The Equation of Motion

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The Equation of Motion in terms of $\tau$

控制方程通式：

1.直角坐标系($ x, y, z $)

直角坐标系Cartesian coordinates ($\textit { x,y,z }$): NO.
$\rho \left (\dfrac{\partial v_x}{\partial t}+v_x \dfrac{\partial v_x}{\partial x}+v_y \dfrac{\partial v_x}{\partial y} + v_z \dfrac{\partial v_x}{\partial z}\right)=- \dfrac{\partial p}{\partial x}-\left [\dfrac {\partial}{\partial x}\tau_{xx}+\dfrac {\partial}{\partial y}\tau_{yx}+\dfrac {\partial}{\partial z}\tau_{zx}\right]+\rho g_x$ 1-1
$\rho \left (\dfrac{\partial v_y}{\partial t}+v_x \dfrac{\partial v_y}{\partial x}+v_y \dfrac{\partial v_y}{\partial y} + v_z \dfrac{\partial v_y}{\partial z}\right)=- \dfrac{\partial p}{\partial y}-\left [\dfrac {\partial}{\partial x}\tau_{xy}+\dfrac {\partial}{\partial y}\tau_{yy}+\dfrac {\partial}{\partial z}\tau_{zy}\right]+\rho g_y$ 1-2
$\rho \left (\dfrac{\partial v_z}{\partial t}+v_x \dfrac{\partial v_z}{\partial x}+v_y \dfrac{\partial v_z}{\partial y} + v_z \dfrac{\partial v_z}{\partial z}\right)=- \dfrac{\partial p}{\partial z}-\left [\dfrac {\partial}{\partial x}\tau_{xz}+\dfrac {\partial}{\partial y}\tau_{yz}+\dfrac {\partial}{\partial z}\tau_{zz}\right]+\rho g_z$ 1-3

2.圆柱坐标系（$r,\theta, z$)

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圆柱坐标系Cylindrical coordinates coordinates ($\textit {r, $\theta$, z }$): NO.
$\rho \left (\dfrac{\partial v_r}{\partial t}+v_r \dfrac{\partial v_r}{\partial r}+\dfrac {v_\theta}{r} \dfrac{\partial v_r}{\partial \theta} + v_z \dfrac{\partial v_r}{\partial z}-\dfrac {v_\theta^2}{r}\right)=- \dfrac{\partial p}{\partial r}-\left [\dfrac{1}{r}\dfrac {\partial}{\partial r} (r\tau_{rr})+\dfrac{1}{r}\dfrac {\partial}{\partial \theta}\tau_{\theta r}+\dfrac {\partial}{\partial z}\tau_{zr}-\dfrac{\tau_{\theta \theta}}{r}\right]+\rho g_r$ 2-1
$\rho \left (\dfrac{\partial v_\theta}{\partial t}+v_r \dfrac{\partial v_\theta}{\partial r}+\dfrac {v_\theta}{r} \dfrac{\partial v_\theta}{\partial \theta} + v_z \dfrac{\partial v_\theta}{\partial z}+\dfrac {v_r v_\theta}{r}\right)=- \dfrac{1}{r}\dfrac{\partial p}{\partial \theta}-\left [\dfrac{1}{r^2}\dfrac {\partial}{\partial r} (r^2\tau_{r\theta})+\dfrac{1}{r}\dfrac {\partial}{\partial \theta}\tau_{\theta \theta}+\dfrac {\partial}{\partial z}\tau_{z\theta}+\dfrac{\tau_{\theta r}-\tau_{r \theta}}{r}\right]+\rho g_\theta$ 2-2
$\rho \left (\dfrac{\partial v_z}{\partial t}+v_r \dfrac{\partial v_z}{\partial r}+\dfrac {v_\theta}{r} \dfrac{\partial v_z}{\partial \theta} + v_z \dfrac{\partial v_z}{\partial z}\right)=- \dfrac{\partial p}{\partial z}-\left [\dfrac{1}{r}\dfrac {\partial}{\partial r} (r\tau_{zz})+\dfrac{1}{r}\dfrac {\partial}{\partial \theta}\tau_{\theta z}+\dfrac {\partial}{\partial z}\tau_{zz}\right]+\rho g_z$ 2-3

3.球坐标系（$r, \theta, \phi $)

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球坐标系Spherical coordinates（$\textit {r, $\theta$, $\phi$ }$): NO.
$\rho \left (\dfrac{\partial v_r}{\partial t}+v_r \dfrac{\partial v_r}{\partial r}+\dfrac {v_\theta}{r} \dfrac{\partial v_r}{\partial \theta} + \dfrac{v_\phi}{r sin\theta} \dfrac{\partial v_r}{\partial \phi}-\dfrac {v_\theta^2+v_\phi^2}{r}\right)=- \dfrac{\partial p}{\partial r}-\left [\dfrac{1}{r^2}\dfrac {\partial}{\partial r} (r^2\tau_{rr})+\dfrac{1}{r sin\theta}\dfrac {\partial}{\partial \theta}(\tau_{\theta r}sin\theta)+\dfrac{1}{r sin\theta}\dfrac {\partial}{\partial \phi}\tau_{\phi r}-\dfrac{\tau_{\theta \theta}+\tau_{\phi \phi}}{r}\right]+\rho g_r$ 3-1
$\rho \left (\dfrac{\partial v_\theta}{\partial t}+v_r \dfrac{\partial v_\theta}{\partial r}+\dfrac {v_\theta}{r} \dfrac{\partial v_\theta}{\partial \theta} + \dfrac{v_\phi}{r sin\theta} \dfrac{\partial v_\theta}{\partial \phi} + \dfrac {v_r v_\theta - v_\phi^2 cot \theta}{r}\right)=- \dfrac {1}{r}\dfrac{\partial p}{\partial \theta}-\left [\dfrac{1}{r^3}\dfrac {\partial}{\partial r} (r^3\tau_{r\theta})+\dfrac{1}{r sin\theta}\dfrac {\partial}{\partial \theta}(\tau_{\theta \theta}sin\theta)+\dfrac{1}{r sin\theta}\dfrac {\partial}{\partial \phi}\tau_{\phi \theta} + \dfrac{(\tau_{\theta r}-\tau_{r \theta})-\tau_{\phi \phi}cot \theta }{r}\right]+\rho g_\theta$ 3-2
$\rho \left (\dfrac{\partial v_\phi}{\partial t}+v_r \dfrac{\partial v_\phi}{\partial r}+\dfrac {v_\theta}{r} \dfrac{\partial v_\phi}{\partial \theta} + \dfrac{v_\phi}{r sin\theta} \dfrac{\partial v_\phi}{\partial \phi} + \dfrac {v_\phi v_r + v_\theta v_\phi cot \theta}{r}\right)=- \dfrac {1}{r sin\theta}\dfrac{\partial p}{\partial \phi}-\left [\dfrac{1}{r^3}\dfrac {\partial}{\partial r} (r^3\tau_{r\phi})+\dfrac{1}{r sin\theta}\dfrac {\partial}{\partial \theta}(\tau_{\theta \phi}sin\theta)+\dfrac{1}{r sin\theta}\dfrac {\partial}{\partial \phi}\tau_{\phi \phi} + \dfrac{(\tau_{\phi r}-\tau_{r \phi})+\tau_{\phi \theta}cot \theta }{r}\right]+\rho g_\phi$ 3-3

注：如果$\pmb \tau $ 具有对称性，那么$\tau_{r \theta}-\tau_{\theta r}=0$

Equation of Motion for a Newtonian Fluid with Constant $\rho$ and $\mu$

控制方程通式：

1.直角坐标系($ x, y, z $)

直角坐标系Cartesian coordinates ($\textit { x,y,z }$): NO.
$\rho \left (\dfrac{\partial v_x}{\partial t}+v_x \dfrac{\partial v_x}{\partial x}+v_y \dfrac{\partial v_x}{\partial y} + v_z \dfrac{\partial v_x}{\partial z}\right)=- \dfrac{\partial p}{\partial x} + \mu\left [\dfrac {\partial^2 v_x}{\partial x}+\dfrac {\partial^2 v_x}{\partial y}+\dfrac {\partial^2 v_x}{\partial z}\right]+\rho g_x$ 1-1
$\rho \left (\dfrac{\partial v_y}{\partial t}+v_x \dfrac{\partial v_y}{\partial x}+v_y \dfrac{\partial v_y}{\partial y} + v_z \dfrac{\partial v_y}{\partial z}\right)=- \dfrac{\partial p}{\partial y} + \mu\left [\dfrac {\partial^2 v_y}{\partial x}+\dfrac {\partial^2 v_y}{\partial y}+\dfrac {\partial^2 v_y}{\partial z}\right]+\rho g_y$ 1-2
$\rho \left (\dfrac{\partial v_z}{\partial t}+v_x \dfrac{\partial v_z}{\partial x}+v_y \dfrac{\partial v_z}{\partial y} + v_z \dfrac{\partial v_z}{\partial z}\right)=- \dfrac{\partial p}{\partial z} + \mu\left [\dfrac {\partial^2 v_z}{\partial x}+\dfrac {\partial^2 v_z}{\partial y}+\dfrac {\partial^2 v_z}{\partial z}\right]+\rho g_z$ 1-3

2.圆柱坐标系（$r,\theta, z$)

​

圆柱坐标系Cylindrical coordinates coordinates ($\textit {r, $\theta$, z }$): NO.
$\rho \left (\dfrac{\partial v_r}{\partial t}+v_r \dfrac{\partial v_r}{\partial r}+\dfrac {v_\theta}{r} \dfrac{\partial v_r}{\partial \theta} + v_z \dfrac{\partial v_r}{\partial z}-\dfrac {v_\theta^2}{r}\right)=- \dfrac{\partial p}{\partial r} + \mu \left [\dfrac {\partial}{\partial r} (\dfrac{1}{r} \dfrac {\partial}{\partial r} (rv_r))+\dfrac{1}{r^2}\dfrac {\partial^2 v_r}{\partial \theta^2}+\dfrac {\partial^2 v_r}{\partial z^2}-\dfrac {2}{r^2}\dfrac{\partial v_\theta}{\partial \theta}\right]+\rho g_r$ 2-1
$\rho \left (\dfrac{\partial v_\theta}{\partial t}+v_r \dfrac{\partial v_\theta}{\partial r}+\dfrac {v_\theta}{r} \dfrac{\partial v_\theta}{\partial \theta} + v_z \dfrac{\partial v_\theta}{\partial z}+\dfrac {v_r v_\theta}{r}\right)=- \dfrac{1}{r}\dfrac{\partial p}{\partial \theta} + \mu \left [\dfrac {\partial}{\partial r} (\dfrac{1}{r} \dfrac {\partial}{\partial r} (rv_\theta))+\dfrac {1}{r^2} \dfrac {\partial^2 v_\theta}{\partial \theta^2}+\dfrac {\partial^2 v_\theta}{\partial z^2} + \dfrac {2}{r^2} \dfrac {\partial v_r}{\partial \theta}\right]+\rho g_\theta$ 2-2
$\rho \left (\dfrac{\partial v_z}{\partial t}+v_r \dfrac{\partial v_z}{\partial r}+\dfrac {v_\theta}{r} \dfrac{\partial v_z}{\partial \theta} + v_z \dfrac{\partial v_z}{\partial z}\right)=- \dfrac{\partial p}{\partial z} + \mu \left [\dfrac{1}{r}\dfrac {\partial}{\partial r} (r \dfrac {\partial v_z}{\partial r})+\dfrac{1}{r^2}\dfrac {\partial^2 v_z}{\partial \theta^2}+\dfrac {\partial^2 v_z}{\partial z^2}\right]+\rho g_z$ 2-3

3.球坐标系（$r, \theta, \phi $)

​

球坐标系Spherical coordinates（$\textit {r, $\theta$, $\phi$ }$): NO.
$\rho \left (\dfrac{\partial v_r}{\partial t}+v_r \dfrac{\partial v_r}{\partial r}+\dfrac {v_\theta}{r} \dfrac{\partial v_r}{\partial \theta} + \dfrac{v_\phi}{r sin\theta} \dfrac{\partial v_r}{\partial \phi}-\dfrac {v_\theta^2+v_\phi^2}{r}\right)=- \dfrac{\partial p}{\partial r} + \mu \left [\dfrac{1}{r^2}\dfrac {\partial^2}{\partial r^2} (r^2 v_r)+\dfrac{1}{r sin\theta}\dfrac {\partial}{\partial \theta}(sin\theta \dfrac {\partial v_r}{\partial \theta})+\dfrac{1}{r^2 sin^2\theta}\dfrac {\partial^2 v_r}{\partial \phi^2}\right]+\rho g_r$ 3-1
$\rho \left (\dfrac{\partial v_\theta}{\partial t}+v_r \dfrac{\partial v_\theta}{\partial r}+\dfrac {v_\theta}{r} \dfrac{\partial v_\theta}{\partial \theta} + \dfrac{v_\phi}{r sin\theta} \dfrac{\partial v_\theta}{\partial \phi} + \dfrac {v_r v_\theta - v_\phi^2 cot \theta}{r}\right)=- \dfrac {1}{r}\dfrac{\partial p}{\partial \theta} + \mu \left [\dfrac{1}{r^2}\dfrac {\partial}{\partial r} (r^2 \dfrac {\partial v_\theta}{\partial r})+\dfrac{1}{r^2}\dfrac {\partial}{\partial \theta} (\dfrac {1}{sin \theta} \dfrac {\partial}{\partial \theta}(v_\theta sin \theta))+ \dfrac{1}{r^2 sin^2\theta}\dfrac {\partial^2 v_\theta}{\partial \phi^2}+\dfrac {2}{r^2} \dfrac{\partial v_r}{\partial \theta} - \dfrac {2cot\theta}{r^2 sin \theta} \dfrac{\partial v_\phi}{\partial \phi}\right]+\rho g_\theta$ 3-2
$\rho \left (\dfrac{\partial v_\phi}{\partial t}+v_r \dfrac{\partial v_\phi}{\partial r}+\dfrac {v_\theta}{r} \dfrac{\partial v_\phi}{\partial \theta} + \dfrac{v_\phi}{r sin\theta} \dfrac{\partial v_\phi}{\partial \phi} + \dfrac {v_\phi v_r + v_\theta v_\phi cot \theta}{r}\right)=- \dfrac {1}{r sin\theta}\dfrac{\partial p}{\partial \phi} + \mu \left [\dfrac{1}{r^2}\dfrac {\partial}{\partial r} (r^2 \dfrac {\partial v_\phi}{\partial r})+\dfrac{1}{r^2}\dfrac {\partial}{\partial \theta} (\dfrac {1}{sin \theta} \dfrac {\partial}{\partial \theta}(v_\phi sin \theta))+ \dfrac{1}{r^2 sin^2\theta}\dfrac {\partial^2 v_\phi}{\partial \phi^2} + \dfrac {2}{r^2 sin \theta} \dfrac{\partial v_r}{\partial \phi} + \dfrac {2cot\theta}{r^2 sin \theta} \dfrac{\partial v_\theta}{\partial \phi}\right]+\rho g_\phi$ 3-3

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参考文献

  1. R. Byron Bird, Warren E. stewart, Edwin N. Lightfoot. Transport phenomena:Revised second edition John Wiely &Sons, Inc.
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