Non-isothermal, non-Newtonian Hele–Shaw flows within Cattaneo’s heat flux law

In this paper, we consider a non-isothermal, non-Newtonian injection process. This leads to the study of a novel elliptic–hyperbolic system. The hyperbolic nature of the system arises because we replace the infinite speed of propagation from classical, thermal elasticity by a finite propagation velocity. We present a formal derivation of the elliptic–hyperbolic system starting from conservations of mass, momentum, and energy in a three-dimensional domain, where the removal of the infinite propagation speed is achieved using Cattaneo’s law for heat conduction. The existence of weak solutions to certain elliptic–hyperbolic problems associated with the resulting equations is proved.