牛顿流体耗散方程

Dissipation Function for Newtonian Fluids

1.直角坐标系($ x, y, z $)

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

直角坐标系Cartesian coordinates ($\textit { x,y,z }$):

NO.

$\pmb \Phi_v = 2 \left [\left(\dfrac {\partial v_x}{\partial x}\right)^2 + \left(\dfrac {\partial v_y}{\partial y}\right)^2 +\left(\dfrac {\partial v_z}{\partial z}\right)^2 \right] + \left [\dfrac {\partial v_y}{\partial x} + \dfrac {\partial v_x}{\partial y} \right]^2 + \left [\dfrac {\partial v_z}{\partial y} + \dfrac {\partial v_y}{\partial z} \right]^2 + \left [\dfrac {\partial v_x}{\partial z} + \dfrac {\partial v_z}{\partial x} \right]^2 - \dfrac {2}{3} \left[\dfrac {\partial v_x}{\partial x} + \dfrac {\partial v_y}{\partial y} ++ \dfrac {\partial v_z}{\partial z} \right]^2$

1-1

2.圆柱坐标系（$r,\theta, z$)

圆柱坐标系Cylindrical coordinates coordinates ($\textit {r, $\theta$, z }$):	NO.
$\pmb \Phi_v = 2 \left [\left(\dfrac {\partial v_r}{\partial r}\right)^2 + \left(\dfrac {1}{r}\dfrac {\partial v_\theta}{\partial \theta}+\dfrac {v_r}{r}\right)^2 +\left(\dfrac {\partial v_z}{\partial z}\right)^2 \right] + \left [r\dfrac {\partial }{\partial r} \left (\dfrac {v_\theta}{r}\right) + \dfrac {1}{r} \dfrac {\partial v_r}{\partial \theta} \right]^2 + \left [\dfrac {1}{r}\dfrac {\partial v_z}{\partial \theta} + \dfrac {\partial v_\theta}{\partial z} \right]^2 + \left [\dfrac {\partial v_r}{\partial z} + \dfrac {\partial v_z}{\partial r} \right]^2 - \dfrac {2}{3} \left[\dfrac {1}{r} \dfrac {\partial }{\partial r} \left(r v_r\right) + \dfrac {1}{r} \dfrac {\partial v_\theta}{\partial \theta} + \dfrac {\partial v_z}{\partial z} \right]^2$	2-1

圆柱坐标系Cylindrical coordinates coordinates ($\textit {r, $\theta$, z }$):

NO.

$\pmb \Phi_v = 2 \left [\left(\dfrac {\partial v_r}{\partial r}\right)^2 + \left(\dfrac {1}{r}\dfrac {\partial v_\theta}{\partial \theta}+\dfrac {v_r}{r}\right)^2 +\left(\dfrac {\partial v_z}{\partial z}\right)^2 \right] + \left [r\dfrac {\partial }{\partial r} \left (\dfrac {v_\theta}{r}\right) + \dfrac {1}{r} \dfrac {\partial v_r}{\partial \theta} \right]^2 + \left [\dfrac {1}{r}\dfrac {\partial v_z}{\partial \theta} + \dfrac {\partial v_\theta}{\partial z} \right]^2 + \left [\dfrac {\partial v_r}{\partial z} + \dfrac {\partial v_z}{\partial r} \right]^2 - \dfrac {2}{3} \left[\dfrac {1}{r} \dfrac {\partial }{\partial r} \left(r v_r\right) + \dfrac {1}{r} \dfrac {\partial v_\theta}{\partial \theta} + \dfrac {\partial v_z}{\partial z} \right]^2$

2-1

3.球坐标系（$r, \theta, \phi $)

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

球坐标系Spherical coordinates（$\textit {r, $\theta$, $\phi$ }$):

NO.

$\pmb \Phi_v = 2 \left [\left(\dfrac {\partial v_r}{\partial r}\right)^2 + \left(\dfrac {1}{r}\dfrac {\partial v_\theta}{\partial \theta}+\dfrac {v_r}{r}\right)^2 + \left (\dfrac {1}{r sin\theta} \dfrac {\partial v_\phi}{\partial \phi}+\dfrac {v_r+v_\theta cot \theta}{r}\right)^2 \right] + \left [r\dfrac {\partial }{\partial r} \left (\dfrac {v_\theta}{r}\right) + \dfrac {1}{r} \dfrac {\partial v_r}{\partial \theta} \right]^2 + \left [\dfrac {sin\theta}{r}\dfrac {\partial}{\partial \theta} \left (\dfrac {v_\phi}{sin \theta}\right) + \dfrac {1}{rsin\theta}\dfrac {\partial v_\theta}{\partial \phi} \right]^2 + \left [\dfrac {1}{rsin\theta} \dfrac {\partial v_r}{\partial \phi} + r \dfrac {\partial}{\partial r} \left(\dfrac {v_\phi}{r}\right) \right]^2 - \dfrac {2}{3} \left[\dfrac {1}{r^2} \dfrac {\partial }{\partial r} \left(r^2 v_r\right) + \dfrac {1}{r sin \theta} \dfrac {\partial}{\partial \theta} (v_\theta sin \theta) + \dfrac {1}{r sin \theta} \dfrac {\partial v_\phi}{\partial \phi} \right]^2$

3-1

参考文献

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