χ_{1}

to be one in phase 1 and zero in phase 2, including its dependence on both transverse coordinates even though the averaged fields depend only on

x, t

. With no interphase mass transfer, its distributional phase-mass equation is

\partial_{t} (ρ χ_{1}) + \nabla \cdot (ρ v χ_{1}) = 0

. At a pipe cross section

Σ

of area

A

, set

α = \frac{1}{A} \int_{Σ} χ_{1} d A, ρ_{1} = \frac{1}{A α} \int_{Σ} ρ χ_{1} d A, u_{1} = \frac{\int _{Σ} ρ v _{x} χ _{1} d A}{A α ρ _{1}} .

(1)

Thus

ρ_{1}

is an intrinsic phase density and

u_{1}

is the density-weighted mean axial velocity. This weighting is necessary when density varies within the phase; it becomes the ordinary intrinsic mean for an incompressible phase of constant density.

Integrate over the cross section. The transverse divergence contributes no boundary mass flux at the impermeable pipe wall, and the material interface has no mass leakage. The section area is constant, so