有限要素法・流体力学による数値計算

\[ \phi= \frac{\partial\Theta}{\partial\tau} +V_x\frac{\partial\Theta}{\partial X} +V_y\frac{\partial\Theta}{\partial Y} +V_z\frac{\partial\Theta}{\partial Z} +\frac{\partial q_x^{*}}{\partial X} +\frac{\partial q_y^{*}}{\partial Y} +\frac{\partial q_z^{*}}{\partial Z} =0 \]

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

\[ \begin{aligned} V_x&=N_1V_{x,1}+N_2V_{x,2}+N_3V_{x,3}+N_4V_{x,4} =\begin{bmatrix}N_1&N_2&N_3&N_4\end{bmatrix} \begin{Bmatrix}V_{x,1}\\V_{x,2}\\V_{x,3}\\V_{x,4}\end{Bmatrix} =[N]\{V_x\}=0\\[6pt] V_y&=N_1V_{y,1}+N_2V_{y,2}+N_3V_{y,3}+N_4V_{y,4}=0\\[6pt] V_z&=N_1V_{z,1}+N_2V_{z,2}+N_3V_{z,3}+N_4V_{z,4}=0\\[6pt] \Theta&=N_1\Theta_1+N_2\Theta_2+N_3\Theta_3+N_4\Theta_4 \end{aligned} \]

\[ \begin{aligned} &\int_V \begin{bmatrix}N_1\\N_2\\N_3\\N_4\end{bmatrix} \left( \frac{\partial\Theta}{\partial\tau} +V_x\frac{\partial\Theta}{\partial X} +V_y\frac{\partial\Theta}{\partial Y} +V_z\frac{\partial\Theta}{\partial Z} +\frac{\partial q_x^{*}}{\partial X} +\frac{\partial q_y^{*}}{\partial Y} +\frac{\partial q_z^{*}}{\partial Z} \right)dV\\[6pt] &= \int_V\begin{bmatrix}N_1\\N_2\\N_3\\N_4\end{bmatrix} \frac{\partial\Theta}{\partial\tau}dV +V_x\int_V\begin{bmatrix}N_1\\N_2\\N_3\\N_4\end{bmatrix} \frac{\partial\Theta}{\partial X}dV +V_y\int_V\begin{bmatrix}N_1\\N_2\\N_3\\N_4\end{bmatrix} \frac{\partial\Theta}{\partial Y}dV\\[6pt] &\quad +V_z\int_V\begin{bmatrix}N_1\\N_2\\N_3\\N_4\end{bmatrix} \frac{\partial\Theta}{\partial Z}dV +\int_V\begin{bmatrix}N_1\\N_2\\N_3\\N_4\end{bmatrix} \frac{\partial q_x^{*}}{\partial X}dV +\int_V\begin{bmatrix}N_1\\N_2\\N_3\\N_4\end{bmatrix} \frac{\partial q_y^{*}}{\partial Y}dV\\[6pt] &\quad +\int_V\begin{bmatrix}N_1\\N_2\\N_3\\N_4\end{bmatrix} \frac{\partial q_z^{*}}{\partial Z}dV \end{aligned} \]

\[ \begin{aligned} &\int_V \begin{bmatrix}N_1\\N_2\\N_3\\N_4\end{bmatrix} \left( \frac{\partial\Theta}{\partial\tau} +V_x\frac{\partial\Theta}{\partial X} +V_y\frac{\partial\Theta}{\partial Y} +V_z\frac{\partial\Theta}{\partial Z} +\frac{\partial q_x^{*}}{\partial X} +\frac{\partial q_y^{*}}{\partial Y} +\frac{\partial q_z^{*}}{\partial Z} \right)dV\\[6pt] &= \int_V\begin{bmatrix}N_1\\N_2\\N_3\\N_4\end{bmatrix} \frac{\partial\Theta}{\partial\tau}dV +V_x\int_V\begin{bmatrix}N_1\\N_2\\N_3\\N_4\end{bmatrix} \frac{\partial\Theta}{\partial X}dV +V_y\int_V\begin{bmatrix}N_1\\N_2\\N_3\\N_4\end{bmatrix} \frac{\partial\Theta}{\partial Y}dV\\[6pt] &\quad +V_z\int_V\begin{bmatrix}N_1\\N_2\\N_3\\N_4\end{bmatrix} \frac{\partial\Theta}{\partial Z}dV +\int_V\begin{bmatrix}N_1\\N_2\\N_3\\N_4\end{bmatrix} \frac{\partial q_x^{*}}{\partial X}dV +\int_V\begin{bmatrix}N_1\\N_2\\N_3\\N_4\end{bmatrix} \frac{\partial q_y^{*}}{\partial Y}dV\\[6pt] &\quad +\int_V\begin{bmatrix}N_1\\N_2\\N_3\\N_4\end{bmatrix} \frac{\partial q_z^{*}}{\partial Z}dV \end{aligned} \]

\[ \begin{aligned} &={} \int_V[N]^T\frac{\partial\Theta}{\partial\tau}dV +V_x\int_V[N]^T\frac{\partial\Theta}{\partial X}dV +V_y\int_V[N]^T\frac{\partial\Theta}{\partial Y}dV +V_z\int_V[N]^T\frac{\partial\Theta}{\partial Z}dV\\[6pt] &+\int_V[N]^T\frac{\partial q_x^{*}}{\partial X}dV +\int_V[N]^T\frac{\partial q_y^{*}}{\partial Y}dV +\int_V[N]^T\frac{\partial q_z^{*}}{\partial Z}dV \end{aligned} \]

\[ \begin{aligned} &={} \int_V[N]^T \frac{[N]\left(\{\Theta\}^{\tau+\Delta\tau}-\{\Theta\}^{\tau}\right)} {\Delta\tau}dV\\[6pt] &+V_x\int_V[N]^T\frac{\partial[N]\{\Theta\}}{\partial X}dV +V_y\int_V[N]^T\frac{\partial[N]\{\Theta\}}{\partial Y}dV +V_z\int_V[N]^T\frac{\partial[N]\{\Theta\}}{\partial Z}dV\\[6pt] &+\int_S[N]^Tq_x^{*}n_x\,dS -\int_V\left[\frac{\partial N}{\partial X}\right]^Tq_x^{*}dV\\[6pt] &+\int_S[N]^Tq_y^{*}n_y\,dS -\int_V\left[\frac{\partial N}{\partial Y}\right]^Tq_y^{*}dV\\[6pt] &+\int_S[N]^Tq_z^{*}n_z\,dS -\int_V\left[\frac{\partial N}{\partial Z}\right]^Tq_z^{*}dV \end{aligned} \]

\[ \begin{aligned} &={} \int_V[N]^T[N]dV \frac{\{\Theta\}^{\tau+\Delta\tau}-\{\Theta\}^{\tau}} {\Delta\tau}\\[6pt] &+V_x\int_V[N]^T\left[\frac{\partial N}{\partial X}\right]dV\{\Theta\} +V_y\int_V[N]^T\left[\frac{\partial N}{\partial Y}\right]dV\{\Theta\} +V_z\int_V[N]^T\left[\frac{\partial N}{\partial Z}\right]dV\{\Theta\}\\[6pt] &-\int_V\left[\frac{\partial N}{\partial X}\right]^Tq_x^{*}dV -\int_V\left[\frac{\partial N}{\partial Y}\right]^Tq_y^{*}dV -\int_V\left[\frac{\partial N}{\partial Z}\right]^Tq_z^{*}dV\\[6pt] &+\int_S[N]^T(q_x^{*}n_x+q_y^{*}n_y+q_z^{*}n_z)dS \end{aligned} \]

\[ \begin{aligned} &={} \int_V[N]^T[N]dV \frac{\{\Theta\}^{\tau+\Delta\tau}-\{\Theta\}^{\tau}} {\Delta\tau}\\[6pt] &+V_x\int_V[N]^T\left[\frac{\partial N}{\partial X}\right]dV\{\Theta\} +V_y\int_V[N]^T\left[\frac{\partial N}{\partial Y}\right]dV\{\Theta\} +V_z\int_V[N]^T\left[\frac{\partial N}{\partial Z}\right]dV\{\Theta\}\\[6pt] &-\int_V\left[\frac{\partial N}{\partial X}\right]^T \left(-\frac{1}{Pe}\frac{\partial\Theta}{\partial X}\right)dV\\[6pt] &-\int_V\left[\frac{\partial N}{\partial Y}\right]^T \left(-\frac{1}{Pe}\frac{\partial\Theta}{\partial Y}\right)dV\\[6pt] &-\int_V\left[\frac{\partial N}{\partial Z}\right]^T \left(-\frac{1}{Pe}\frac{\partial\Theta}{\partial Z}\right)dV\\[6pt] &+Nu(\Theta-\Theta_h)\int_S[N]^T dS \end{aligned} \]

\[ \begin{aligned} &={} \int_V[N]^T[N]dV \frac{\{\Theta\}^{\tau+\Delta\tau}-\{\Theta\}^{\tau}} {\Delta\tau}\\[6pt] &+V_x\int_V[N]^T\left[\frac{\partial N}{\partial X}\right]dV\{\Theta\} +V_y\int_V[N]^T\left[\frac{\partial N}{\partial Y}\right]dV\{\Theta\} +V_z\int_V[N]^T\left[\frac{\partial N}{\partial Z}\right]dV\{\Theta\}\\[6pt] &+\frac{1}{Pe}\int_V\left[\frac{\partial N}{\partial X}\right]^T \frac{\partial\Theta}{\partial X}dV +\frac{1}{Pe}\int_V\left[\frac{\partial N}{\partial Y}\right]^T \frac{\partial\Theta}{\partial Y}dV +\frac{1}{Pe}\int_V\left[\frac{\partial N}{\partial Z}\right]^T \frac{\partial\Theta}{\partial Z}dV\\[6pt] &+Nu(\Theta-\Theta_h)\int_S[N]^T dS \end{aligned} \]

\[ \begin{aligned} &={} \int_V[N]^T[N]dV \frac{\{\Theta\}^{\tau+\Delta\tau}-\{\Theta\}^{\tau}} {\Delta\tau}\\[6pt] &+V_x\int_V[N]^T\left[\frac{\partial N}{\partial X}\right]dV\{\Theta\} +V_y\int_V[N]^T\left[\frac{\partial N}{\partial Y}\right]dV\{\Theta\} +V_z\int_V[N]^T\left[\frac{\partial N}{\partial Z}\right]dV\{\Theta\}\\[6pt] &+\frac{1}{Pe}\int_V\left[\frac{\partial N}{\partial X}\right]^T \frac{\partial[N]\{\Theta\}}{\partial X}dV +\frac{1}{Pe}\int_V\left[\frac{\partial N}{\partial Y}\right]^T \frac{\partial[N]\{\Theta\}}{\partial Y}dV +\frac{1}{Pe}\int_V\left[\frac{\partial N}{\partial Z}\right]^T \frac{\partial[N]\{\Theta\}}{\partial Z}dV\\[6pt] &+Nu(\Theta-\Theta_h)\int_S[N]^T dS \end{aligned} \]

\[ \begin{aligned} &={} \int_V[N]^T[N]dV \frac{\{\Theta\}^{\tau+\Delta\tau}-\{\Theta\}^{\tau}} {\Delta\tau}\\[6pt] &+V_x\int_V[N]^T\left[\frac{\partial N}{\partial X}\right]dV\{\Theta\} +V_y\int_V[N]^T\left[\frac{\partial N}{\partial Y}\right]dV\{\Theta\} +V_z\int_V[N]^T\left[\frac{\partial N}{\partial Z}\right]dV\{\Theta\}\\[6pt] &+\frac{1}{Pe}\int_V\left[\frac{\partial N}{\partial X}\right]^T \left[\frac{\partial N}{\partial X}\right]dV\{\Theta\} +\frac{1}{Pe}\int_V\left[\frac{\partial N}{\partial Y}\right]^T \left[\frac{\partial N}{\partial Y}\right]dV\{\Theta\}\\[6pt] &+\frac{1}{Pe}\int_V\left[\frac{\partial N}{\partial Z}\right]^T \left[\frac{\partial N}{\partial Z}\right]dV\{\Theta\} +Nu(\Theta-\Theta_h)\int_S[N]^T dS \end{aligned} \]

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

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

\[ \begin{aligned} &+\frac{1}{Pe}\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\{\Theta\}\\[6pt] &+\frac{1}{Pe}\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\{\Theta\}\\[6pt] &+\frac{1}{Pe}\int_V \begin{bmatrix} \frac{\partial N_1}{\partial Z}\frac{\partial N_1}{\partial Z}&\frac{\partial N_1}{\partial Z}\frac{\partial N_2}{\partial Z}&\frac{\partial N_1}{\partial Z}\frac{\partial N_3}{\partial Z}&\frac{\partial N_1}{\partial Z}\frac{\partial N_4}{\partial Z}\\ \frac{\partial N_2}{\partial Z}\frac{\partial N_1}{\partial Z}&\frac{\partial N_2}{\partial Z}\frac{\partial N_2}{\partial Z}&\frac{\partial N_2}{\partial Z}\frac{\partial N_3}{\partial Z}&\frac{\partial N_2}{\partial Z}\frac{\partial N_4}{\partial Z}\\ \frac{\partial N_3}{\partial Z}\frac{\partial N_1}{\partial Z}&\frac{\partial N_3}{\partial Z}\frac{\partial N_2}{\partial Z}&\frac{\partial N_3}{\partial Z}\frac{\partial N_3}{\partial Z}&\frac{\partial N_3}{\partial Z}\frac{\partial N_4}{\partial Z}\\ \frac{\partial N_4}{\partial Z}\frac{\partial N_1}{\partial Z}&\frac{\partial N_4}{\partial Z}\frac{\partial N_2}{\partial Z}&\frac{\partial N_4}{\partial Z}\frac{\partial N_3}{\partial Z}&\frac{\partial N_4}{\partial Z}\frac{\partial N_4}{\partial Z} \end{bmatrix}dV\{\Theta\} \end{aligned} \]

\[ +Nu(\Theta-\Theta_h)\int_S \begin{bmatrix} N_1\\N_2\\N_3 \end{bmatrix}dS \]

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

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

\[ \begin{aligned} &+\frac{1}{Pe}\int_V\frac{1}{36V^2} \begin{bmatrix} c_{1x}c_{1x}&c_{1x}c_{2x}&c_{1x}c_{3x}&c_{1x}c_{4x}\\ c_{2x}c_{1x}&c_{2x}c_{2x}&c_{2x}c_{3x}&c_{2x}c_{4x}\\ c_{3x}c_{1x}&c_{3x}c_{2x}&c_{3x}c_{3x}&c_{3x}c_{4x}\\ c_{4x}c_{1x}&c_{4x}c_{2x}&c_{4x}c_{3x}&c_{4x}c_{4x} \end{bmatrix}dV\{\Theta\}\\[6pt] &+\frac{1}{Pe}\int_V\frac{1}{36V^2} \begin{bmatrix} c_{1y}c_{1y}&c_{1y}c_{2y}&c_{1y}c_{3y}&c_{1y}c_{4y}\\ c_{2y}c_{1y}&c_{2y}c_{2y}&c_{2y}c_{3y}&c_{2y}c_{4y}\\ c_{3y}c_{1y}&c_{3y}c_{2y}&c_{3y}c_{3y}&c_{3y}c_{4y}\\ c_{4y}c_{1y}&c_{4y}c_{2y}&c_{4y}c_{3y}&c_{4y}c_{4y} \end{bmatrix}dV\{\Theta\}\\[6pt] &+\frac{1}{Pe}\int_V\frac{1}{36V^2} \begin{bmatrix} c_{1z}c_{1z}&c_{1z}c_{2z}&c_{1z}c_{3z}&c_{1z}c_{4z}\\ c_{2z}c_{1z}&c_{2z}c_{2z}&c_{2z}c_{3z}&c_{2z}c_{4z}\\ c_{3z}c_{1z}&c_{3z}c_{2z}&c_{3z}c_{3z}&c_{3z}c_{4z}\\ c_{4z}c_{1z}&c_{4z}c_{2z}&c_{4z}c_{3z}&c_{4z}c_{4z} \end{bmatrix}dV\{\Theta\} \end{aligned} \]

\[ +Nu(\Theta-\Theta_h)\int_S \begin{bmatrix} L_1\\L_2\\L_3 \end{bmatrix}dS \]

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

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

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

\[ \begin{aligned} &+\frac{1}{Pe}\frac{1}{36V^2}V \begin{bmatrix} c_{1x}c_{1x}&c_{1x}c_{2x}&c_{1x}c_{3x}&c_{1x}c_{4x}\\ c_{2x}c_{1x}&c_{2x}c_{2x}&c_{2x}c_{3x}&c_{2x}c_{4x}\\ c_{3x}c_{1x}&c_{3x}c_{2x}&c_{3x}c_{3x}&c_{3x}c_{4x}\\ c_{4x}c_{1x}&c_{4x}c_{2x}&c_{4x}c_{3x}&c_{4x}c_{4x} \end{bmatrix}\{\Theta\}\\[6pt] &+\frac{1}{Pe}\frac{1}{36V^2}V \begin{bmatrix} c_{1y}c_{1y}&c_{1y}c_{2y}&c_{1y}c_{3y}&c_{1y}c_{4y}\\ c_{2y}c_{1y}&c_{2y}c_{2y}&c_{2y}c_{3y}&c_{2y}c_{4y}\\ c_{3y}c_{1y}&c_{3y}c_{2y}&c_{3y}c_{3y}&c_{3y}c_{4y}\\ c_{4y}c_{1y}&c_{4y}c_{2y}&c_{4y}c_{3y}&c_{4y}c_{4y} \end{bmatrix}\{\Theta\}\\[6pt] &+\frac{1}{Pe}\frac{1}{36V^2}V \begin{bmatrix} c_{1z}c_{1z}&c_{1z}c_{2z}&c_{1z}c_{3z}&c_{1z}c_{4z}\\ c_{2z}c_{1z}&c_{2z}c_{2z}&c_{2z}c_{3z}&c_{2z}c_{4z}\\ c_{3z}c_{1z}&c_{3z}c_{2z}&c_{3z}c_{3z}&c_{3z}c_{4z}\\ c_{4z}c_{1z}&c_{4z}c_{2z}&c_{4z}c_{3z}&c_{4z}c_{4z} \end{bmatrix}\{\Theta\} \end{aligned} \]

\[ +Nu(\Theta-\Theta_h)\frac{S}{3} \begin{bmatrix} 1\\1\\1 \end{bmatrix} \]

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

\[ \begin{aligned} &+\frac{1}{Pe}\frac{1}{36V} \begin{bmatrix} c_{1x}c_{1x}&c_{1x}c_{2x}&c_{1x}c_{3x}&c_{1x}c_{4x}\\ c_{2x}c_{1x}&c_{2x}c_{2x}&c_{2x}c_{3x}&c_{2x}c_{4x}\\ c_{3x}c_{1x}&c_{3x}c_{2x}&c_{3x}c_{3x}&c_{3x}c_{4x}\\ c_{4x}c_{1x}&c_{4x}c_{2x}&c_{4x}c_{3x}&c_{4x}c_{4x} \end{bmatrix}\{\Theta\}\\[6pt] &+\frac{1}{Pe}\frac{1}{36V} \begin{bmatrix} c_{1y}c_{1y}&c_{1y}c_{2y}&c_{1y}c_{3y}&c_{1y}c_{4y}\\ c_{2y}c_{1y}&c_{2y}c_{2y}&c_{2y}c_{3y}&c_{2y}c_{4y}\\ c_{3y}c_{1y}&c_{3y}c_{2y}&c_{3y}c_{3y}&c_{3y}c_{4y}\\ c_{4y}c_{1y}&c_{4y}c_{2y}&c_{4y}c_{3y}&c_{4y}c_{4y} \end{bmatrix}\{\Theta\}\\[6pt] &+\frac{1}{Pe}\frac{1}{36V} \begin{bmatrix} c_{1z}c_{1z}&c_{1z}c_{2z}&c_{1z}c_{3z}&c_{1z}c_{4z}\\ c_{2z}c_{1z}&c_{2z}c_{2z}&c_{2z}c_{3z}&c_{2z}c_{4z}\\ c_{3z}c_{1z}&c_{3z}c_{2z}&c_{3z}c_{3z}&c_{3z}c_{4z}\\ c_{4z}c_{1z}&c_{4z}c_{2z}&c_{4z}c_{3z}&c_{4z}c_{4z} \end{bmatrix}\{\Theta\} \end{aligned} \]

\[ +Nu(\Theta-\Theta_h)\frac{S}{3} \begin{bmatrix} 1\\1\\1 \end{bmatrix} \]

\[ \begin{aligned} &={} [C]\frac{\{\Theta\}^{\tau+\Delta\tau}-\{\Theta\}^{\tau}}{\Delta\tau}\\[6pt] &+V_x[C_x]\{\Theta\} +V_y[C_y]\{\Theta\} +V_z[C_z]\{\Theta\}\\[6pt] &+\frac{1}{Pe}[S_{xx}]\{\Theta\} +\frac{1}{Pe}[S_{yy}]\{\Theta\} +\frac{1}{Pe}[S_{zz}]\{\Theta\}\\[6pt] &+Nu(\Theta-\Theta_h)\frac{S}{3} \begin{bmatrix}1\\1\\1\end{bmatrix} \end{aligned} \]

\[ \begin{aligned} &={} [C]\frac{\{\Theta\}^{\tau+\Delta\tau}-\{\Theta\}^{\tau}}{\Delta\tau}\\[6pt] &+\left(V_x[C_x]+V_y[C_y]+V_z[C_z]\right)\{\Theta\}\\[6pt] &+\frac{1}{Pe}\left([S_{xx}]+[S_{yy}]+[S_{zz}]\right)\{\Theta\}\\[6pt] &+Nu(\Theta-\Theta_h)\frac{S}{3} \begin{bmatrix}1\\1\\1\end{bmatrix}\\[6pt] &={} \begin{bmatrix}0\\0\\0\\0\end{bmatrix} \end{aligned} \]

\[ \begin{aligned} &[C]\frac{\{\Theta\}^{\tau+\Delta\tau}-\{\Theta\}^{\tau}}{\Delta\tau}\\[6pt] &+\left(V_x[C_x]+V_y[C_y]+V_z[C_z]\right)\{\Theta\}\\[6pt] &+\frac{1}{Pe}\left([S_{xx}]+[S_{yy}]+[S_{zz}]\right)\{\Theta\}\\[6pt] &+Nu(\Theta-\Theta_h)\frac{S}{3} \begin{bmatrix}1\\1\\1\end{bmatrix} = \begin{bmatrix}0\\0\\0\\0\end{bmatrix} \end{aligned} \]

\[ \begin{aligned} &\frac{[C]}{\Delta\tau}\{\Theta\}^{\tau+\Delta\tau} +\left(V_x[C_x]+V_y[C_y]+V_z[C_z]\right)\{\Theta\}^{\tau+\Delta\tau}\\[6pt] &+\frac{1}{Pe}\left([S_{xx}]+[S_{yy}]+[S_{zz}]\right)\{\Theta\}^{\tau+\Delta\tau}\\[6pt] &={} \frac{[C]}{\Delta\tau}\{\Theta\}^{\tau} -Nu(\Theta-\Theta_h)\frac{S}{3} \begin{bmatrix}1\\1\\1\end{bmatrix} \end{aligned} \]