Cross-section averaged phase momentum balance (${\partial }_t(\alpha \bar{\rho }\bar{v})+{\partial }_x(\alpha \bar{\rho }\overline{v^2})=M/A-{\partial }_x(\alpha \bar{p})$)

For a phase occupying area $\alpha A$ of a constant-area pipe, define $\bar{\rho }=(\alpha A{)}^{-1}\int \rho dA$, $\bar{v}=\int \rho vdA/(\alpha A\bar{\rho })$ and $\overline{v^2}=\int \rho v^{2}dA/(\alpha A\bar{\rho })$. The momentum per length is $A\alpha \bar{\rho }\bar{v}$, the advected momentum flux is $A\alpha \bar{\rho }\overline{v^2}$, and end pressures contribute $-A{\partial }_x(\alpha \bar{p})$. Applying Newton's second law to a slice proves the displayed balance, with $M$ all other phase forces per length. Interphase pressure terms can cancel the derivative of the area fraction. Equality of the second moment and squared mean requires a separate closure.