有限要素法・流体力学による数値計算

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

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

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

\[ \begin{aligned} V_x &=N_1V_{x,1}+N_2V_{x,2}+N_3V_{x,3}\\[6pt] &=\begin{bmatrix}N_1&N_2&N_3\end{bmatrix} \begin{bmatrix}V_{x,1}\\V_{x,2}\\V_{x,3}\end{bmatrix}\\[6pt] &=[N]\{V_x\}=0,\\[6pt] V_y&=N_1V_{y,1}+N_2V_{y,2}+N_3V_{y,3}=0,\\[6pt] P&=N_1P_{x,1}+N_2P_{x,2}+N_3P_{x,3} \end{aligned} \]

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

\[ \begin{aligned} &={}\int_S\begin{bmatrix}N_1\\N_2\\N_3\end{bmatrix}\frac{\partial V_i}{\partial\tau}dS +V_x\int_S\begin{bmatrix}N_1\\N_2\\N_3\end{bmatrix}\frac{\partial V_i}{\partial X}dS +V_y\int_S\begin{bmatrix}N_1\\N_2\\N_3\end{bmatrix}\frac{\partial V_i}{\partial Y}dS\\[6pt] &-\int_S\begin{bmatrix}N_1\\N_2\\N_3\end{bmatrix}\frac{\partial\sigma^*_{xi}}{\partial X}dS -\int_S\begin{bmatrix}N_1\\N_2\\N_3\end{bmatrix}\frac{\partial\sigma^*_{yi}}{\partial Y}dS\\[6pt] &-\int_S\begin{bmatrix}N_1\\N_2\\N_3\end{bmatrix}g_i^*dS \end{aligned} \]

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

\[ \begin{aligned} &={}\int_S[N]^T\frac{[N](\{V_i\}^{\tau+\Delta\tau}-\{V_i\}^{\tau})}{\Delta\tau}dS\\[6pt] &+V_x\int_S[N]^T\frac{\partial[N]\{V_i\}}{\partial X}dS +V_y\int_S[N]^T\frac{\partial[N]\{V_i\}}{\partial Y}dS\\[6pt] &-\int_L[N]^T\sigma^*_{xi}n_xdL +\int_S\left[\frac{\partial N}{\partial X}\right]^T\sigma^*_{xi}dS\\[6pt] &-\int_L[N]^T\sigma^*_{yi}n_ydL +\int_S\left[\frac{\partial N}{\partial Y}\right]^T\sigma^*_{yi}dS\\[6pt] &-\int_S[N]^Tg_i^*dS \end{aligned} \]

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

\[ \begin{aligned} &={}\int_S[N]^T[N]dS\frac{\{V_i\}^{\tau+\Delta\tau}-\{V_i\}^{\tau}}{\Delta\tau}\\[6pt] &+V_x\int_S[N]^T\left[\frac{\partial N}{\partial X}\right]dS\{V_i\} +V_y\int_S[N]^T\left[\frac{\partial N}{\partial Y}\right]dS\{V_i\}\\[6pt] &+\int_S\left[\frac{\partial N}{\partial X}\right]^T \left\{-\delta_{xi}P+\frac1{Re}\left(\frac{\partial V_i}{\partial X}+\frac{\partial V_x}{\partial X_i}\right)\right\}dS\\[6pt] &+\int_S\left[\frac{\partial N}{\partial Y}\right]^T \left\{-\delta_{yi}P+\frac1{Re}\left(\frac{\partial V_i}{\partial Y}+\frac{\partial V_y}{\partial X_i}\right)\right\}dS\\[6pt] &-\frac{2K^*}{We}n_i\int_L[N]^TdL\\[6pt] &-\int_S[N]^Tg_i^*dS \end{aligned} \]

\[ \begin{aligned} &={}\int_S[N]^T[N]dS\frac{\{V_i\}^{\tau+\Delta\tau}-\{V_i\}^{\tau}}{\Delta\tau}\\[6pt] &+V_x\int_S[N]^T\left[\frac{\partial N}{\partial X}\right]dS\{V_i\} +V_y\int_S[N]^T\left[\frac{\partial N}{\partial Y}\right]dS\{V_i\}\\[6pt] &+\int_S\left[\frac{\partial N}{\partial X}\right]^T \left\{-\delta_{xi}P+\frac1{Re}\left(\frac{\partial V_i}{\partial X}+\frac{\partial V_x}{\partial X_i}\right)\right\}dS\\[6pt] &+\int_S\left[\frac{\partial N}{\partial Y}\right]^T \left\{-\delta_{yi}P+\frac1{Re}\left(\frac{\partial V_i}{\partial Y}+\frac{\partial V_y}{\partial X_i}\right)\right\}dS\\[6pt] &+\frac{2K^*}{We}n_i\int_L[N]^TdL\\[6pt] &-\int_S[N]^Tg_i^*dS \end{aligned} \]

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

\[ \begin{aligned} &={}\int_S[N]^T[N]dS\frac{\{V_i\}^{\tau+\Delta\tau}-\{V_i\}^{\tau}}{\Delta\tau}\\[6pt] &+V_x\int_S[N]^T\left[\frac{\partial N}{\partial X}\right]dS\{V_i\} +V_y\int_S[N]^T\left[\frac{\partial N}{\partial Y}\right]dS\{V_i\}\\[6pt] &+\frac1{Re}\int_S\left[\frac{\partial N}{\partial X}\right]^T\frac{\partial[N]\{V_i\}}{\partial X}dS +\frac1{Re}\int_S\left[\frac{\partial N}{\partial X}\right]^T\frac{\partial[N]\{V_x\}}{\partial X_i}dS -\delta_{xi}\int_S\left[\frac{\partial N}{\partial X}\right]^T[N]\{P\}dS\\[6pt] &+\frac1{Re}\int_S\left[\frac{\partial N}{\partial Y}\right]^T\frac{\partial[N]\{V_i\}}{\partial Y}dS +\frac1{Re}\int_S\left[\frac{\partial N}{\partial Y}\right]^T\frac{\partial[N]\{V_y\}}{\partial X_i}dS -\delta_{yi}\int_S\left[\frac{\partial N}{\partial Y}\right]^T[N]\{P\}dS\\[6pt] &+\frac{2K^*}{We}n_iL\int_L[N]^TdL\\[6pt] &-g_i^*\int_S[N]^TdS \end{aligned} \]

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

\[ =\int_S \begin{bmatrix} N_1N_1&N_1N_2&N_1N_3\\ N_2N_1&N_2N_2&N_2N_3\\ N_3N_1&N_3N_2&N_3N_3 \end{bmatrix}dS\, \frac{\{V_i\}^{\tau+\Delta\tau}-\{V_i\}^{\tau}}{\Delta\tau} \]

\[ \begin{aligned} &+V_x\int_S \begin{bmatrix} N_1\frac{\partial N_1}{\partial X}&N_1\frac{\partial N_2}{\partial X}&N_1\frac{\partial N_3}{\partial X}\\ N_2\frac{\partial N_1}{\partial X}&N_2\frac{\partial N_2}{\partial X}&N_2\frac{\partial N_3}{\partial X}\\ N_3\frac{\partial N_1}{\partial X}&N_3\frac{\partial N_2}{\partial X}&N_3\frac{\partial N_3}{\partial X} \end{bmatrix}dS\{V_i\}\\[6pt] &+V_y\int_S \begin{bmatrix} N_1\frac{\partial N_1}{\partial Y}&N_1\frac{\partial N_2}{\partial Y}&N_1\frac{\partial N_3}{\partial Y}\\ N_2\frac{\partial N_1}{\partial Y}&N_2\frac{\partial N_2}{\partial Y}&N_2\frac{\partial N_3}{\partial Y}\\ N_3\frac{\partial N_1}{\partial Y}&N_3\frac{\partial N_2}{\partial Y}&N_3\frac{\partial N_3}{\partial Y} \end{bmatrix}dS\{V_i\} \end{aligned} \]

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

\[ \begin{aligned} &+\frac1{Re}\int_S \begin{bmatrix} \frac{\partial N_1}{\partial Y}\frac{\partial N_1}{\partial Y}&\frac{\partial N_1}{\partial Y}\frac{\partial N_2}{\partial Y}&\frac{\partial N_1}{\partial Y}\frac{\partial N_3}{\partial Y}\\ \frac{\partial N_2}{\partial Y}\frac{\partial N_1}{\partial Y}&\frac{\partial N_2}{\partial Y}\frac{\partial N_2}{\partial Y}&\frac{\partial N_2}{\partial Y}\frac{\partial N_3}{\partial Y}\\ \frac{\partial N_3}{\partial Y}\frac{\partial N_1}{\partial Y}&\frac{\partial N_3}{\partial Y}\frac{\partial N_2}{\partial Y}&\frac{\partial N_3}{\partial Y}\frac{\partial N_3}{\partial Y} \end{bmatrix}dS\{V_i\}\\[6pt] &+\frac1{Re}\int_S \begin{bmatrix} \frac{\partial N_1}{\partial Y}\frac{\partial N_1}{\partial X_i}&\frac{\partial N_1}{\partial Y}\frac{\partial N_2}{\partial X_i}&\frac{\partial N_1}{\partial Y}\frac{\partial N_3}{\partial X_i}\\ \frac{\partial N_2}{\partial Y}\frac{\partial N_1}{\partial X_i}&\frac{\partial N_2}{\partial Y}\frac{\partial N_2}{\partial X_i}&\frac{\partial N_2}{\partial Y}\frac{\partial N_3}{\partial X_i}\\ \frac{\partial N_3}{\partial Y}\frac{\partial N_1}{\partial X_i}&\frac{\partial N_3}{\partial Y}\frac{\partial N_2}{\partial X_i}&\frac{\partial N_3}{\partial Y}\frac{\partial N_3}{\partial X_i} \end{bmatrix}dS\{V_y\}\\[6pt] &-\int_S \begin{bmatrix} \frac{\partial N_1}{\partial Y}N_1&\frac{\partial N_1}{\partial Y}N_2&\frac{\partial N_1}{\partial Y}N_3\\ \frac{\partial N_2}{\partial Y}N_1&\frac{\partial N_2}{\partial Y}N_2&\frac{\partial N_2}{\partial Y}N_3\\ \frac{\partial N_3}{\partial Y}N_1&\frac{\partial N_3}{\partial Y}N_2&\frac{\partial N_3}{\partial Y}N_3 \end{bmatrix}dS\{P\} \end{aligned} \]

\[ +\frac{2K^*}{We}n_i\int_L \begin{bmatrix}N_1\\N_2\end{bmatrix}dL \]

\[ -g_i^*\int_S \begin{bmatrix}N_1\\N_2\\N_3\end{bmatrix}dS, \qquad (i=1,2) \]

\[ =\int_S \begin{bmatrix} L_1L_1&L_1L_2&L_1L_3\\ L_2L_1&L_2L_2&L_2L_3\\ L_3L_1&L_3L_2&L_3L_3 \end{bmatrix}dS\, \frac{\{V_i\}^{\tau+\Delta\tau}-\{V_i\}^{\tau}}{\Delta\tau} \]

\[ \begin{aligned} &+V_x\int_S\frac1{2A} \begin{bmatrix} L_1c_{1x}&L_1c_{2x}&L_1c_{3x}\\ L_2c_{1x}&L_2c_{2x}&L_2c_{3x}\\ L_3c_{1x}&L_3c_{2x}&L_3c_{3x} \end{bmatrix}dS\{V_i\}\\[6pt] &+V_y\int_S\frac1{2A} \begin{bmatrix} L_1c_{1y}&L_1c_{2y}&L_1c_{3y}\\ L_2c_{1y}&L_2c_{2y}&L_2c_{3y}\\ L_3c_{1y}&L_3c_{2y}&L_3c_{3y} \end{bmatrix}dS\{V_i\} \end{aligned} \]

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

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

\[ +\frac{2K^*}{We}n_i\int_L \begin{bmatrix}L_1\\L_2\end{bmatrix}dL \]

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

\[ \int_S L_1^pL_2^qL_3^r\,dS =\frac{p!q!r!}{(p+q+r+2)!}\,2A \] \[ \int_S L_iL_j\,dS= \begin{cases} \displaystyle \frac{1!1!}{(1+1+2)!}\,2A=\frac{2}{4!}A=\frac1{12}A,&(i\ne j),\\[6pt] \displaystyle \frac{2!}{(1+1+2)!}\,2A=\frac{4}{4!}V=\frac1{6}A,&(i=j) \end{cases} \] \[ \int_S L_i\,dS=\frac{1!}{(1+2)!}\,2A=\frac{2}{3!}A=\frac13A \] \[ \int_L L_1^pL_2^q\,dL=\frac{p!q!}{(p+q)!}L \] \[ \int_L L_i\,dL=\frac1{1!}L=L \]

\[ =\frac1{12}A \begin{bmatrix} 2&1&1\\ 1&2&1\\ 1&1&2 \end{bmatrix} \frac{\{V_i\}^{\tau+\Delta\tau}-\{V_i\}^{\tau}}{\Delta\tau} \]

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

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

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

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

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

\[ =\frac1{12}A \begin{bmatrix} 2&1&1\\ 1&2&1\\ 1&1&2 \end{bmatrix} \frac{\{V_i\}^{\tau+\Delta\tau}-\{V_i\}^{\tau}}{\Delta\tau} \]

\[ \begin{aligned} &+V_x\frac16 \begin{bmatrix} c_{1x}&c_{2x}&c_{3x}\\ c_{1x}&c_{2x}&c_{3x}\\ c_{1x}&c_{2x}&c_{3x} \end{bmatrix}\{V_i\}\\[6pt] &+V_y\frac16 \begin{bmatrix} c_{1y}&c_{2y}&c_{3y}\\ c_{1y}&c_{2y}&c_{3y}\\ c_{1y}&c_{2y}&c_{3y} \end{bmatrix}\{V_i\} \end{aligned} \]

\[ \begin{aligned} &+\frac1{Re}\frac1{4A} \begin{bmatrix} c_{1x}c_{1x}&c_{1x}c_{2x}&c_{1x}c_{3x}\\ c_{2x}c_{1x}&c_{2x}c_{2x}&c_{2x}c_{3x}\\ c_{3x}c_{1x}&c_{3x}c_{2x}&c_{3x}c_{3x} \end{bmatrix}\{V_i\}\\[6pt] &+\frac1{Re}\frac1{4A} \begin{bmatrix} c_{1x}c_{1i}&c_{1x}c_{2i}&c_{1x}c_{3i}\\ c_{2x}c_{1i}&c_{2x}c_{2i}&c_{2x}c_{3i}\\ c_{3x}c_{1i}&c_{3x}c_{2i}&c_{3x}c_{3i} \end{bmatrix}\{V_x\}\\[6pt] &-\delta_{xi}\frac16 \begin{bmatrix} c_{1x}&c_{1x}&c_{1x}\\ c_{2x}&c_{2x}&c_{2x}\\ c_{3x}&c_{3x}&c_{3x} \end{bmatrix}\{P\} \end{aligned} \]

\[ \begin{aligned} &+\frac1{Re}\frac1{4A} \begin{bmatrix} c_{1y}c_{1y}&c_{1y}c_{2y}&c_{1y}c_{3y}\\ c_{2y}c_{1y}&c_{2y}c_{2y}&c_{2y}c_{3y}\\ c_{3y}c_{1y}&c_{3y}c_{2y}&c_{3y}c_{3y} \end{bmatrix}\{V_i\}\\[6pt] &+\frac1{Re}\frac1{4A} \begin{bmatrix} c_{1y}c_{1i}&c_{1y}c_{2i}&c_{1y}c_{3i}\\ c_{2y}c_{1i}&c_{2y}c_{2i}&c_{2y}c_{3i}\\ c_{3y}c_{1i}&c_{3y}c_{2i}&c_{3y}c_{3i} \end{bmatrix}\{V_y\}\\[6pt] &-\delta_{yi}\frac16 \begin{bmatrix} c_{1y}&c_{1y}&c_{1y}\\ c_{2y}&c_{2y}&c_{2y}\\ c_{3y}&c_{3y}&c_{3y} \end{bmatrix}\{P\} \end{aligned} \]

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

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

\[ \begin{aligned} &={}[C]\frac{\{V_i\}^{\tau+\Delta\tau}-\{V_i\}^{\tau}}{\Delta\tau} +V_x[C_x]\{V_i\}+V_y[C_y]\{V_i\}\\[6pt] &+\frac1{Re}[S_{xx}]\{V_i\}+\frac1{Re}[S_{xi}]\{V_x\}-\delta_{xi}[H_x]\{P\}\\[6pt] &+\frac1{Re}[S_{yy}]\{V_i\}+\frac1{Re}[S_{yi}]\{V_y\}-\delta_{yi}[H_y]\{P\}\\[6pt] &+\frac{2K^*}{We}Ln_i\begin{bmatrix}1\\1\end{bmatrix} -g_i^*\frac{A}{3}\begin{bmatrix}1\\1\\1\end{bmatrix}, \qquad (i=1,2) \end{aligned} \]

\[ \begin{aligned} &={}[C]\frac{\{V_i\}^{\tau+\Delta\tau}-\{V_i\}^{\tau}}{\Delta\tau} +(V_x[C_x]+V_y[C_y])\{V_i\}\\[6pt] &+\frac1{Re}([S_{xx}]+[S_{yy}])\{V_i\}\\[6pt] &+\frac1{Re}([S_{xi}]\{V_x\}+[S_{yi}]\{V_y\})\\[6pt] &-(\delta_{xi}[H_x]+\delta_{yi}[H_y])\{P\}\\[6pt] &+\frac{2K^*}{We}Ln_i\begin{bmatrix}1\\1\end{bmatrix}\\[6pt] &-g_i^*\frac{A}{3}\begin{bmatrix}1\\1\\1\end{bmatrix}, \qquad (i=1,2,3)\\[6pt] &={}\begin{bmatrix}0\\0\\0\end{bmatrix} \end{aligned} \]

\[ \begin{aligned} [C]\frac{\{V_i\}^{\tau+\Delta\tau}-\{V_i\}^{\tau}}{\Delta\tau} &+(V_x[C_x]+V_y[C_y])\{V_i\}\\[6pt] &+\frac1{Re}([S_{xx}]+[S_{yy}])\{V_i\}\\[6pt] &+\frac1{Re}([S_{xi}]\{V_x\}+[S_{yi}]\{V_y\})\\[6pt] &-(\delta_{xi}[H_x]+\delta_{yi}[H_y])\{P\}\\[6pt] &+\frac{2K^*}{We}Ln_i\begin{bmatrix}1\\1\end{bmatrix} -g_i^*\frac{A}{3}\begin{bmatrix}1\\1\\1\end{bmatrix} =\begin{bmatrix}0\\0\\0\end{bmatrix}, \qquad (i=1,2) \end{aligned} \]

\[ \begin{aligned} \frac{[C]}{\Delta\tau}\{V_i\}^{\tau+\Delta\tau} &+(V_x[C_x]+V_y[C_y])\{V_i\}^{\tau+\Delta\tau}\\[6pt] &+\frac1{Re}([S_{xx}]+[S_{yy}])\{V_i\}^{\tau+\Delta\tau}\\[6pt] &+\frac1{Re}([S_{xi}]\{V_x\}^{\tau+\Delta\tau}+[S_{yi}]\{V_y\}^{\tau+\Delta\tau})\\[6pt] &-(\delta_{xi}[H_x]+\delta_{yi}[H_y])\{P\}^{\tau+\Delta\tau}\\[6pt] &={}\frac{[C]}{\Delta\tau}\{V_i\}^{\tau} -\frac{2K^*}{We}Ln_i\begin{bmatrix}1\\1\end{bmatrix} +g_i^*\frac{A}{3}\begin{bmatrix}1\\1\\1\end{bmatrix}, \qquad (i=1,2) \end{aligned} \]

\[ \begin{aligned} \frac{[C]}{\Delta\tau}\{V_x\}^{\tau+\Delta\tau} &+(V_x[C_x]+V_y[C_y])\{V_x\}^{\tau+\Delta\tau}\\[6pt] &+\frac1{Re}\left\{(2[S_{xx}]+[S_{yy}])\{V_x\}^{\tau+\Delta\tau}+[S_{yx}]\{V_y\}^{\tau+\Delta\tau}-[H_x]\{P\}^{\tau+\Delta\tau}\right\}\\[6pt] &={}\frac{[C]}{\Delta\tau}\{V_x\}^{\tau} -\frac{2K^*}{We}Ln_x\begin{bmatrix}1\\1\end{bmatrix} +g_x^*\frac{A}{3}\begin{bmatrix}1\\1\\1\end{bmatrix} \end{aligned} \]

\[ \begin{aligned} \frac{[C]}{\Delta\tau}\{V_y\}^{\tau+\Delta\tau} &+(V_x[C_x]+V_y[C_y])\{V_y\}^{\tau+\Delta\tau}\\[6pt] &+\frac1{Re}\left\{[S_{xy}]\{V_x\}^{\tau+\Delta\tau}+([S_{xx}]+2[S_{yy}])\{V_y\}^{\tau+\Delta\tau}-[H_y]\{P\}^{\tau+\Delta\tau}\right\}\\[6pt] &={}\frac{[C]}{\Delta\tau}\{V_y\}^{\tau} -\frac{2K^*}{We}Ln_y\begin{bmatrix}1\\1\end{bmatrix} +g_y^*\frac{A}{3}\begin{bmatrix}1\\1\\1\end{bmatrix} \end{aligned} \]