Application of the local volume-averaging theorems [Eqs. (2.4.1a), (2.4.1b), (2.4.2), and (2.4.7)] to the phasic conservation equations given in leads to the following set of local volume-averaged conservation equations for multiphase flow. These equations are rigorous and subject only to the length-scale restriction, Eq. (2.4.3), which is inherent in the local volume-averaging theorems. Unless otherwise stated, all solid structures are stationary, nonporous, and nonreacting; Uk and Wk vanish in Awk. Both volume porosities (γv) and directional surface porosities ($\gamma _{\hbox{\scriptsize\scitshape a}x} $, $\gamma _{\hbox{\scriptsize\scitshape a}y} $, and $\gamma _{\hbox{\scriptsize\scitshape a}z} $) are invariant in time and in space, and they are functions of their initial structure locations, sizes, and shapes.

Local volume-averaged mass conservation equation of a phase and its interfacial balance equation

The continuity equation is written as