有限要素法・流体力学による数値計算

\[ \phi_x=\frac{\partial V_x}{\partial\tau} +V_x\frac{\partial V_x}{\partial X} +V_y\frac{\partial V_x}{\partial Y} +V_z\frac{\partial V_x}{\partial Z} -\frac{\partial\sigma^*_{xx}}{\partial X} -\frac{\partial\sigma^*_{yx}}{\partial Y} -\frac{\partial\sigma^*_{zx}}{\partial Z} -g_x^*=0 \]

\[ \phi_y=\frac{\partial V_y}{\partial\tau} +V_x\frac{\partial V_y}{\partial X} +V_y\frac{\partial V_y}{\partial Y} +V_z\frac{\partial V_y}{\partial Z} -\frac{\partial\sigma^*_{xy}}{\partial X} -\frac{\partial\sigma^*_{yy}}{\partial Y} -\frac{\partial\sigma^*_{zy}}{\partial Z} -g_y^*=0 \]

\[ \phi_z=\frac{\partial V_z}{\partial\tau} +V_x\frac{\partial V_z}{\partial X} +V_y\frac{\partial V_z}{\partial Y} +V_z\frac{\partial V_z}{\partial Z} -\frac{\partial\sigma^*_{xz}}{\partial X} -\frac{\partial\sigma^*_{yz}}{\partial Y} -\frac{\partial\sigma^*_{zz}}{\partial Z} -g_z^*=0 \]

\[ \int_V \begin{bmatrix} N_1\\N_2\\N_3\\N_4 \end{bmatrix} \phi_i\,dV = \int_V \begin{bmatrix} 0\\0\\0\\0 \end{bmatrix} dV = \begin{bmatrix} 0\\0\\0\\0 \end{bmatrix} \]

\[ \begin{aligned} V_x&={}N_1V_{x,1}+N_2V_{x,2}+N_3V_{x,3}+N_4V_{x,4} = \begin{bmatrix}N_1&N_2&N_3&N_4\end{bmatrix} \begin{bmatrix}V_{x,1}\\V_{x,2}\\V_{x,3}\\V_{x,4}\end{bmatrix} =[N]\{V_x\}\\[6pt] V_y&={}N_1V_{y,1}+N_2V_{y,2}+N_3V_{y,3}+N_4V_{y,4}\\[6pt] V_z&={}N_1V_{z,1}+N_2V_{z,2}+N_3V_{z,3}+N_4V_{z,4}\\[6pt] P&={}N_1P_{z,1}+N_2P_{z,2}+N_3P_{z,3}+N_4P_{z,4} \end{aligned} \]

\[ \begin{aligned} &\int_V \begin{bmatrix}N_1\\N_2\\N_3\\N_4\end{bmatrix} \phi_i\,dV\\[6pt] &={}\int_V \begin{bmatrix}N_1\\N_2\\N_3\\N_4\end{bmatrix} \left( \frac{\partial V_i}{\partial\tau} +V_x\frac{\partial V_i}{\partial X} +V_y\frac{\partial V_i}{\partial Y} +V_z\frac{\partial V_i}{\partial Z} -\frac{\partial\sigma^*_{xi}}{\partial X} -\frac{\partial\sigma^*_{yi}}{\partial Y} -\frac{\partial\sigma^*_{zi}}{\partial Z} -g_i^* \right)dV \end{aligned} \]

\[ \begin{aligned} &={}\int_V \begin{bmatrix}N_1\\N_2\\N_3\\N_4\end{bmatrix} \frac{\partial V_i}{\partial\tau}\,dV\\[6pt] &+V_x\int_V \begin{bmatrix}N_1\\N_2\\N_3\\N_4\end{bmatrix} \frac{\partial V_i}{\partial X}\,dV +V_y\int_V \begin{bmatrix}N_1\\N_2\\N_3\\N_4\end{bmatrix} \frac{\partial V_i}{\partial Y}\,dV +V_z\int_V \begin{bmatrix}N_1\\N_2\\N_3\\N_4\end{bmatrix} \frac{\partial V_i}{\partial Z}\,dV\\[6pt] &-\int_V \begin{bmatrix}N_1\\N_2\\N_3\\N_4\end{bmatrix} \frac{\partial\sigma^*_{xi}}{\partial X}\,dV -\int_V \begin{bmatrix}N_1\\N_2\\N_3\\N_4\end{bmatrix} \frac{\partial\sigma^*_{yi}}{\partial Y}\,dV -\int_V \begin{bmatrix}N_1\\N_2\\N_3\\N_4\end{bmatrix} \frac{\partial\sigma^*_{zi}}{\partial Z}\,dV\\[6pt] &-\int_V \begin{bmatrix}N_1\\N_2\\N_3\\N_4\end{bmatrix} g_i^*\,dV \end{aligned} \]

\[ \begin{aligned} &={}\int_V[N]^T\frac{\partial V_i}{\partial\tau}dV\\[6pt] &+V_x\int_V[N]^T\frac{\partial V_i}{\partial X}dV +V_y\int_V[N]^T\frac{\partial V_i}{\partial Y}dV +V_z\int_V[N]^T\frac{\partial V_i}{\partial Z}dV\\[6pt] &-\int_V[N]^T\frac{\partial\sigma^*_{xi}}{\partial X}dV -\int_V[N]^T\frac{\partial\sigma^*_{yi}}{\partial Y}dV -\int_V[N]^T\frac{\partial\sigma^*_{zi}}{\partial Z}dV\\[6pt] &-\int_V[N]^Tg_i^*dV \end{aligned} \]

\[ \begin{aligned} &={}\int_V[N]^T \frac{[N](\{V_i\}^{\tau+\Delta\tau}-\{V_i\}^{\tau})}{\Delta\tau}dV\\[6pt] &+V_x\int_V[N]^T\frac{\partial[N]\{V_i\}}{\partial X}dV +V_y\int_V[N]^T\frac{\partial[N]\{V_i\}}{\partial Y}dV +V_z\int_V[N]^T\frac{\partial[N]\{V_i\}}{\partial Z}dV\\[6pt] &-\int_S[N]^T\sigma^*_{xi}n_xdS +\int_V\left[\frac{\partial N}{\partial X}\right]^T\sigma^*_{xi}dV\\[6pt] &-\int_S[N]^T\sigma^*_{yi}n_ydS +\int_V\left[\frac{\partial N}{\partial Y}\right]^T\sigma^*_{yi}dV\\[6pt] &-\int_S[N]^T\sigma^*_{zi}n_zdS +\int_V\left[\frac{\partial N}{\partial Z}\right]^T\sigma^*_{zi}dV\\[6pt] &-\int_V[N]^Tg_i^*dV \end{aligned} \]

\[ \begin{aligned} &={}\int_V[N]^T[N]dV\, \frac{\{V_i\}^{\tau+\Delta\tau}-\{V_i\}^{\tau}}{\Delta\tau}\\[6pt] &+V_x\int_V[N]^T\left[\frac{\partial N}{\partial X}\right]dV\{V_i\} +V_y\int_V[N]^T\left[\frac{\partial N}{\partial Y}\right]dV\{V_i\} +V_z\int_V[N]^T\left[\frac{\partial N}{\partial Z}\right]dV\{V_i\}\\[6pt] &+\int_V\left[\frac{\partial N}{\partial X}\right]^T\sigma^*_{xi}dV +\int_V\left[\frac{\partial N}{\partial Y}\right]^T\sigma^*_{yi}dV +\int_V\left[\frac{\partial N}{\partial Z}\right]^T\sigma^*_{zi}dV\\[6pt] &-\int_S[N]^T\left(\sigma^*_{xi}n_x+\sigma^*_{yi}n_y+\sigma^*_{zi}n_z\right)dS\\[6pt] &-\int_V[N]^Tg_i^*dV \end{aligned} \]

\[ \begin{aligned} &={}\int_V[N]^T[N]dV\, \frac{\{V_i\}^{\tau+\Delta\tau}-\{V_i\}^{\tau}}{\Delta\tau}\\[6pt] &+V_x\int_V[N]^T\left[\frac{\partial N}{\partial X}\right]dV\{V_i\} +V_y\int_V[N]^T\left[\frac{\partial N}{\partial Y}\right]dV\{V_i\} +V_z\int_V[N]^T\left[\frac{\partial N}{\partial Z}\right]dV\{V_i\}\\[6pt] &+\int_V\left[\frac{\partial N}{\partial X}\right]^T \left\{-\delta_{xi}P+\frac{1}{Re}\left(\frac{\partial V_i}{\partial X}+\frac{\partial V_x}{\partial X_i}\right)\right\}dV\\[6pt] &+\int_V\left[\frac{\partial N}{\partial Y}\right]^T \left\{-\delta_{yi}P+\frac{1}{Re}\left(\frac{\partial V_i}{\partial Y}+\frac{\partial V_y}{\partial X_i}\right)\right\}dV\\[6pt] &+\int_V\left[\frac{\partial N}{\partial Z}\right]^T \left\{-\delta_{zi}P+\frac{1}{Re}\left(\frac{\partial V_i}{\partial Z}+\frac{\partial V_z}{\partial X_i}\right)\right\}dV\\[6pt] &-\frac{2K^*}{We}n_i\int_S[N]^T dS\\[6pt] &-\int_V[N]^Tg_i^*dV \end{aligned} \]

\[ \begin{aligned} &={}\int_V[N]^T[N]dV\, \frac{\{V_i\}^{\tau+\Delta\tau}-\{V_i\}^{\tau}}{\Delta\tau}\\[6pt] &+V_x\int_V[N]^T\left[\frac{\partial N}{\partial X}\right]dV\{V_i\} +V_y\int_V[N]^T\left[\frac{\partial N}{\partial Y}\right]dV\{V_i\} +V_z\int_V[N]^T\left[\frac{\partial N}{\partial Z}\right]dV\{V_i\}\\[6pt] &+\int_V\left[\frac{\partial N}{\partial X}\right]^T \left\{-\delta_{xi}P+\frac{1}{Re}\left(\frac{\partial V_i}{\partial X}+\frac{\partial V_x}{\partial X_i}\right)\right\}dV\\[6pt] &+\int_V\left[\frac{\partial N}{\partial Y}\right]^T \left\{-\delta_{yi}P+\frac{1}{Re}\left(\frac{\partial V_i}{\partial Y}+\frac{\partial V_y}{\partial X_i}\right)\right\}dV\\[6pt] &+\int_V\left[\frac{\partial N}{\partial Z}\right]^T \left\{-\delta_{zi}P+\frac{1}{Re}\left(\frac{\partial V_i}{\partial Z}+\frac{\partial V_z}{\partial X_i}\right)\right\}dV\\[6pt] &-\frac{2K^*}{We}n_i\int_S[N]^T dS\\[6pt] &-\int_V[N]^Tg_i^*dV \end{aligned} \]

\[ \begin{aligned} &={}\int_V[N]^T[N]dV\, \frac{\{V_i\}^{\tau+\Delta\tau}-\{V_i\}^{\tau}}{\Delta\tau}\\[6pt] &+V_x\int_V[N]^T\left[\frac{\partial N}{\partial X}\right]dV\{V_i\} +V_y\int_V[N]^T\left[\frac{\partial N}{\partial Y}\right]dV\{V_i\} +V_z\int_V[N]^T\left[\frac{\partial N}{\partial Z}\right]dV\{V_i\}\\[6pt] &+\frac1{Re}\int_V\left[\frac{\partial N}{\partial X}\right]^T\frac{\partial V_i}{\partial X}dV +\frac1{Re}\int_V\left[\frac{\partial N}{\partial X}\right]^T\frac{\partial V_x}{\partial X_i}dV -\int_V\left[\frac{\partial N}{\partial X}\right]^T\delta_{xi}P\,dV\\[6pt] &+\frac1{Re}\int_V\left[\frac{\partial N}{\partial Y}\right]^T\frac{\partial V_i}{\partial Y}dV +\frac1{Re}\int_V\left[\frac{\partial N}{\partial Y}\right]^T\frac{\partial V_y}{\partial X_i}dV -\int_V\left[\frac{\partial N}{\partial Y}\right]^T\delta_{yi}P\,dV\\[6pt] &+\frac1{Re}\int_V\left[\frac{\partial N}{\partial Z}\right]^T\frac{\partial V_i}{\partial Z}dV +\frac1{Re}\int_V\left[\frac{\partial N}{\partial Z}\right]^T\frac{\partial V_z}{\partial X_i}dV -\int_V\left[\frac{\partial N}{\partial Z}\right]^T\delta_{zi}P\,dV\\[6pt] &+\frac{2K^*}{We}n_i\int_S[N]^T dS\\[6pt] &-g_i^*\int_V[N]^T dV \end{aligned} \]

\[ \begin{aligned} &={}\int_V[N]^T[N]dV\, \frac{\{V_i\}^{\tau+\Delta\tau}-\{V_i\}^{\tau}}{\Delta\tau}\\[6pt] &+V_x\int_V[N]^T\left[\frac{\partial N}{\partial X}\right]dV\{V_i\} +V_y\int_V[N]^T\left[\frac{\partial N}{\partial Y}\right]dV\{V_i\} +V_z\int_V[N]^T\left[\frac{\partial N}{\partial Z}\right]dV\{V_i\}\\[6pt] &+\frac1{Re}\int_V\left[\frac{\partial N}{\partial X}\right]^T \frac{\partial[N]\{V_i\}}{\partial X}dV +\frac1{Re}\int_V\left[\frac{\partial N}{\partial X}\right]^T \frac{\partial[N]\{V_x\}}{\partial X_i}dV -\delta_{xi}\int_V\left[\frac{\partial N}{\partial X}\right]^T[N]\{P\}dV\\[6pt] &+\frac1{Re}\int_V\left[\frac{\partial N}{\partial Y}\right]^T \frac{\partial[N]\{V_i\}}{\partial Y}dV +\frac1{Re}\int_V\left[\frac{\partial N}{\partial Y}\right]^T \frac{\partial[N]\{V_y\}}{\partial X_i}dV -\delta_{yi}\int_V\left[\frac{\partial N}{\partial Y}\right]^T[N]\{P\}dV\\[6pt] &+\frac1{Re}\int_V\left[\frac{\partial N}{\partial Z}\right]^T \frac{\partial[N]\{V_i\}}{\partial Z}dV +\frac1{Re}\int_V\left[\frac{\partial N}{\partial Z}\right]^T \frac{\partial[N]\{V_z\}}{\partial X_i}dV -\delta_{zi}\int_V\left[\frac{\partial N}{\partial Z}\right]^T[N]\{P\}dV\\[6pt] &+\frac{2K^*}{We}n_i\int_S[N]^T dS\\[6pt] &-g_i^*\int_V[N]^T dV \end{aligned} \]

\[ \begin{aligned} &={}\int_V[N]^T[N]dV\, \frac{\{V_i\}^{\tau+\Delta\tau}-\{V_i\}^{\tau}}{\Delta\tau}\\[6pt] &+V_x\int_V[N]^T\left[\frac{\partial N}{\partial X}\right]dV\{V_i\} +V_y\int_V[N]^T\left[\frac{\partial N}{\partial Y}\right]dV\{V_i\} +V_z\int_V[N]^T\left[\frac{\partial N}{\partial Z}\right]dV\{V_i\}\\[6pt] &+\frac1{Re}\int_V\left[\frac{\partial N}{\partial X}\right]^T\left[\frac{\partial N}{\partial X}\right]dV\{V_i\} +\frac1{Re}\int_V\left[\frac{\partial N}{\partial X}\right]^T\left[\frac{\partial N}{\partial X_i}\right]dV\{V_x\} -\delta_{xi}\int_V\left[\frac{\partial N}{\partial X}\right]^T[N]dV\{P\}\\[6pt] &+\frac1{Re}\int_V\left[\frac{\partial N}{\partial Y}\right]^T\left[\frac{\partial N}{\partial Y}\right]dV\{V_i\} +\frac1{Re}\int_V\left[\frac{\partial N}{\partial Y}\right]^T\left[\frac{\partial N}{\partial X_i}\right]dV\{V_y\} -\delta_{yi}\int_V\left[\frac{\partial N}{\partial Y}\right]^T[N]dV\{P\}\\[6pt] &+\frac1{Re}\int_V\left[\frac{\partial N}{\partial Z}\right]^T\left[\frac{\partial N}{\partial Z}\right]dV\{V_i\} +\frac1{Re}\int_V\left[\frac{\partial N}{\partial Z}\right]^T\left[\frac{\partial N}{\partial X_i}\right]dV\{V_z\} -\delta_{zi}\int_V\left[\frac{\partial N}{\partial Z}\right]^T[N]dV\{P\}\\[6pt] &+\frac{2K^*}{We}n_i\int_S[N]^T dS\\[6pt] &-g_i^*\int_V[N]^T dV \qquad (i=1,2,3) \end{aligned} \]

\[ =\int_V \begin{bmatrix} N_1N_1&N_1N_2&N_1N_3&N_1N_4\\ N_2N_1&N_2N_2&N_2N_3&N_2N_4\\ N_3N_1&N_3N_2&N_3N_3&N_3N_4\\ N_4N_1&N_4N_2&N_4N_3&N_4N_4 \end{bmatrix} dV\, \frac{\{V_i\}^{\tau+\Delta\tau}-\{V_i\}^{\tau}}{\Delta\tau} \]

\[ \begin{aligned} &+V_x\int_V \begin{bmatrix} N_1\frac{\partial N_1}{\partial X}&N_1\frac{\partial N_2}{\partial X}&N_1\frac{\partial N_3}{\partial X}&N_1\frac{\partial N_4}{\partial X}\\ N_2\frac{\partial N_1}{\partial X}&N_2\frac{\partial N_2}{\partial X}&N_2\frac{\partial N_3}{\partial X}&N_2\frac{\partial N_4}{\partial X}\\ N_3\frac{\partial N_1}{\partial X}&N_3\frac{\partial N_2}{\partial X}&N_3\frac{\partial N_3}{\partial X}&N_3\frac{\partial N_4}{\partial X}\\ N_4\frac{\partial N_1}{\partial X}&N_4\frac{\partial N_2}{\partial X}&N_4\frac{\partial N_3}{\partial X}&N_4\frac{\partial N_4}{\partial X} \end{bmatrix}dV\{V_i\}\\[6pt] &+V_y\int_V \begin{bmatrix} N_1\frac{\partial N_1}{\partial Y}&N_1\frac{\partial N_2}{\partial Y}&N_1\frac{\partial N_3}{\partial Y}&N_1\frac{\partial N_4}{\partial Y}\\ N_2\frac{\partial N_1}{\partial Y}&N_2\frac{\partial N_2}{\partial Y}&N_2\frac{\partial N_3}{\partial Y}&N_2\frac{\partial N_4}{\partial Y}\\ N_3\frac{\partial N_1}{\partial Y}&N_3\frac{\partial N_2}{\partial Y}&N_3\frac{\partial N_3}{\partial Y}&N_3\frac{\partial N_4}{\partial Y}\\ N_4\frac{\partial N_1}{\partial Y}&N_4\frac{\partial N_2}{\partial Y}&N_4\frac{\partial N_3}{\partial Y}&N_4\frac{\partial N_4}{\partial Y} \end{bmatrix}dV\{V_i\}\\[6pt] &+V_z\int_V \begin{bmatrix} N_1\frac{\partial N_1}{\partial Z}&N_1\frac{\partial N_2}{\partial Z}&N_1\frac{\partial N_3}{\partial Z}&N_1\frac{\partial N_4}{\partial Z}\\ N_2\frac{\partial N_1}{\partial Z}&N_2\frac{\partial N_2}{\partial Z}&N_2\frac{\partial N_3}{\partial Z}&N_2\frac{\partial N_4}{\partial Z}\\ N_3\frac{\partial N_1}{\partial Z}&N_3\frac{\partial N_2}{\partial Z}&N_3\frac{\partial N_3}{\partial Z}&N_3\frac{\partial N_4}{\partial Z}\\ N_4\frac{\partial N_1}{\partial Z}&N_4\frac{\partial N_2}{\partial Z}&N_4\frac{\partial N_3}{\partial Z}&N_4\frac{\partial N_4}{\partial Z} \end{bmatrix}dV\{V_i\} \end{aligned} \]

\[ \begin{aligned} &+\frac1{Re}\int_V \begin{bmatrix} \frac{\partial N_1}{\partial X}\frac{\partial N_1}{\partial X}&\frac{\partial N_1}{\partial X}\frac{\partial N_2}{\partial X}&\frac{\partial N_1}{\partial X}\frac{\partial N_3}{\partial X}&\frac{\partial N_1}{\partial X}\frac{\partial N_4}{\partial X}\\ \frac{\partial N_2}{\partial X}\frac{\partial N_1}{\partial X}&\frac{\partial N_2}{\partial X}\frac{\partial N_2}{\partial X}&\frac{\partial N_2}{\partial X}\frac{\partial N_3}{\partial X}&\frac{\partial N_2}{\partial X}\frac{\partial N_4}{\partial X}\\ \frac{\partial N_3}{\partial X}\frac{\partial N_1}{\partial X}&\frac{\partial N_3}{\partial X}\frac{\partial N_2}{\partial X}&\frac{\partial N_3}{\partial X}\frac{\partial N_3}{\partial X}&\frac{\partial N_3}{\partial X}\frac{\partial N_4}{\partial X}\\ \frac{\partial N_4}{\partial X}\frac{\partial N_1}{\partial X}&\frac{\partial N_4}{\partial X}\frac{\partial N_2}{\partial X}&\frac{\partial N_4}{\partial X}\frac{\partial N_3}{\partial X}&\frac{\partial N_4}{\partial X}\frac{\partial N_4}{\partial X} \end{bmatrix}dV\{V_i\}\\[6pt] &+\frac1{Re}\int_V \begin{bmatrix} \frac{\partial N_1}{\partial X}\frac{\partial N_1}{\partial X_i}&\frac{\partial N_1}{\partial X}\frac{\partial N_2}{\partial X_i}&\frac{\partial N_1}{\partial X}\frac{\partial N_3}{\partial X_i}&\frac{\partial N_1}{\partial X}\frac{\partial N_4}{\partial X_i}\\ \frac{\partial N_2}{\partial X}\frac{\partial N_1}{\partial X_i}&\frac{\partial N_2}{\partial X}\frac{\partial N_2}{\partial X_i}&\frac{\partial N_2}{\partial X}\frac{\partial N_3}{\partial X_i}&\frac{\partial N_2}{\partial X}\frac{\partial N_4}{\partial X_i}\\ \frac{\partial N_3}{\partial X}\frac{\partial N_1}{\partial X_i}&\frac{\partial N_3}{\partial X}\frac{\partial N_2}{\partial X_i}&\frac{\partial N_3}{\partial X}\frac{\partial N_3}{\partial X_i}&\frac{\partial N_3}{\partial X}\frac{\partial N_4}{\partial X_i}\\ \frac{\partial N_4}{\partial X}\frac{\partial N_1}{\partial X_i}&\frac{\partial N_4}{\partial X}\frac{\partial N_2}{\partial X_i}&\frac{\partial N_4}{\partial X}\frac{\partial N_3}{\partial X_i}&\frac{\partial N_4}{\partial X}\frac{\partial N_4}{\partial X_i} \end{bmatrix}dV\{V_x\}\\[6pt] &-\delta_{xi}\int_V \begin{bmatrix} \frac{\partial N_1}{\partial X}N_1&\frac{\partial N_1}{\partial X}N_2&\frac{\partial N_1}{\partial X}N_3&\frac{\partial N_1}{\partial X}N_4\\ \frac{\partial N_2}{\partial X}N_1&\frac{\partial N_2}{\partial X}N_2&\frac{\partial N_2}{\partial X}N_3&\frac{\partial N_2}{\partial X}N_4\\ \frac{\partial N_3}{\partial X}N_1&\frac{\partial N_3}{\partial X}N_2&\frac{\partial N_3}{\partial X}N_3&\frac{\partial N_3}{\partial X}N_4\\ \frac{\partial N_4}{\partial X}N_1&\frac{\partial N_4}{\partial X}N_2&\frac{\partial N_4}{\partial X}N_3&\frac{\partial N_4}{\partial X}N_4 \end{bmatrix}dV\{P\} \end{aligned} \]

\[ \begin{aligned} &+\frac1{Re}\int_V \begin{bmatrix} \frac{\partial N_1}{\partial Y}\frac{\partial N_1}{\partial Y}&\frac{\partial N_1}{\partial Y}\frac{\partial N_2}{\partial Y}&\frac{\partial N_1}{\partial Y}\frac{\partial N_3}{\partial Y}&\frac{\partial N_1}{\partial Y}\frac{\partial N_4}{\partial Y}\\ \frac{\partial N_2}{\partial Y}\frac{\partial N_1}{\partial Y}&\frac{\partial N_2}{\partial Y}\frac{\partial N_2}{\partial Y}&\frac{\partial N_2}{\partial Y}\frac{\partial N_3}{\partial Y}&\frac{\partial N_2}{\partial Y}\frac{\partial N_4}{\partial Y}\\ \frac{\partial N_3}{\partial Y}\frac{\partial N_1}{\partial Y}&\frac{\partial N_3}{\partial Y}\frac{\partial N_2}{\partial Y}&\frac{\partial N_3}{\partial Y}\frac{\partial N_3}{\partial Y}&\frac{\partial N_3}{\partial Y}\frac{\partial N_4}{\partial Y}\\ \frac{\partial N_4}{\partial Y}\frac{\partial N_1}{\partial Y}&\frac{\partial N_4}{\partial Y}\frac{\partial N_2}{\partial Y}&\frac{\partial N_4}{\partial Y}\frac{\partial N_3}{\partial Y}&\frac{\partial N_4}{\partial Y}\frac{\partial N_4}{\partial Y} \end{bmatrix}dV\{V_i\}\\[6pt] &+\frac1{Re}\int_V \begin{bmatrix} \frac{\partial N_1}{\partial Y}\frac{\partial N_1}{\partial X_i}&\frac{\partial N_1}{\partial Y}\frac{\partial N_2}{\partial X_i}&\frac{\partial N_1}{\partial Y}\frac{\partial N_3}{\partial X_i}&\frac{\partial N_1}{\partial Y}\frac{\partial N_4}{\partial X_i}\\ \frac{\partial N_2}{\partial Y}\frac{\partial N_1}{\partial X_i}&\frac{\partial N_2}{\partial Y}\frac{\partial N_2}{\partial X_i}&\frac{\partial N_2}{\partial Y}\frac{\partial N_3}{\partial X_i}&\frac{\partial N_2}{\partial Y}\frac{\partial N_4}{\partial X_i}\\ \frac{\partial N_3}{\partial Y}\frac{\partial N_1}{\partial X_i}&\frac{\partial N_3}{\partial Y}\frac{\partial N_2}{\partial X_i}&\frac{\partial N_3}{\partial Y}\frac{\partial N_3}{\partial X_i}&\frac{\partial N_3}{\partial Y}\frac{\partial N_4}{\partial X_i}\\ \frac{\partial N_4}{\partial Y}\frac{\partial N_1}{\partial X_i}&\frac{\partial N_4}{\partial Y}\frac{\partial N_2}{\partial X_i}&\frac{\partial N_4}{\partial Y}\frac{\partial N_3}{\partial X_i}&\frac{\partial N_4}{\partial Y}\frac{\partial N_4}{\partial X_i} \end{bmatrix}dV\{V_y\}\\[6pt] &-\delta_{yi}\int_V \begin{bmatrix} \frac{\partial N_1}{\partial Y}N_1&\frac{\partial N_1}{\partial Y}N_2&\frac{\partial N_1}{\partial Y}N_3&\frac{\partial N_1}{\partial Y}N_4\\ \frac{\partial N_2}{\partial Y}N_1&\frac{\partial N_2}{\partial Y}N_2&\frac{\partial N_2}{\partial Y}N_3&\frac{\partial N_2}{\partial Y}N_4\\ \frac{\partial N_3}{\partial Y}N_1&\frac{\partial N_3}{\partial Y}N_2&\frac{\partial N_3}{\partial Y}N_3&\frac{\partial N_3}{\partial Y}N_4\\ \frac{\partial N_4}{\partial Y}N_1&\frac{\partial N_4}{\partial Y}N_2&\frac{\partial N_4}{\partial Y}N_3&\frac{\partial N_4}{\partial Y}N_4 \end{bmatrix}dV\{P\} \end{aligned} \]

\[ \begin{aligned} &+\frac1{Re}\int_V \begin{bmatrix} \dfrac{\partial N_1}{\partial Z}\dfrac{\partial N_1}{\partial Z}&\dfrac{\partial N_1}{\partial Z}\dfrac{\partial N_2}{\partial Z}&\dfrac{\partial N_1}{\partial Z}\dfrac{\partial N_3}{\partial Z}&\dfrac{\partial N_1}{\partial Z}\dfrac{\partial N_4}{\partial Z}\\ \dfrac{\partial N_2}{\partial Z}\dfrac{\partial N_1}{\partial Z}&\dfrac{\partial N_2}{\partial Z}\dfrac{\partial N_2}{\partial Z}&\dfrac{\partial N_2}{\partial Z}\dfrac{\partial N_3}{\partial Z}&\dfrac{\partial N_2}{\partial Z}\dfrac{\partial N_4}{\partial Z}\\ \dfrac{\partial N_3}{\partial Z}\dfrac{\partial N_1}{\partial Z}&\dfrac{\partial N_3}{\partial Z}\dfrac{\partial N_2}{\partial Z}&\dfrac{\partial N_3}{\partial Z}\dfrac{\partial N_3}{\partial Z}&\dfrac{\partial N_3}{\partial Z}\dfrac{\partial N_4}{\partial Z}\\ \dfrac{\partial N_4}{\partial Z}\dfrac{\partial N_1}{\partial Z}&\dfrac{\partial N_4}{\partial Z}\dfrac{\partial N_2}{\partial Z}&\dfrac{\partial N_4}{\partial Z}\dfrac{\partial N_3}{\partial Z}&\dfrac{\partial N_4}{\partial Z}\dfrac{\partial N_4}{\partial Z} \end{bmatrix}dV\{V_i\}\\[6pt] &+\frac1{Re}\int_V \begin{bmatrix} \dfrac{\partial N_1}{\partial Z}\dfrac{\partial N_1}{\partial X_i}&\dfrac{\partial N_1}{\partial Z}\dfrac{\partial N_2}{\partial X_i}&\dfrac{\partial N_1}{\partial Z}\dfrac{\partial N_3}{\partial X_i}&\dfrac{\partial N_1}{\partial Z}\dfrac{\partial N_4}{\partial X_i}\\ \dfrac{\partial N_2}{\partial Z}\dfrac{\partial N_1}{\partial X_i}&\dfrac{\partial N_2}{\partial Z}\dfrac{\partial N_2}{\partial X_i}&\dfrac{\partial N_2}{\partial Z}\dfrac{\partial N_3}{\partial X_i}&\dfrac{\partial N_2}{\partial Z}\dfrac{\partial N_4}{\partial X_i}\\ \dfrac{\partial N_3}{\partial Z}\dfrac{\partial N_1}{\partial X_i}&\dfrac{\partial N_3}{\partial Z}\dfrac{\partial N_2}{\partial X_i}&\dfrac{\partial N_3}{\partial Z}\dfrac{\partial N_3}{\partial X_i}&\dfrac{\partial N_3}{\partial Z}\dfrac{\partial N_4}{\partial X_i}\\ \dfrac{\partial N_4}{\partial Z}\dfrac{\partial N_1}{\partial X_i}&\dfrac{\partial N_4}{\partial Z}\dfrac{\partial N_2}{\partial X_i}&\dfrac{\partial N_4}{\partial Z}\dfrac{\partial N_3}{\partial X_i}&\dfrac{\partial N_4}{\partial Z}\dfrac{\partial N_4}{\partial X_i} \end{bmatrix}dV\{V_z\}\\[6pt] &-\delta_{zi}\int_V \begin{bmatrix} \dfrac{\partial N_1}{\partial Z}N_1&\dfrac{\partial N_1}{\partial Z}N_2&\dfrac{\partial N_1}{\partial Z}N_3&\dfrac{\partial N_1}{\partial Z}N_4\\ \dfrac{\partial N_2}{\partial Z}N_1&\dfrac{\partial N_2}{\partial Z}N_2&\dfrac{\partial N_2}{\partial Z}N_3&\dfrac{\partial N_2}{\partial Z}N_4\\ \dfrac{\partial N_3}{\partial Z}N_1&\dfrac{\partial N_3}{\partial Z}N_2&\dfrac{\partial N_3}{\partial Z}N_3&\dfrac{\partial N_3}{\partial Z}N_4\\ \dfrac{\partial N_4}{\partial Z}N_1&\dfrac{\partial N_4}{\partial Z}N_2&\dfrac{\partial N_4}{\partial Z}N_3&\dfrac{\partial N_4}{\partial Z}N_4 \end{bmatrix}dV\{P\} \end{aligned} \]

\[ +\frac{2K^*}{We}n_i\int_S \begin{bmatrix}N_1\\N_2\\N_3\end{bmatrix}dS \]

\[ -g_i^*\int_V \begin{bmatrix}N_1\\N_2\\N_3\\N_4\end{bmatrix}dV \qquad (i=1,2,3) \]

\[ =\int_V \begin{bmatrix} L_1L_1&L_1L_2&L_1L_3&L_1L_4\\ L_2L_1&L_2L_2&L_2L_3&L_2L_4\\ L_3L_1&L_3L_2&L_3L_3&L_3L_4\\ L_4L_1&L_4L_2&L_4L_3&L_4L_4 \end{bmatrix}dV\, \frac{\{V_i\}^{\tau+\Delta\tau}-\{V_i\}^{\tau}}{\Delta\tau} \]

\[ \begin{aligned} &+V_x\int_V\frac1{6V} \begin{bmatrix} L_1c_{1x}&L_1c_{2x}&L_1c_{3x}&L_1c_{4x}\\ L_2c_{1x}&L_2c_{2x}&L_2c_{3x}&L_2c_{4x}\\ L_3c_{1x}&L_3c_{2x}&L_3c_{3x}&L_3c_{4x}\\ L_4c_{1x}&L_4c_{2x}&L_4c_{3x}&L_4c_{4x} \end{bmatrix}dV\{V_i\}\\[6pt] &+V_y\int_V\frac1{6V} \begin{bmatrix} L_1c_{1y}&L_1c_{2y}&L_1c_{3y}&L_1c_{4y}\\ L_2c_{1y}&L_2c_{2y}&L_2c_{3y}&L_2c_{4y}\\ L_3c_{1y}&L_3c_{2y}&L_3c_{3y}&L_3c_{4y}\\ L_4c_{1y}&L_4c_{2y}&L_4c_{3y}&L_4c_{4y} \end{bmatrix}dV\{V_i\}\\[6pt] &+V_z\int_V\frac1{6V} \begin{bmatrix} L_1c_{1z}&L_1c_{2z}&L_1c_{3z}&L_1c_{4z}\\ L_2c_{1z}&L_2c_{2z}&L_2c_{3z}&L_2c_{4z}\\ L_3c_{1z}&L_3c_{2z}&L_3c_{3z}&L_3c_{4z}\\ L_4c_{1z}&L_4c_{2z}&L_4c_{3z}&L_4c_{4z} \end{bmatrix}dV\{V_i\} \end{aligned} \]

\[ \begin{aligned} &+\frac1{Re}\int_V\frac1{36V^2} \begin{bmatrix} c_{1x}c_{1x}&c_{1x}c_{2x}&c_{1x}c_{3x}&c_{1x}c_{4x}\\ c_{2x}c_{1x}&c_{2x}c_{2x}&c_{2x}c_{3x}&c_{2x}c_{4x}\\ c_{3x}c_{1x}&c_{3x}c_{2x}&c_{3x}c_{3x}&c_{3x}c_{4x}\\ c_{4x}c_{1x}&c_{4x}c_{2x}&c_{4x}c_{3x}&c_{4x}c_{4x} \end{bmatrix}dV\{V_i\}\\[6pt] &+\frac1{Re}\int_V\frac1{36V^2} \begin{bmatrix} c_{1x}c_{1i}&c_{1x}c_{2i}&c_{1x}c_{3i}&c_{1x}c_{4i}\\ c_{2x}c_{1i}&c_{2x}c_{2i}&c_{2x}c_{3i}&c_{2x}c_{4i}\\ c_{3x}c_{1i}&c_{3x}c_{2i}&c_{3x}c_{3i}&c_{3x}c_{4i}\\ c_{4x}c_{1i}&c_{4x}c_{2i}&c_{4x}c_{3i}&c_{4x}c_{4i} \end{bmatrix}dV\{V_x\}\\[6pt] &-\delta_{xi}\int_V\frac1{6V} \begin{bmatrix} c_{1x}L_1&c_{1x}L_2&c_{1x}L_3&c_{1x}L_4\\ c_{2x}L_1&c_{2x}L_2&c_{2x}L_3&c_{2x}L_4\\ c_{3x}L_1&c_{3x}L_2&c_{3x}L_3&c_{3x}L_4\\ c_{4x}L_1&c_{4x}L_2&c_{4x}L_3&c_{4x}L_4 \end{bmatrix}dV\{P\} \end{aligned} \]

\[ \begin{aligned} &+\frac1{Re}\int_V\frac1{36V^2} \begin{bmatrix} c_{1y}c_{1y}&c_{1y}c_{2y}&c_{1y}c_{3y}&c_{1y}c_{4y}\\ c_{2y}c_{1y}&c_{2y}c_{2y}&c_{2y}c_{3y}&c_{2y}c_{4y}\\ c_{3y}c_{1y}&c_{3y}c_{2y}&c_{3y}c_{3y}&c_{3y}c_{4y}\\ c_{4y}c_{1y}&c_{4y}c_{2y}&c_{4y}c_{3y}&c_{4y}c_{4y} \end{bmatrix}dV\{V_i\}\\[6pt] &+\frac1{Re}\int_V\frac1{36V^2} \begin{bmatrix} c_{1y}c_{1i}&c_{1y}c_{2i}&c_{1y}c_{3i}&c_{1y}c_{4i}\\ c_{2y}c_{1i}&c_{2y}c_{2i}&c_{2y}c_{3i}&c_{2y}c_{4i}\\ c_{3y}c_{1i}&c_{3y}c_{2i}&c_{3y}c_{3i}&c_{3y}c_{4i}\\ c_{4y}c_{1i}&c_{4y}c_{2i}&c_{4y}c_{3i}&c_{4y}c_{4i} \end{bmatrix}dV\{V_y\}\\[6pt] &-\delta_{yi}\int_V\frac1{6V} \begin{bmatrix} c_{1y}L_1&c_{1y}L_2&c_{1y}L_3&c_{1y}L_4\\ c_{2y}L_1&c_{2y}L_2&c_{2y}L_3&c_{2y}L_4\\ c_{3y}L_1&c_{3y}L_2&c_{3y}L_3&c_{3y}L_4\\ c_{4y}L_1&c_{4y}L_2&c_{4y}L_3&c_{4y}L_4 \end{bmatrix}dV\{P\} \end{aligned} \]

\[ \begin{aligned} &+\frac1{Re}\int_V\frac1{36V^2} \begin{bmatrix} c_{1z}c_{1z}&c_{1z}c_{2z}&c_{1z}c_{3z}&c_{1z}c_{4z}\\ c_{2z}c_{1z}&c_{2z}c_{2z}&c_{2z}c_{3z}&c_{2z}c_{4z}\\ c_{3z}c_{1z}&c_{3z}c_{2z}&c_{3z}c_{3z}&c_{3z}c_{4z}\\ c_{4z}c_{1z}&c_{4z}c_{2z}&c_{4z}c_{3z}&c_{4z}c_{4z} \end{bmatrix}dV\{V_i\}\\[6pt] &+\frac1{Re}\int_V\frac1{36V^2} \begin{bmatrix} c_{1z}c_{1i}&c_{1z}c_{2i}&c_{1z}c_{3i}&c_{1z}c_{4i}\\ c_{2z}c_{1i}&c_{2z}c_{2i}&c_{2z}c_{3i}&c_{2z}c_{4i}\\ c_{3z}c_{1i}&c_{3z}c_{2i}&c_{3z}c_{3i}&c_{3z}c_{4i}\\ c_{4z}c_{1i}&c_{4z}c_{2i}&c_{4z}c_{3i}&c_{4z}c_{4i} \end{bmatrix}dV\{V_z\}\\[6pt] &-\delta_{zi}\int_V\frac1{6V} \begin{bmatrix} c_{1z}L_1&c_{1z}L_2&c_{1z}L_3&c_{1z}L_4\\ c_{2z}L_1&c_{2z}L_2&c_{2z}L_3&c_{2z}L_4\\ c_{3z}L_1&c_{3z}L_2&c_{3z}L_3&c_{3z}L_4\\ c_{4z}L_1&c_{4z}L_2&c_{4z}L_3&c_{4z}L_4 \end{bmatrix}dV\{P\} \end{aligned} \]

\[ +\frac{2K^*}{We}n_i\int_S \begin{bmatrix}L_1\\L_2\\L_3\end{bmatrix}dS \]

\[ -g_i^*\int_V \begin{bmatrix}L_1\\L_2\\L_3\\L_4\end{bmatrix}^{T}dV \qquad (i=1,2,3) \]

\[ \int_V L_1^pL_2^qL_3^rL_4^s\,dV =\frac{p!q!r!s!}{(p+q+r+s+3)!}\,6V \] \[ \int_VL_iL_j\,dV =\begin{cases} \displaystyle \frac{1!1!}{(1+1+3)!}\,6V =\frac{6}{5!}V =\frac{1}{20}V & (i\ne j)\\[6pt] \displaystyle \frac{2!}{(1+1+3)!}\,6V =\frac{12}{5!}V =\frac{1}{10}V & (i=j) \end{cases} \] \[ \int_VL_i\,dV =\frac{1!}{(1+3)!}\,6V =\frac{6}{4!}V =\frac14V \] \[ \int_SL_1^pL_2^qL_3^r\,dS =\frac{p!q!r!}{(p+q+r+2)!}\,2S \] \[ \int_SL_i\,dS =\frac{1!}{(1+2)!}\,2S =\frac13S \]

\[ =\frac1{20}V \begin{bmatrix} 2&1&1&1\\ 1&2&1&1\\ 1&1&2&1\\ 1&1&1&2 \end{bmatrix} \frac{\{V_i\}^{\tau+\Delta\tau}-\{V_i\}^{\tau}}{\Delta\tau} \]

\[ \begin{aligned} &+V_x\frac1{6V}\frac{V}{4} \begin{bmatrix} c_{1x}&c_{2x}&c_{3x}&c_{4x}\\ c_{1x}&c_{2x}&c_{3x}&c_{4x}\\ c_{1x}&c_{2x}&c_{3x}&c_{4x}\\ c_{1x}&c_{2x}&c_{3x}&c_{4x} \end{bmatrix}\{V_i\}\\[6pt] &+V_y\frac1{6V}\frac{V}{4} \begin{bmatrix} c_{1y}&c_{2y}&c_{3y}&c_{4y}\\ c_{1y}&c_{2y}&c_{3y}&c_{4y}\\ c_{1y}&c_{2y}&c_{3y}&c_{4y}\\ c_{1y}&c_{2y}&c_{3y}&c_{4y} \end{bmatrix}\{V_i\}\\[6pt] &+V_z\frac1{6V}\frac{V}{4} \begin{bmatrix} c_{1z}&c_{2z}&c_{3z}&c_{4z}\\ c_{1z}&c_{2z}&c_{3z}&c_{4z}\\ c_{1z}&c_{2z}&c_{3z}&c_{4z}\\ c_{1z}&c_{2z}&c_{3z}&c_{4z} \end{bmatrix}\{V_i\} \end{aligned} \]

\[ \begin{aligned} &+\frac1{Re}\frac1{36V^2}V \begin{bmatrix} c_{1x}c_{1x}&c_{1x}c_{2x}&c_{1x}c_{3x}&c_{1x}c_{4x}\\ c_{2x}c_{1x}&c_{2x}c_{2x}&c_{2x}c_{3x}&c_{2x}c_{4x}\\ c_{3x}c_{1x}&c_{3x}c_{2x}&c_{3x}c_{3x}&c_{3x}c_{4x}\\ c_{4x}c_{1x}&c_{4x}c_{2x}&c_{4x}c_{3x}&c_{4x}c_{4x} \end{bmatrix}\{V_i\}\\[6pt] &+\frac1{Re}\frac1{36V^2}V \begin{bmatrix} c_{1x}c_{1i}&c_{1x}c_{2i}&c_{1x}c_{3i}&c_{1x}c_{4i}\\ c_{2x}c_{1i}&c_{2x}c_{2i}&c_{2x}c_{3i}&c_{2x}c_{4i}\\ c_{3x}c_{1i}&c_{3x}c_{2i}&c_{3x}c_{3i}&c_{3x}c_{4i}\\ c_{4x}c_{1i}&c_{4x}c_{2i}&c_{4x}c_{3i}&c_{4x}c_{4i} \end{bmatrix}\{V_x\}\\[6pt] &-\delta_{xi}\frac1{6V}\frac{V}{4} \begin{bmatrix} c_{1x}&c_{1x}&c_{1x}&c_{1x}\\ c_{2x}&c_{2x}&c_{2x}&c_{2x}\\ c_{3x}&c_{3x}&c_{3x}&c_{3x}\\ c_{4x}&c_{4x}&c_{4x}&c_{4x} \end{bmatrix}\{P\} \end{aligned} \]

\[ \begin{aligned} &+\frac1{Re}\frac1{36V^2}V \begin{bmatrix} c_{1y}c_{1y}&c_{1y}c_{2y}&c_{1y}c_{3y}&c_{1y}c_{4y}\\ c_{2y}c_{1y}&c_{2y}c_{2y}&c_{2y}c_{3y}&c_{2y}c_{4y}\\ c_{3y}c_{1y}&c_{3y}c_{2y}&c_{3y}c_{3y}&c_{3y}c_{4y}\\ c_{4y}c_{1y}&c_{4y}c_{2y}&c_{4y}c_{3y}&c_{4y}c_{4y} \end{bmatrix}\{V_i\}\\[6pt] &+\frac1{Re}\frac1{36V^2}V \begin{bmatrix} c_{1y}c_{1i}&c_{1y}c_{2i}&c_{1y}c_{3i}&c_{1y}c_{4i}\\ c_{2y}c_{1i}&c_{2y}c_{2i}&c_{2y}c_{3i}&c_{2y}c_{4i}\\ c_{3y}c_{1i}&c_{3y}c_{2i}&c_{3y}c_{3i}&c_{3y}c_{4i}\\ c_{4y}c_{1i}&c_{4y}c_{2i}&c_{4y}c_{3i}&c_{4y}c_{4i} \end{bmatrix}\{V_y\}\\[6pt] &-\delta_{yi}\frac1{6V}\frac{V}{4} \begin{bmatrix} c_{1y}&c_{1y}&c_{1y}&c_{1y}\\ c_{2y}&c_{2y}&c_{2y}&c_{2y}\\ c_{3y}&c_{3y}&c_{3y}&c_{3y}\\ c_{4y}&c_{4y}&c_{4y}&c_{4y} \end{bmatrix}\{P\} \end{aligned} \]

\[ \begin{aligned} &+\frac1{Re}\frac1{36V^2}V \begin{bmatrix} c_{1z}c_{1z}&c_{1z}c_{2z}&c_{1z}c_{3z}&c_{1z}c_{4z}\\ c_{2z}c_{1z}&c_{2z}c_{2z}&c_{2z}c_{3z}&c_{2z}c_{4z}\\ c_{3z}c_{1z}&c_{3z}c_{2z}&c_{3z}c_{3z}&c_{3z}c_{4z}\\ c_{4z}c_{1z}&c_{4z}c_{2z}&c_{4z}c_{3z}&c_{4z}c_{4z} \end{bmatrix}\{V_i\}\\[6pt] &+\frac1{Re}\frac1{36V^2}V \begin{bmatrix} c_{1z}c_{1i}&c_{1z}c_{2i}&c_{1z}c_{3i}&c_{1z}c_{4i}\\ c_{2z}c_{1i}&c_{2z}c_{2i}&c_{2z}c_{3i}&c_{2z}c_{4i}\\ c_{3z}c_{1i}&c_{3z}c_{2i}&c_{3z}c_{3i}&c_{3z}c_{4i}\\ c_{4z}c_{1i}&c_{4z}c_{2i}&c_{4z}c_{3i}&c_{4z}c_{4i} \end{bmatrix}\{V_z\}\\[6pt] &-\delta_{zi}\frac1{6V}\frac{V}{4} \begin{bmatrix} c_{1z}&c_{1z}&c_{1z}&c_{1z}\\ c_{2z}&c_{2z}&c_{2z}&c_{2z}\\ c_{3z}&c_{3z}&c_{3z}&c_{3z}\\ c_{4z}&c_{4z}&c_{4z}&c_{4z} \end{bmatrix}\{P\} \end{aligned} \]

\[ +\frac{2K^*}{We}n_i\frac{S}{3} \begin{bmatrix}1\\1\\1\end{bmatrix} \]

\[ -g_i^*\frac{V}{4} \begin{bmatrix}1\\1\\1\\1\end{bmatrix} \qquad(i=1,2,3) \]

\[ =\frac1{20}V \begin{bmatrix} 2&1&1&1\\ 1&2&1&1\\ 1&1&2&1\\ 1&1&1&2 \end{bmatrix} \frac{\{V_i\}^{\tau+\Delta\tau}-\{V_i\}^{\tau}}{\Delta\tau} \]

\[ \begin{aligned} &+V_x\frac1{24} \begin{bmatrix} c_{1x}&c_{2x}&c_{3x}&c_{4x}\\ c_{1x}&c_{2x}&c_{3x}&c_{4x}\\ c_{1x}&c_{2x}&c_{3x}&c_{4x}\\ c_{1x}&c_{2x}&c_{3x}&c_{4x} \end{bmatrix}\{V_i\}\\[6pt] &+V_y\frac1{24} \begin{bmatrix} c_{1y}&c_{2y}&c_{3y}&c_{4y}\\ c_{1y}&c_{2y}&c_{3y}&c_{4y}\\ c_{1y}&c_{2y}&c_{3y}&c_{4y}\\ c_{1y}&c_{2y}&c_{3y}&c_{4y} \end{bmatrix}\{V_i\}\\[6pt] &+V_z\frac1{24} \begin{bmatrix} c_{1z}&c_{2z}&c_{3z}&c_{4z}\\ c_{1z}&c_{2z}&c_{3z}&c_{4z}\\ c_{1z}&c_{2z}&c_{3z}&c_{4z}\\ c_{1z}&c_{2z}&c_{3z}&c_{4z} \end{bmatrix}\{V_i\} \end{aligned} \]

\[ \begin{aligned} &+\frac1{Re}\frac1{36V} \begin{bmatrix} c_{1x}c_{1x}&c_{1x}c_{2x}&c_{1x}c_{3x}&c_{1x}c_{4x}\\ c_{2x}c_{1x}&c_{2x}c_{2x}&c_{2x}c_{3x}&c_{2x}c_{4x}\\ c_{3x}c_{1x}&c_{3x}c_{2x}&c_{3x}c_{3x}&c_{3x}c_{4x}\\ c_{4x}c_{1x}&c_{4x}c_{2x}&c_{4x}c_{3x}&c_{4x}c_{4x} \end{bmatrix}\{V_i\}\\[6pt] &+\frac1{Re}\frac1{36V} \begin{bmatrix} c_{1x}c_{1i}&c_{1x}c_{2i}&c_{1x}c_{3i}&c_{1x}c_{4i}\\ c_{2x}c_{1i}&c_{2x}c_{2i}&c_{2x}c_{3i}&c_{2x}c_{4i}\\ c_{3x}c_{1i}&c_{3x}c_{2i}&c_{3x}c_{3i}&c_{3x}c_{4i}\\ c_{4x}c_{1i}&c_{4x}c_{2i}&c_{4x}c_{3i}&c_{4x}c_{4i} \end{bmatrix}\{V_x\}\\[6pt] &-\delta_{xi}\frac1{24} \begin{bmatrix} c_{1x}&c_{1x}&c_{1x}&c_{1x}\\ c_{2x}&c_{2x}&c_{2x}&c_{2x}\\ c_{3x}&c_{3x}&c_{3x}&c_{3x}\\ c_{4x}&c_{4x}&c_{4x}&c_{4x} \end{bmatrix}\{P\} \end{aligned} \]

\[ \begin{aligned} &+\frac1{Re}\frac1{36V} \begin{bmatrix} c_{1y}c_{1y}&c_{1y}c_{2y}&c_{1y}c_{3y}&c_{1y}c_{4y}\\ c_{2y}c_{1y}&c_{2y}c_{2y}&c_{2y}c_{3y}&c_{2y}c_{4y}\\ c_{3y}c_{1y}&c_{3y}c_{2y}&c_{3y}c_{3y}&c_{3y}c_{4y}\\ c_{4y}c_{1y}&c_{4y}c_{2y}&c_{4y}c_{3y}&c_{4y}c_{4y} \end{bmatrix}\{V_i\}\\[6pt] &+\frac1{Re}\frac1{36V} \begin{bmatrix} c_{1y}c_{1i}&c_{1y}c_{2i}&c_{1y}c_{3i}&c_{1y}c_{4i}\\ c_{2y}c_{1i}&c_{2y}c_{2i}&c_{2y}c_{3i}&c_{2y}c_{4i}\\ c_{3y}c_{1i}&c_{3y}c_{2i}&c_{3y}c_{3i}&c_{3y}c_{4i}\\ c_{4y}c_{1i}&c_{4y}c_{2i}&c_{4y}c_{3i}&c_{4y}c_{4i} \end{bmatrix}\{V_y\}\\[6pt] &-\delta_{yi}\frac1{24} \begin{bmatrix} c_{1y}&c_{1y}&c_{1y}&c_{1y}\\ c_{2y}&c_{2y}&c_{2y}&c_{2y}\\ c_{3y}&c_{3y}&c_{3y}&c_{3y}\\ c_{4y}&c_{4y}&c_{4y}&c_{4y} \end{bmatrix}\{P\} \end{aligned} \]

\[ \begin{aligned} &+\frac1{Re}\frac1{36V} \begin{bmatrix} c_{1z}c_{1z}&c_{1z}c_{2z}&c_{1z}c_{3z}&c_{1z}c_{4z}\\ c_{2z}c_{1z}&c_{2z}c_{2z}&c_{2z}c_{3z}&c_{2z}c_{4z}\\ c_{3z}c_{1z}&c_{3z}c_{2z}&c_{3z}c_{3z}&c_{3z}c_{4z}\\ c_{4z}c_{1z}&c_{4z}c_{2z}&c_{4z}c_{3z}&c_{4z}c_{4z} \end{bmatrix}\{V_i\}\\[6pt] &+\frac1{Re}\frac1{36V} \begin{bmatrix} c_{1z}c_{1i}&c_{1z}c_{2i}&c_{1z}c_{3i}&c_{1z}c_{4i}\\ c_{2z}c_{1i}&c_{2z}c_{2i}&c_{2z}c_{3i}&c_{2z}c_{4i}\\ c_{3z}c_{1i}&c_{3z}c_{2i}&c_{3z}c_{3i}&c_{3z}c_{4i}\\ c_{4z}c_{1i}&c_{4z}c_{2i}&c_{4z}c_{3i}&c_{4z}c_{4i} \end{bmatrix}\{V_z\}\\[6pt] &-\delta_{zi}\frac1{24} \begin{bmatrix} c_{1z}&c_{1z}&c_{1z}&c_{1z}\\ c_{2z}&c_{2z}&c_{2z}&c_{2z}\\ c_{3z}&c_{3z}&c_{3z}&c_{3z}\\ c_{4z}&c_{4z}&c_{4z}&c_{4z} \end{bmatrix}\{P\} \end{aligned} \]

\[ +\frac{2K^*}{3We}n_iS \begin{bmatrix} 1\\ 1\\ 1 \end{bmatrix} \]

\[ -g_i^*\frac{V}{4} \begin{bmatrix}1\\1\\1\\1\end{bmatrix} \qquad(i=1,2,3) \]

\[ \begin{aligned} &={}[C]\frac{\{V_i\}^{\tau+\Delta\tau}-\{V_i\}^{\tau}}{\Delta\tau} +V_x[C_x]\{V_i\}+V_y[C_y]\{V_i\}+V_z[C_z]\{V_i\}\\[6pt] &+\frac{1}{Re}[S_{xx}]\{V_i\}+\frac{1}{Re}[S_{xi}]\{V_x\}-\delta_{xi}[H_x]\{P\}\\[6pt] &+\frac{1}{Re}[S_{yy}]\{V_i\}+\frac{1}{Re}[S_{yi}]\{V_y\}-\delta_{yi}[H_y]\{P\}\\[6pt] &+\frac{1}{Re}[S_{zz}]\{V_i\}+\frac{1}{Re}[S_{zi}]\{V_z\}-\delta_{zi}[H_z]\{P\}\\[6pt] &+\frac{2K^*}{3We}n_iS \begin{bmatrix}1\\1\\1\end{bmatrix}\\[6pt] &-g_i^*\frac{V}{4} \begin{bmatrix}1\\1\\1\\1\end{bmatrix} \qquad (i=1,2,3) \end{aligned} \]

\[ \begin{aligned} &={}[C]\frac{\{V_i\}^{\tau+\Delta\tau}-\{V_i\}^{\tau}}{\Delta\tau}\\[6pt] &+(V_x[C_x]+V_y[C_y]+V_z[C_z])\{V_i\}\\[6pt] &+\frac{1}{Re}([S_{xx}]+[S_{yy}]+[S_{zz}])\{V_i\}\\[6pt] &+\frac{1}{Re}\left([S_{xi}]\{V_x\}+[S_{yi}]\{V_y\}+[S_{zi}]\{V_z\}\right)\\[6pt] &-(\delta_{xi}[H_x]+\delta_{yi}[H_y]+\delta_{zi}[H_z])\{P\}\\[6pt] &+\frac{2K^*}{3We}n_iS \begin{bmatrix}1\\1\\1\end{bmatrix}\\[6pt] &-g_i^*\frac{V}{4} \begin{bmatrix}1\\1\\1\\1\end{bmatrix} \qquad (i=1,2,3)\\[6pt] &={}\begin{bmatrix}0\\0\\0\\0\end{bmatrix} \end{aligned} \]

\[ \begin{aligned} &[C]\frac{\{V_i\}^{\tau+\Delta\tau}-\{V_i\}^{\tau}}{\Delta\tau} +(V_x[C_x]+V_y[C_y]+V_z[C_z])\{V_i\}\\[6pt] &+\frac{1}{Re}([S_{xx}]+[S_{yy}]+[S_{zz}])\{V_i\}\\[6pt] &+\frac{1}{Re}\left([S_{xi}]\{V_x\}+[S_{yi}]\{V_y\}+[S_{zi}]\{V_z\}\right)\\[6pt] &-(\delta_{xi}[H_x]+\delta_{yi}[H_y]+\delta_{zi}[H_z])\{P\}\\[6pt] &+\frac{2K^*}{3We}n_iS \begin{bmatrix}1\\1\\1\end{bmatrix} -g_i^*\frac{V}{4} \begin{bmatrix}1\\1\\1\\1\end{bmatrix} =\begin{bmatrix}0\\0\\0\\0\end{bmatrix} \qquad (i=1,2,3) \end{aligned} \]

\[ \begin{aligned} \frac{[C]}{\Delta\tau}\{V_i\}^{\tau+\Delta\tau} &+(V_x[C_x]+V_y[C_y]+V_z[C_z])\{V_i\}^{\tau+\Delta\tau}\\[6pt] &+\frac{1}{Re}([S_{xx}]+[S_{yy}]+[S_{zz}])\{V_i\}^{\tau+\Delta\tau}\\[6pt] &+\frac{1}{Re}\left([S_{xi}]\{V_x\}^{\tau+\Delta\tau} +[S_{yi}]\{V_y\}^{\tau+\Delta\tau} +[S_{zi}]\{V_z\}^{\tau+\Delta\tau}\right)\\[6pt] &-(\delta_{xi}[H_x]+\delta_{yi}[H_y]+\delta_{zi}[H_z])\{P\}^{\tau+\Delta\tau}\\[6pt] &={}\frac{[C]}{\Delta\tau}\{V_i\}^{\tau} -\frac{2K^*}{3We}n_iS \begin{bmatrix}1\\1\\1\end{bmatrix} +g_i^*\frac{V}{4} \begin{bmatrix}1\\1\\1\\1\end{bmatrix} \qquad (i=1,2,3) \end{aligned} \]

\[ \begin{aligned} \frac{[C]}{\Delta\tau}\{V_x\}^{\tau+\Delta\tau} &+(V_x[C_x]+V_y[C_y]+V_z[C_z])\{V_x\}^{\tau+\Delta\tau}\\[6pt] &+\frac{1}{Re}\left\{ (2[S_{xx}]+[S_{yy}]+[S_{zz}])\{V_x\}^{\tau+\Delta\tau} +[S_{yx}]\{V_y\}^{\tau+\Delta\tau} +[S_{zx}]\{V_z\}^{\tau+\Delta\tau} \right\} -[H_x]\{P\}^{\tau+\Delta\tau}\\[6pt] &={}\frac{[C]}{\Delta\tau}\{V_x\}^{\tau} -\frac{2K^*}{3We}n_xS \begin{bmatrix}1\\1\\1\end{bmatrix} +g_x^*\frac{V}{4} \begin{bmatrix}1\\1\\1\\1\end{bmatrix} \end{aligned} \]

\[ \begin{aligned} \frac{[C]}{\Delta\tau}\{V_y\}^{\tau+\Delta\tau} &+(V_x[C_x]+V_y[C_y]+V_z[C_z])\{V_y\}^{\tau+\Delta\tau}\\[6pt] &+\frac{1}{Re}\left\{ [S_{xy}]\{V_x\}^{\tau+\Delta\tau} +([S_{xx}]+2[S_{yy}]+[S_{zz}])\{V_y\}^{\tau+\Delta\tau} +[S_{zy}]\{V_z\}^{\tau+\Delta\tau} \right\} -[H_y]\{P\}^{\tau+\Delta\tau}\\[6pt] &={}\frac{[C]}{\Delta\tau}\{V_y\}^{\tau} -\frac{2K^*}{3We}n_yS \begin{bmatrix}1\\1\\1\end{bmatrix} +g_y^*\frac{V}{4} \begin{bmatrix}1\\1\\1\\1\end{bmatrix} \end{aligned} \]

\[ \begin{aligned} \frac{[C]}{\Delta\tau}\{V_z\}^{\tau+\Delta\tau} &+(V_x[C_x]+V_y[C_y]+V_z[C_z])\{V_z\}^{\tau+\Delta\tau}\\[6pt] &+\frac{1}{Re}\left\{ [S_{xz}]\{V_x\}^{\tau+\Delta\tau} +[S_{yz}]\{V_y\}^{\tau+\Delta\tau} +([S_{xx}]+[S_{yy}]+2[S_{zz}])\{V_z\}^{\tau+\Delta\tau} \right\} -[H_z]\{P\}^{\tau+\Delta\tau}\\[6pt] &={}\frac{[C]}{\Delta\tau}\{V_z\}^{\tau} -\frac{2K^*}{3We}n_zS \begin{bmatrix}1\\1\\1\end{bmatrix} +g_z^*\frac{V}{4} \begin{bmatrix}1\\1\\1\\1\end{bmatrix} \qquad (i=1,2,3) \end{aligned} \]