# Momentum equation

The general multi-ﬁeld momentum balance equation for the phase k is expressed as

$\frac{\partial}{\partial t} (\alpha_k \rho_k U_{k,i}) + \frac{\partial}{\partial x_j} (\alpha_k \rho_k U_{k,i} U_{k,j}) = \frac{\partial}{\partial x_j} (\alpha_k \tau_{k,ij}+\sum^{Re}_{k,ij})-\alpha_k\frac{\partial P}{\partial x_i}+\alpha_k \rho_k g_{i} +I_{p→k,i}+\alpha_k S_k$   (11)

Where $g_i$ represents acceleration due to gravity and P the mean pressure. The viscous stress tensor is deﬁned by $\tau_{k,ij}=\mu_k(\frac{\partial U_i}{\partial x_j}+\frac{\partial U_j}{\partial x_i}- \frac{2}{3}div(U)\delta_{ij})$ where $\mu_k$ is the dynamic viscosity and $\delta_{ij}$, the Kronecker delta.​

$\sum^{Re}_{k,ij}$ is the turbulent stress tensor, $S_k$ is the external source term and $I_{p→k,i}$ represents the average interface momentum transfer rate from phase p to phase k, that accounts for mass transfer, drag force, added mass force, lift and satisfies the local balance equations

$I_{p→k,i}+ I_{k→p,i}=0$   (12)

NEPTUNE_CFD code includes several assumptions:

1. the surface tension force is neglected,
2. the average pressure of the two phases on either side of the interface are equal.

Furthermore, in this study, the pressure drop and turbulent stress tensor are neglected, the last one because the ﬂow is considered laminar.