\begin{equation}
\frac{{\p \left( {{\alpha_k }{\rho_k}{\bfU_k }} \right)}}{{\p t}} + \nabla \cdot \left( {{\alpha_k}{\rho_k } {{\bfU_k} {\bfU_k}} } \right) - \nabla \cdot \left( {{\alpha_k}{ \rho_k}{\tau_k}} \right)
= - {\alpha_k} \nabla p_k + {\alpha_k}{\rho_k} \bfg + \sum {\bfM_{ij}},
\end{equation}
\begin{equation}
\frac{{\p \left( {{\alpha_k }{\rho_k}{ }} \right)}}{{\p t}} + \nabla \cdot \left( {{\alpha_k}{\rho_k } { {\bfU_k}} } \right) =0
\end{equation}
\begin{equation}
\nabla \cdot \left( {{\alpha_k}{\rho_k } { {\bfU_k}} } \right) =0
\end{equation}
\begin{equation}
\sum {{\alpha_k}{\rho_k } { {\bfU_k}} }\cdot\bfS_f =0
\end{equation}
\begin{equation}
\tau_k=-\nu_\mathrm{k,eff}\left(\nabla \bfU_k+\nabla^\rT \bfU_k\right)+\frac{2}{3}\nu_\mathrm{k,eff}\nabla \cdot \left(\bfU_k \cdot\bfI\right),
\label{taud}
\end{equation}
\begin{equation}
\bfM_{\mathrm{drag}}=\frac{3}{4}\alpha_k\rho_\rc C_\rD\frac{1}{d_k} \left|\bfU_\rc-\bfU_k\right| \left(\bfU_\rc-\bfU_k\right),
\end{equation}
\begin{equation}
Re=\frac{d_k|\bfU_k-\bfU_\rc|}{\nu_\rc}
\end{equation}
\begin{equation}
\bfM_\lift=\alpha_\rd C_\rL\rho_\rc\bfU_\rr\times\left(\nabla\times\bfU_\rc\right),
\end{equation}
\begin{equation}
\bfM_\wall=C_\wall\rho_\rc\alpha_k|\bfU_\rc-\bfU_k|^2\cdot\bfn
\end{equation}
\begin{equation}
\bfM_\turb=C_\rT\rho_\rc k_\rc\nabla\alpha_\rd,
\end{equation}
\begin{equation}\label{m1}
\frac{{\p \left( {{\alpha_k }{\rho_k }{\bfU_k}} \right)}}{{\p t}} + \nabla \cdot \left( {{\alpha_k}{\rho_k} {{\bfU_k} {\bfU_k}} } \right) - \nabla \cdot \left( {{\alpha_\rd}{ \rho_\rd}{\tau_\rd}} \right)
= -\Kd_k\bfU_k+\bfM_{\lift,k}+\bfM_{\turb,k}+\bfM_{\wall,k},
\end{equation}
\begin{equation}\label{Kd}
\Kd=\frac{3}{4}\alpha_k\rho_\rc C_{\rD,k}\frac{1}{d_k} \left|\bfU_\rc-\bfU_k\right|.
\end{equation}
\begin{equation}
{A_{k,\mathrm{P}}}\mathbf{U}_{k,\mathrm{P}}{\rm{ + }}\sum {A_{k,\mathrm{N}}\mathbf{U}_{k,\mathrm{N}}} = S_{k,\mathrm{P}},
\label{apanmomrd}
\end{equation}
\begin{equation}
\mathbf{HbyA}_{k,\mathrm{P}} = \frac{1}{{{A_{k,\mathrm{P}}}}}\left( { - \sum {{A_{k,\mathrm{N}}}\mathbf{U}_{k,\mathrm{N}}} + S_{k,\mathrm{P}}} \right),
\label{hbyad}
\end{equation}
\begin{equation}
\bfU_{k,\rP} = \bfHbyA_{k,\rP}+\frac{\alpha_{k,\rP}}{A_{k,\rP}}\left(\nabla p_{\mathrm{rgh},\rP}-\alpha_{\rc,\rP}\left(\rho_\rc-\rho_k\right)\bfg-\bfg\cdot\bfh_\rP\nabla\rho_\rP\right)+\frac{\Kd_k}{A_{k,\rP}}\bfU_{\rc,\rP},
\label{hbyad2}
\end{equation}
\begin{equation}\label{incompressiblep}
\sum\alpha_{k,f}\phi_{k}+\alpha_{\rc,f}\phi_{\rc}=\nabla\cdot\left(\left(\sum\alpha_{k,\rP}\frac{\alpha_{k,\rP}}{A_{k,\rP}}+\alpha_{\rc,\rP}\frac{\alpha_{\rc,\rP}}{A_{\rc,\rP}}
\right)\nabla p_{\mathrm{rgh},\rP}\right),
\end{equation}
\begin{equation}
\phi_{k}=\left(\bfHbyA_{k,f}+\frac{\alpha_{k,f}}{A_{k,f}}\left(-\alpha_{\rc,f}\left(\rho_\rc-\rho_\rd\right)\bfg-\bfg\cdot\bfh_f\nabla\rho_f\right)+\frac{\Kd_f}{A_{k,f}}\bfU_{\rc,f}\right)\cdot\bfS_f
\end{equation}
\begin{equation}
\phi_{\rc}=\left(\bfHbyA_{\rc,f}+\frac{\alpha_{\rc,f}}{A_{\rc,f}}\left(-\alpha_{\rd,f}\left(\rho_\rd-\rho_\rc\right)\bfg-\bfg\cdot\bfh_f\nabla\rho_f\right)+\frac{\Kd_f}{A_{\rc,f}}\bfU_{\rd,f}\right)\cdot\bfS_f
\end{equation}
