做个公式记录

[李东岳]

经常有些公式简单到不能包含在任何文章里，但是每次一步一步推导的时候还不能升略，专门写个文档还很麻烦，在这开个贴记录一下把，随时随地可以访问。

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$$\begin{array}{}\text{(1)}& \frac{1}{{\tau }_0{\xi }^{2/3}}\frac{\mathrm{\partial }vn}{\mathrm{\partial }v}\end{array}$$

$$\begin{array}{}\text{(2)}& {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}{\int }_0^{+\mathrm{\infty }}\frac{v^j{\xi }^k}{{\tau }_0{\xi }^{2/3}}\frac{\mathrm{\partial }vn}{\mathrm{\partial }v}\mathrm{d}v\mathrm{d}\xi =\frac{1}{{\tau }_0}{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}{\int }_0^{+\mathrm{\infty }}{v}^j{\xi }^{k-2/3}\frac{\mathrm{\partial }vn}{\mathrm{\partial }v}\mathrm{d}v\mathrm{d}\xi \end{array}$$

$$\begin{array}{}\text{(3)}& =\frac{1}{{\tau }_0}{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}{\int }_0^{+\mathrm{\infty }}{v}^j{\xi }^{k-2/3}\mathrm{d}(vn)\mathrm{d}\xi \end{array}$$

$$\begin{array}{}\text{(4)}& =\frac{1}{{\tau }_0}{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}{\xi }^{k-2/3}({\int }_0^{+\mathrm{\infty }}{v}^j\mathrm{d}(vn))\mathrm{d}\xi \end{array}$$

$$\begin{array}{}\text{(5)}& =\frac{1}{{\tau }_0}{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}{\xi }^{k-2/3}(v^{j+1}{n}_{-\mathrm{\infty }}^{+\mathrm{\infty }}-{\int }_0^{+\mathrm{\infty }}j{v}^{j}n\mathrm{d}(v))\mathrm{d}\xi \end{array}$$

$$\begin{array}{}\text{(6)}& =-\frac{1}{{\tau }_0}{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}{\int }_0^{+\mathrm{\infty }}j{v}^j{\xi }^{k-2/3}n\mathrm{d}(v)\mathrm{d}\xi \end{array}$$

$$\begin{array}{}\text{(7)}& =-\frac{j}{{\tau }_0}{m}_{j,k-2/3}\end{array}$$

[李东岳]

$$\begin{array}{}\text{(8)}& {\int }_0^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{u}_d^j{\xi }^k\frac{\mathrm{\partial }n}{\mathrm{\partial }t}\mathrm{d}\xi \mathrm{d}{u}_{\mathrm{d}}=\frac{\mathrm{\partial }{m}_{j,k}}{\mathrm{\partial }t}\end{array}$$

$$\begin{array}{}\text{(9)}& {\int }_0^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{u}_d^j{\xi }^k\frac{\mathrm{\partial }n{u}_{\mathrm{d}}}{\mathrm{\partial }x}\mathrm{d}\xi \mathrm{d}{u}_{\mathrm{d}}=\frac{\mathrm{\partial }{m}_{j+1,k}}{\mathrm{\partial }x}\end{array}$$

$$\begin{array}{}\text{(10)}& {\int }_0^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{u}_d^j{\xi }^k\frac{\mathrm{\partial }}{\mathrm{\partial }{u}_{\mathrm{d}}}\left(\frac{1}{{\rho }_{\mathrm{d}}}\frac{\mathrm{\partial }{p}_c}{\mathrm{\partial }x}n\right)\mathrm{d}\xi \mathrm{d}{u}_{\mathrm{d}}=\frac{1}{{\rho }_{\mathrm{d}}}\frac{\mathrm{\partial }{p}_c}{\mathrm{\partial }x}{\int }_0^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{u}_d^j{\xi }^k\frac{\mathrm{\partial }n}{\mathrm{\partial }{u}_{\mathrm{d}}}\mathrm{d}\xi \mathrm{d}{u}_{\mathrm{d}}=\frac{1}{{\rho }_{\mathrm{d}}}\frac{\mathrm{\partial }{p}_c}{\mathrm{\partial }x}{\int }_0^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{u}_d^j{\xi }^k\mathrm{d}n\mathrm{d}\xi =\frac{1}{{\rho }_{\mathrm{d}}}\frac{\mathrm{\partial }{p}_c}{\mathrm{\partial }x}{\int }_0^{\mathrm{\infty }}{\xi }^k\left({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{u}_d^j\mathrm{d}n\right)\mathrm{d}\xi \end{array}$$
$$\begin{array}{}\text{(11)}& =\frac{1}{{\rho }_{\mathrm{d}}}\frac{\mathrm{\partial }{p}_c}{\mathrm{\partial }x}{\int }_0^{\mathrm{\infty }}{\xi }^k(u_d^j{n}_{-\mathrm{\infty }}^{\mathrm{\infty }}-{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}n\mathrm{d}{u}^j)\mathrm{d}\xi =-\frac{1}{{\rho }_{\mathrm{d}}}\frac{\mathrm{\partial }{p}_c}{\mathrm{\partial }x}{\int }_0^{\mathrm{\infty }}{\xi }^k\left(j{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}n{u}^{j-1}\mathrm{d}u\right)\mathrm{d}\xi =-j\frac{1}{{\rho }_{\mathrm{d}}}\frac{\mathrm{\partial }{p}_c}{\mathrm{\partial }x}{\int }_0^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}n{u}^{j-1}{\xi }^k\mathrm{d}u\mathrm{d}\xi =-j\frac{1}{{\rho }_{\mathrm{d}}}\frac{\mathrm{\partial }{p}_c}{\mathrm{\partial }x}{m}_{j-1,k}\end{array}$$

[李东岳]

$$\rho =\begin{array}{cc}a& b\\ c& d\end{array}$$

$$\rho =\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$$

[李东岳]

$$\begin{array}{}\text{(12)}& \frac{\mathrm{d}{u}_d}{\mathrm{d}t}=g+\frac{1}{\tau }(u_c-u_d)\end{array}$$

$$\begin{array}{}\text{(13)}& \frac{\mathrm{d}{u}_d}{u_d-\tau g-u_c}=-\frac{1}{\tau }\mathrm{d}t\end{array}$$

$$\begin{array}{}\text{(14)}& \mathrm{l}\mathrm{n}|u_d-\tau g-u_c|=-\frac{t}{\tau }+C\end{array}$$

$$\begin{array}{}\text{(15)}& u_d-\tau g-u_c=e^{-t/\tau +C},t=0,u_d=u_d^0\end{array}$$

$$\begin{array}{}\text{(16)}& u_d^0-\tau g-u_c=e^C\end{array}$$

$$\begin{array}{}\text{(17)}& C=\mathrm{l}\mathrm{n}|u_d^0-\tau g-u_c|\end{array}$$

$$\begin{gathered} \begin{array}{}\text{(18)}& u_d=e^{-t/\tau }(u_d^0-\tau g-u_c)+\tau g+u_c \\ =e^{-t/\tau }{u}_d^0+(1-e^{-t/\tau })(\tau g+u_c)\end{array} \end{gathered}$$

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$$\begin{array}{}\text{(19)}& S={\int }_0^t{u}_d\mathrm{d}t=(u_d^0-\tau g-u_c){\int }_0^t{e}^{-t/\tau }\mathrm{d}t+(\tau g+u_c)t\end{array}$$

[李东岳]

$$\begin{array}{}\text{(20)}& \frac{\mathrm{\partial }\left({\alpha }_k{\rho }_k{\mathbf{U}}_k\right)}{\mathrm{\partial }t}+\mathrm{\nabla }\cdot \left({\alpha }_k{\rho }_k{\mathbf{U}}_k{\mathbf{U}}_k\right)-\mathrm{\nabla }\cdot \left({\alpha }_k{\rho }_k{\tau }_k\right)=-{\alpha }_k\mathrm{\nabla }{p}_k+{\alpha }_k{\rho }_k\mathbf{g}+\sum {\mathbf{M}}_{ij},\end{array}$$

$$\begin{array}{}\text{(21)}& \frac{\mathrm{\partial }\left({\alpha }_k{\rho }_k\right)}{\mathrm{\partial }t}+\mathrm{\nabla }\cdot \left({\alpha }_k{\rho }_k{\mathbf{U}}_k\right)=0\end{array}$$

$$\begin{array}{}\text{(22)}& \mathrm{\nabla }\cdot \left({\alpha }_k{\rho }_k{\mathbf{U}}_k\right)=0\end{array}$$

$$\begin{array}{}\text{(23)}& \sum {\alpha }_k{\rho }_k{\mathbf{U}}_k\cdot {\mathbf{S}}_f=0\end{array}$$

$$\begin{array}{}\text{(24)}& {\tau }_k=-{\nu }_{\mathrm{k},\mathrm{e}\mathrm{f}\mathrm{f}}(\mathrm{\nabla }{\mathbf{U}}_k+{\mathrm{\nabla }}^{\mathrm{T}}{\mathbf{U}}_k)+\frac{2}{3}{\nu }_{\mathrm{k},\mathrm{e}\mathrm{f}\mathrm{f}}\mathrm{\nabla }\cdot ({\mathbf{U}}_k\cdot \mathbf{I}),\end{array}$$

$$\begin{array}{}\text{(25)}& {\mathbf{M}}_{\mathrm{d}\mathrm{r}\mathrm{a}\mathrm{g}}=\frac{3}{4}{\alpha }_k{\rho }_{\mathrm{c}}{C}_{\mathrm{D}}\frac{1}{d_k}|{\mathbf{U}}_{\mathrm{c}}-{\mathbf{U}}_k|({\mathbf{U}}_{\mathrm{c}}-{\mathbf{U}}_k),\end{array}$$

$$\begin{array}{}\text{(26)}& Re=\frac{d_k|{\mathbf{U}}_k-{\mathbf{U}}_{\mathrm{c}}|}{{\nu }_{\mathrm{c}}}\end{array}$$

$$\begin{array}{}\text{(27)}& {\mathbf{M}}_{\mathrm{l}\mathrm{i}\mathrm{f}\mathrm{t}}={\alpha }_{\mathrm{d}}{C}_{\mathrm{L}}{\rho }_{\mathrm{c}}{\mathbf{U}}_{\mathrm{r}}\times (\mathrm{\nabla }\times {\mathbf{U}}_{\mathrm{c}}),\end{array}$$

$$\begin{array}{}\text{(28)}& {\mathbf{M}}_{\mathrm{w}\mathrm{a}\mathrm{l}\mathrm{l}}=C_{\mathrm{w}\mathrm{a}\mathrm{l}\mathrm{l}}{\rho }_{\mathrm{c}}{\alpha }_k|{\mathbf{U}}_{\mathrm{c}}-{\mathbf{U}}_k{|}^2\cdot \mathbf{n}\end{array}$$

$$\begin{array}{}\text{(29)}& {\mathbf{M}}_{\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{b}}=C_{\mathrm{T}}{\rho }_{\mathrm{c}}{k}_{\mathrm{c}}\mathrm{\nabla }{\alpha }_{\mathrm{d}},\end{array}$$

$$\begin{array}{}\text{(30)}& \frac{\mathrm{\partial }\left({\alpha }_k{\rho }_k{\mathbf{U}}_k\right)}{\mathrm{\partial }t}+\mathrm{\nabla }\cdot \left({\alpha }_k{\rho }_k{\mathbf{U}}_k{\mathbf{U}}_k\right)-\mathrm{\nabla }\cdot \left({\alpha }_{\mathrm{d}}{\rho }_{\mathrm{d}}{\tau }_{\mathrm{d}}\right)=-{\mathrm{K}\mathrm{d}}_k{\mathbf{U}}_k+{\mathbf{M}}_{\mathrm{l}\mathrm{i}\mathrm{f}\mathrm{t},k}+{\mathbf{M}}_{\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{b},k}+{\mathbf{M}}_{\mathrm{w}\mathrm{a}\mathrm{l}\mathrm{l},k},\end{array}$$

$$\begin{array}{}\text{(31)}& \mathrm{K}\mathrm{d}=\frac{3}{4}{\alpha }_k{\rho }_{\mathrm{c}}{C}_{\mathrm{D},k}\frac{1}{d_k}|{\mathbf{U}}_{\mathrm{c}}-{\mathbf{U}}_k|.\end{array}$$

$$\begin{array}{}\text{(32)}& A_{k,\mathrm{P}}{\mathbf{U}}_{k,\mathrm{P}}+\sum A_{k,\mathrm{N}}{\mathbf{U}}_{k,\mathrm{N}}=S_{k,\mathrm{P}},\end{array}$$

$$\begin{array}{}\text{(33)}& {\mathbf{H}\mathbf{b}\mathbf{y}\mathbf{A}}_{k,\mathrm{P}}=\frac{1}{A_{k,\mathrm{P}}}\left(-\sum A_{k,\mathrm{N}}{\mathbf{U}}_{k,\mathrm{N}}+S_{k,\mathrm{P}}\right),\end{array}$$

$$\begin{array}{}\text{(34)}& {\mathbf{U}}_{k,\mathrm{P}}={\mathbf{H}\mathbf{b}\mathbf{y}\mathbf{A}}_{k,\mathrm{P}}+\frac{{\alpha }_{k,\mathrm{P}}}{A_{k,\mathrm{P}}}(\mathrm{\nabla }{p}_{\mathrm{r}\mathrm{g}\mathrm{h},\mathrm{P}}-{\alpha }_{\mathrm{c},\mathrm{P}}({\rho }_{\mathrm{c}}-{\rho }_k)\mathbf{g}-\mathbf{g}\cdot {\mathbf{h}}_{\mathrm{P}}\mathrm{\nabla }{\rho }_{\mathrm{P}})+\frac{{\mathrm{K}\mathrm{d}}_k}{A_{k,\mathrm{P}}}{\mathbf{U}}_{\mathrm{c},\mathrm{P}},\end{array}$$

$$\begin{array}{}\text{(35)}& \sum {\alpha }_{k,f}{\varphi }_k+{\alpha }_{\mathrm{c},f}{\varphi }_{\mathrm{c}}=\mathrm{\nabla }\cdot \left((\sum {\alpha }_{k,\mathrm{P}}\frac{{\alpha }_{k,\mathrm{P}}}{A_{k,\mathrm{P}}}+{\alpha }_{\mathrm{c},\mathrm{P}}\frac{{\alpha }_{\mathrm{c},\mathrm{P}}}{A_{\mathrm{c},\mathrm{P}}})\mathrm{\nabla }{p}_{\mathrm{r}\mathrm{g}\mathrm{h},\mathrm{P}}\right),\end{array}$$

$$\begin{array}{}\text{(36)}& {\varphi }_k=({\mathbf{H}\mathbf{b}\mathbf{y}\mathbf{A}}_{k,f}+\frac{{\alpha }_{k,f}}{A_{k,f}}(-{\alpha }_{\mathrm{c},f}({\rho }_{\mathrm{c}}-{\rho }_{\mathrm{d}})\mathbf{g}-\mathbf{g}\cdot {\mathbf{h}}_f\mathrm{\nabla }{\rho }_f)+\frac{{\mathrm{K}\mathrm{d}}_f}{A_{k,f}}{\mathbf{U}}_{\mathrm{c},f})\cdot {\mathbf{S}}_f\end{array}$$

$$\begin{array}{}\text{(37)}& {\varphi }_{\mathrm{c}}=({\mathbf{H}\mathbf{b}\mathbf{y}\mathbf{A}}_{\mathrm{c},f}+\frac{{\alpha }_{\mathrm{c},f}}{A_{\mathrm{c},f}}(-{\alpha }_{\mathrm{d},f}({\rho }_{\mathrm{d}}-{\rho }_{\mathrm{c}})\mathbf{g}-\mathbf{g}\cdot {\mathbf{h}}_f\mathrm{\nabla }{\rho }_f)+\frac{{\mathrm{K}\mathrm{d}}_f}{A_{\mathrm{c},f}}{\mathbf{U}}_{\mathrm{d},f})\cdot {\mathbf{S}}_f\end{array}$$