regimes: shear-thinning fluids enhance thermal redistribution and become increasingly sensitive to convective cooling as the Biot number increases, whereas shear-thickening fluids suppress internal thermal transport, promoting greater thermal retention within the jet core and reducing the influence of free-surface cooling. These findings clarify the interplay between rheology, nonlinear thermal diffusion and free-surface cooling and provide new analytical insight into downstream thermal transport in non-Newtonian liquid jets.