Macroscopic equations of non-Newtonian fluid flow and heat transfer in a porous matrix

This paper presents a rigorous derivation of the macroscopic equations that describe the flow of non-Newtonian fluids in porous media. The equations a re obtained from the corresponding microscopic equations with the method of volume averaging. The traditional averaging procedure leads to a mass cons ervation, a momentum conservation and an energy conservation equations expr essed in terms of the volume-averaged temperature, pressure, and velocity. The momentum equation contains, in addition to the desired volume-averaged values, an integral involving the point value of the pressure, velocity, an d other lower-order terms. The presence of this integral in the volume-aver aged equation gives rise to an interfacial flux term that requires a closur e condition. In the present approach this interface flux is represented in terms of an overall driving force that is based on flow over dilute arrays of spheres. The analysis of this study is based upon the conditions that th e porous material is (i) rigid, (ii) stationary, and (iii) locally isotropi c with respect to average geometrical properties.