Momentum Conservation

$\bgroup\color{blue}$\displaystyle {\frac{{\partial}}{{\partial t}}}$\egroup$ $\bgroup\color{blue}$\displaystyle \left(\vphantom{ \alpha_m\rho_m V_m }\right.$\egroup$ $\bgroup\color{blue}$\displaystyle \alpha_{m}^{}$\egroup$ $\bgroup\color{blue}$\displaystyle \rho_{m}^{}$\egroup$ V_m $\bgroup\color{blue}$\displaystyle \left.\vphantom{ \alpha_m\rho_m V_m }\right)$\egroup$ + $\bgroup\color{red}$\displaystyle {\frac{{\partial}}{{\partial z}}}$\egroup$ $\bgroup\color{red}$\displaystyle \left(\vphantom{ \alpha_m \rho_m V_m^2 }\right.$\egroup$ $\bgroup\color{red}$\displaystyle \alpha_{m}^{}$\egroup$ $\bgroup\color{red}$\displaystyle \rho_{m}^{}$\egroup$ V_m² $\bgroup\color{red}$\displaystyle \left.\vphantom{ \alpha_m \rho_m V_m^2 }\right)$\egroup$ = - $\bgroup\color[rgb]{0,.5,.2}$\displaystyle \alpha_{m}^{}$\egroup$ $\bgroup\color[rgb]{0,.5,.2}$\displaystyle {\frac{{\partial P_m}}{{\partial z}}}$\egroup$ - $\bgroup\color[rgb]{.5,0,1}$\displaystyle \mathcal {F}$\egroup$ _mV_m + $\bgroup\color[rgb]{1,.5,0}$\displaystyle \alpha_{m}^{}$\egroup$ $\bgroup\color[rgb]{1,.5,0}$\displaystyle \rho_{m}^{}$\egroup$ g_z

$\bgroup\color{blue}$\displaystyle \epsilon_{v}^{}$\egroup$ $\bgroup\color{blue}$\displaystyle {\frac{{\partial}}{{\partial t}}}$\egroup$ $\bgroup\color{blue}$\displaystyle \left(\vphantom{ \alpha_l\rho_l V_l }\right.$\egroup$ $\bgroup\color{blue}$\displaystyle \alpha_{l}^{}$\egroup$ $\bgroup\color{blue}$\displaystyle \rho_{l}^{}$\egroup$ V_l $\bgroup\color{blue}$\displaystyle \left.\vphantom{ \alpha_l\rho_l V_l }\right)$\egroup$ + $\bgroup\color{red}$\displaystyle \epsilon_{v}^{}$\egroup$ $\bgroup\color{red}$\displaystyle {\frac{{\partial}}{{\partial z}}}$\egroup$ $\bgroup\color{red}$\displaystyle \left(\vphantom{ \alpha_l \rho_l V_l^2 }\right.$\egroup$ $\bgroup\color{red}$\displaystyle \alpha_{l}^{}$\egroup$ $\bgroup\color{red}$\displaystyle \rho_{l}^{}$\egroup$ V_l² $\bgroup\color{red}$\displaystyle \left.\vphantom{ \alpha_l \rho_l V_l^2 }\right)$\egroup$ = - $\bgroup\color[rgb]{0,.5,.2}$\displaystyle \epsilon_{v}^{}$\egroup$ $\bgroup\color[rgb]{0,.5,.2}$\displaystyle \alpha_{l}^{}$\egroup$ $\bgroup\color[rgb]{0,.5,.2}$\displaystyle {\frac{{\partial P_l}}{{\partial z}}}$\egroup$ - $\bgroup\color[rgb]{.5,0,1}$\displaystyle \mathcal {F}$\egroup$ _lV_l + $\bgroup\color[rgb]{1,.5,0}$\displaystyle \epsilon_{v}^{}$\egroup$ $\bgroup\color[rgb]{1,.5,0}$\displaystyle \alpha_{l}^{}$\egroup$ $\bgroup\color[rgb]{1,.5,0}$\displaystyle \rho_{l}^{}$\egroup$ g_z