不同文献中动量方程的表达形式也多种多样，现总结如下：

1. 守恒形式：
   $$\frac{\partial (\rho \mathbf{U})}{\partial t}=-\nabla \cdot (\rho \mathbf{U}\mathbf{U})-\nabla \cdot \tau -\nabla p+\rho \mathbf{g}$$

2. 引入切应力张量的守恒形式：
   $$\frac{\partial (\rho \mathbf{U})}{\partial t}=-\nabla \cdot (\rho \mathbf{U}\mathbf{U})-\nabla \cdot \left(2\mu \mathbf{D}-\frac{2}{3}\mu (\nabla \cdot \mathbf{U})\mathbf{I}\right)-\nabla p+\rho \mathbf{g}$$

3. 引入secondary viscosity 的守恒形式( $\lambda =-\frac{2}{3}\mu$ )：
   $$\frac{\partial (\rho \mathbf{U})}{\partial t}=-\nabla \cdot (\rho \mathbf{U}\mathbf{U})-\nabla \cdot \left(2\mu \mathbf{D}+\lambda (\nabla \cdot \mathbf{U})\mathbf{I}\right)-\nabla p+\rho \mathbf{g}$$

4. 包含体积粘度的守恒形式：
   $$\frac{\partial (\rho \mathbf{U})}{\partial t}=-\nabla \cdot (\rho \mathbf{U}\mathbf{U})-\nabla \cdot \left(2\mu \mathbf{D}\left[-\frac{2}{3}\mu +\kappa \right](\nabla \cdot \mathbf{U})\mathbf{I}\right)-\nabla p+\rho \mathbf{g}$$

5. 带有迹tr操作符的守恒形式：
   $$\frac{\partial (\rho \mathbf{U})}{\partial t}=-\nabla \cdot (\rho \mathbf{U}\mathbf{U})-\nabla \cdot \left(2\mu \mathbf{D}\left[-\frac{2}{3}\mu +\kappa \right]tr(\mathbf{D})\mathbf{I}\right)-\nabla p+\rho \mathbf{g}$$

$$tr\left(\mathbf{D}\right)=tr\left(\frac{1}{2}\left[\nabla \mathbf{U}+\nabla {\mathbf{U}}^T\right]\right)=\frac{1}{2}\left(2\frac{\partial u_x}{\partial x}+2\frac{\partial u_y}{\partial y}+2\frac{\partial u_z}{\partial z}\right)=\nabla \cdot \mathbf{U}$$

6. 带有柯西（Cauchy）应力的守恒形式：
   $$\frac{\partial (\rho \mathbf{U})}{\partial t}=-\nabla \cdot (\rho \mathbf{U}\mathbf{U})-\nabla \cdot \sigma +\rho \mathbf{g}$$
7. 非守恒形式：
   $$\rho \frac{D\mathbf{U}}{Dt}=-\nabla \cdot \sigma +\rho \mathbf{g}$$

关于物质导数 $\frac{D}{Dt}$ 运算规则可参考热力学数学运算规则。