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Momentum Conservation

Mixture of Gases Momentum:

\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}$\displaystyle \rho_{m}^{}$\egroupVm\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}$\displaystyle \rho_{m}^{}$\egroupVm2\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}$\displaystyle \mathcal {F}$\egroupmVm + \bgroup\color[rgb]{1,.5,0}$\displaystyle \alpha_{m}^{}$\egroup\bgroup\color[rgb]{1,.5,0}$\displaystyle \rho_{m}^{}$\egroupgz

Liquid Momentum:

\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}$\displaystyle \rho_{l}^{}$\egroupVl\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}$\displaystyle \rho_{l}^{}$\egroupVl2\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}$\displaystyle \mathcal {F}$\egrouplVl + \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}$\displaystyle \rho_{l}^{}$\egroupgz



Michael L. Hall