@wikipedia

In multiphase flow the Darcy friction factor can be calculated as Darcy friction factor Single-phase @model with specific approximation of Reynolds number:

{\rm Re} = \frac{ \sum_\alpha \rho_\alpha \, u_\alpha^2 \, A_\alpha}
{\sum_\alpha \mu_\alpha \, u_\alpha \, \sqrt{A_\alpha} } =
\frac{ \sum_\alpha \rho_\alpha \, q_\alpha^2 / A_\alpha}
{\sum_\alpha \mu_\alpha \, q_\alpha / \sqrt{A_\alpha} } =
\frac{1}{\sqrt{A}} \cdot \frac{ \sum_\alpha \rho_\alpha \, q_\alpha^2 / s_\alpha}
{\sum_\alpha \mu_\alpha \, q_\alpha / \sqrt{s_\alpha} }

where

	-phase fluid density		volume share occupied by -phase
	-phase fluid viscosity		cross-sectional area occupied by -phase
	-phase fluid velocity		total cross-sectional area

Reynolds number represent the ration of intertial forces to viscous forces:

{\rm Re} = \frac{\rm Intertial \ Forces}{\rm Viscocus \ Forces}

Homogeneous Pipe Flow

Homogeneous Pipe Flow is characterized by the same phase velocities: (no slippage) and the multiphase Reynolds number takes simpler form:

{\rm Re} =\frac{ \sum_\alpha \rho_\alpha \, u_\alpha \, A_\alpha}
{\sum_\alpha \mu_\alpha \, \sqrt{A_\alpha} } =\frac{ \dot m}
{\sum_\alpha \mu_\alpha \, \sqrt{A_\alpha} } = \frac{\dot m}{\sqrt{A}} \cdot \frac{1}{ \sum_\alpha \mu_\alpha \, \sqrt{s_\alpha} }

2-phase Gas-Liquid flow

{\rm Re} = \frac{\rho_L \, u_L^2 \, A_L + \rho_g \, u_g^2 \, A_g}{\mu_L \, u_L \, \sqrt{A_L} + \mu_g \, u_g \, \sqrt{A_g}}

where

	liquid density		gas density
	liquid velocity		gas velocity
	cross-sectional area occupied by liquid		cross-sectional area occupied by gas
	liquid viscosity		gas viscosity

Reference

L. E. Ortiz-Vidala, N. Mureithib, and O. M. H. Rodrigueza ,TWO-PHASE FRICTION FACTOR IN GAS-LIQUID PIPE FLOW, Engenharia Térmica (Thermal Engineering), Vol. 13, No. 2, December 2014, p. 81-88

Shannak, B. A., 2008, Frictional Pressure Drop of Gas Liquid Two-Phase Flow in Pipes, Nuclear Engineering and Design, Vol. 238, pp. 3277-3284., doi.org/10.1016/j.nucengdes.2008.08.015