# 流体運動方程式　テンソル形式の導出
Krumholtz(6.2)の導出(磁場の項は考えない)

連続の式のテンソル形式
\begin{equation}
\frac{\partial \rho}{\partial t}=-\nabla \cdot (\rho \boldsymbol{v})=-\frac{\partial}{\partial x_k}(\rho v_k) \tag{1}
\end{equation}
運動方程式
\begin{equation}
\frac{\partial \boldsymbol{v}}{\partial t}+(\boldsymbol{v}\cdot \nabla)\boldsymbol{v}=-\frac{1}{\rho}\nabla p-\nabla \phi \tag{2}
\end{equation}
(2)に$\rho$をかける
\begin{equation}
\rho\frac{\partial \boldsymbol{v}}{\partial t}+\rho(\boldsymbol{v}\cdot \nabla)\boldsymbol{v}=-\nabla p-\rho\nabla \phi \tag{3}
\end{equation}
ここで
\begin{equation}
\frac{\partial}{\partial t}(\rho \boldsymbol{v})=\boldsymbol{v}\frac{\partial \rho}{\partial t}＋\rho\frac{\partial \boldsymbol{v}}{\partial t}
\end{equation}
より(3)の左辺第1項は
\begin{equation}
\rho\frac{\partial \boldsymbol{v}}{\partial t}=\frac{\partial}{\partial t}(\rho \boldsymbol{v})-\boldsymbol{v}\frac{\partial \rho}{\partial t}\tag{4}
\end{equation}
また(3)の左辺第2項をテンソル形式にすると
\begin{equation}
\rho(\boldsymbol{v}\cdot \nabla)\boldsymbol{v}=\rho(v_i\frac{\partial (v_j\boldsymbol{e_j})}{\partial x_i})=\rho v_i\frac{\partial v_j}{\partial x_i}\boldsymbol{e_j}\tag{5}
\end{equation}
ここで
\begin{equation}
\rho\frac{\partial (v_iv_j\boldsymbol{e_j})}{\partial x_i}=\rho v_i\frac{\partial v_j}{\partial x_i}\boldsymbol{e_j}+\rho v_j\frac{\partial v_i}{\partial x_i}\boldsymbol{e_j}
\end{equation}
より(5)式は
\begin{equation}
\rho(\boldsymbol{v}\cdot \nabla)\boldsymbol{v}=\rho v_i\frac{\partial v_j}{\partial x_i}\boldsymbol{e_j}=\rho\frac{\partial (v_iv_j\boldsymbol{e_j})}{\partial x_i}-\rho v_j\frac{\partial v_i}{\partial x_i}\boldsymbol{e_j}\tag{6}
\end{equation}
(4)(6)を(3)の左辺に代入すると
\begin{equation}
\frac{\partial}{\partial t}(\rho \boldsymbol{v})-\boldsymbol{v}\frac{\partial \rho}{\partial t}+\rho\frac{\partial (v_iv_j\boldsymbol{e_j})}{\partial x_i}-\rho v_j\frac{\partial v_i}{\partial x_i}\boldsymbol{e_j}=-\nabla p-\rho\nabla \phi\tag{7}
\end{equation}
\begin{equation}
\frac{\partial}{\partial t}(\rho \boldsymbol{v})=\boldsymbol{v}\frac{\partial \rho}{\partial t}-\rho\frac{\partial (v_iv_j\boldsymbol{e_j})}{\partial x_i}+\rho v_j\frac{\partial v_i}{\partial x_i}\boldsymbol{e_j}-\nabla p-\rho\nabla \phi\\
=-\rho\frac{\partial (v_iv_j\boldsymbol{e_j})}{\partial x_i}+ v_i(\frac{\partial \rho}{\partial t}+\rho\frac{\partial v_i}{\partial x_i})\boldsymbol{e_j}-\nabla p-\rho\nabla \phi\tag{8}
\end{equation}
ここで(1)より(8)の右辺第２項は0になるので
\begin{equation}
\frac{\partial}{\partial t}(\rho \boldsymbol{v})==-\rho\frac{\partial (v_iv_j\boldsymbol{e_j})}{\partial x_i}-\nabla p-\rho\nabla \phi\tag{9}
\end{equation}
$\rho$の空間的な変化は小さいと仮定すれば
\begin{equation}
\frac{\partial}{\partial t}(\rho \boldsymbol{v})=-\nabla\cdot(\rho \boldsymbol{v}\boldsymbol{v})-\nabla p-\rho\nabla \phi\tag{9}
\end{equation}