運動方程式の色々な書き方

流体力学

雨降ってて出かけようと思ったけど、出かける気が萎えたので運動方程式でも書いてみる。
運動方程式と質量保存のベクトル表記は
  \large \frac{\partial\vec{v}}{\partial t}+(\vec{v}\cdot\text{grad})\vec{v}=-\frac{1}{\rho}\text{grad}p+\nu\triangle\vec{v}
  \large \text{div}\vec{v}=0
テンソル表記は、
  \large \frac{\partial v_i}{\partial t}+v_j\frac{\partial v_i}{\partial x_j}=-\frac{\partial p}{\partial x_i}+\nu\frac{\partial}{\partial x_j}\frac{\partial v_i}{\partial x_j}
  \large \frac{\partial v_i}{\partial x_i}=0
で、これを書き下すと、
  \large \left(\array{\frac{\partial v_1}{\partial t}+v_1\frac{\partial v_1}{\partial x_1}+v_2\frac{\partial v_1}{\partial x_2}+v_3\frac{\partial v_1}{\partial x_3}\\ \frac{\partial v_2}{\partial t}+v_1\frac{\partial v_2}{\partial x_1}+v_2\frac{\partial v_2}{\partial x_2}+v_3\frac{\partial v_2}{\partial x_3}\\ \frac{\partial v_3}{\partial t}+v_1\frac{\partial v_3}{\partial x_1}+v_2\frac{\partial v_3}{\partial x_2}+v_3\frac{\partial v_3}{\partial x_3}}\right)=\left(\array{-\frac{1}{\rho}\frac{\partial p}{\partial x_1}+\nu\left(\frac{\partial^2}{\partial x_1^2}+\frac{\partial^2}{\partial x_2^2}+\frac{\partial^2}{\partial x_3^2}\right)v_1\\ -\frac{1}{\rho}\frac{\partial p}{\partial x_2}+\nu\left(\frac{\partial^2}{\partial x_1^2}+\frac{\partial^2}{\partial x_2^2}+\frac{\partial^2}{\partial x_3^2}\right)v_2\\ -\frac{1}{\rho}\frac{\partial p}{\partial x_3}+\nu\left(\frac{\partial^2}{\partial x_1^2}+\frac{\partial^2}{\partial x_2^2}+\frac{\partial^2}{\partial x_3^2}\right)v_3}\right)
  \large \frac{\partial v_1}{\partial x_1}+\frac{\partial v_2}{\partial x_2}+\frac{\partial v_3}{\partial x_3}=0
ですね。レポート用なので解像度高めです。