流体力学方程组

\begin{equation} \begin{aligned} & \frac{\partial \rho}{\partial t} +\nabla\cdot(\rho \boldsymbol{\mathbf{u}} )=0\\ & \rho \frac{\mathrm{d}{u_j}}{\mathrm{d}{t}} =- \frac{\partial p}{\partial x_j} +\rho g_j+\mu \frac{\partial^{2}{u_j}}{\partial{x_i}^{2}} + \left(\mu_\nu+\frac{1}{3}\mu \right) \frac{\partial}{\partial{x_j}} \frac{\partial u_m}{\partial x_m} \\ &\rho \frac{\mathrm{d}{}}{\mathrm{d}{t}} \left(\frac{1}{2}| \boldsymbol{\mathbf{u}} |^2 \right) =\rho g_iu_i+u_j \frac{\partial T_{ij}}{\partial x_i} =\rho g_iu_i+u_j \left(- \frac{\partial p}{\partial x_j} + \frac{\partial \tau_{ij}}{\partial x_i} \right) \\ &\rho \frac{\mathrm{d}{e}}{\mathrm{d}{t}} =-p \frac{\partial u_m}{\partial x_m} +2\mu \left(S_{ij}-\frac{1}{3} \frac{\partial u_m}{\partial x_m} \delta_{ij} \right) ^2+\mu_\nu \left( \frac{\partial u_m}{\partial x_m} \right) ^2+ \frac{\partial}{\partial{x_i}} \left(k \frac{\partial T}{\partial x_i} \right) \\ &T_{ij}=-p\delta_{ij}+\tau_{ij}=-p\delta_{ij}+2\mu \left(S_{ij}-\frac{2}{3}S_{mm}\delta_{ij} \right) +\mu_\nu S_{mm} \delta_{ij}\\ &\rho=\rho(p,T)~. \end{aligned} \end{equation}