On the non-isothermal, non-Newtonian Hele-Shaw flows in a domain with rough boundary

In this paper, we investigate the roughness-induced effects on the nonlinear system describing the non-isothermal Hele-Shaw flow of a non-Newtonian fluid. Allowing different orders between the amplitude and the period of the boundary roughness, we rigorously determine the limit problems and also provide the corresponding rates of convergence in the critical case.