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

$\begin{array}{l}\displaystyle {\rm Re} = \frac{\rho_L \, u_L^2 \, A_L + \rho_g \, u_g^2 \, A_g}{\mu_L \, u_L \, \sqrt{A_L} + \mu_g \, u_g \, \sqrt{A_g}}\end{array}$

where

$\begin{array}{l}\rho_L\end{array}$	liquid density	$\begin{array}{l}\rho_g\end{array}$	gas density
$\begin{array}{l}u_L\end{array}$	liquid velocity	$\begin{array}{l}u_g\end{array}$	gas velocity
$\begin{array}{l}A_L\end{array}$	cross-sectional area occupied by liquid	$\begin{array}{l}A_g\end{array}$	cross-sectional area occupied by gas
$\begin{array}{l}\mu_L\end{array}$	liquid viscosity	$\begin{array}{l}\mu_g\end{array}$	gas viscosity

Derivation

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

$\begin{array}{l}\displaystyle {\rm Re} = \frac{\rm Intertial \ Forces}{\rm Viscocus \ Forces}\end{array}$

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

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Reynolds Number for Multiphase Flow @model

Reference