牛顿黏度定律

Newton’s Law of Viscosity

先定义矢量$\pmb{\tau }$

$\pmb \tau=-\mu(\nabla \pmb v+(\nabla \pmb v)^{\dagger} ) +(\frac{2}{3}\mu-\kappa)(\nabla\cdot \pmb v)\delta$

$\tau_{yx}$物理的意义：在垂直于y方向的单位面积的面上所受到x方向上的力，可以表达为

$\tau_{yx}=-\mu \frac {d v_x}{dy}$

其中
$\tau$是流体所受的剪应力$[Pa]$
$ \mu $是流体的黏度 $[Pa·s]$
$ \frac {dv_x}{dy} $是$x$方向上速度的分量在$y$方向上的梯度$ [s^{−1}]$

1.直角坐标系($ x, y, z $)

直角坐标系Cartesian coordinates ($\textit { x,y,z }$):	NO.
$\tau_{xx}=-\mu[2\dfrac{\partial v_x}{\partial x}]+(\dfrac{2}{3}\mu-\kappa)(\nabla\cdot \pmb v)$	1-1
$\tau_{yy}=-\mu[2\dfrac{\partial v_y}{\partial y}]+(\dfrac{2}{3}\mu-\kappa)(\nabla\cdot \pmb v)$	1-2
$\tau_{zz}=-\mu[2\dfrac{\partial v_z}{\partial z}]+(\dfrac{2}{3}\mu-\kappa)(\nabla\cdot \pmb v)$	1-3
$\tau_{xy}=\tau_{yx}=-\mu[\dfrac{\partial v_y}{\partial x}+\dfrac{\partial v_x}{\partial y}]$	1-4
$\tau_{yz}=\tau_{zy}=-\mu[\dfrac{\partial v_z}{\partial y}+\dfrac{\partial v_y}{\partial z}]$	1-5
$\tau_{zx}=\tau_{xz}=-\mu[\dfrac{\partial v_x}{\partial z}+\dfrac{\partial v_z}{\partial x}]$	1-6

其中

$\nabla \cdot \pmb v=\frac {\partial v_x}{\partial x} + \frac {\partial v_y}{\partial y}+ \frac {\partial v_z}{\partial z}$

2.圆柱坐标系（$r,\theta, z$)

圆柱坐标系Cylindrical coordinates coordinates ($\textit {r, $\theta$, z }$):	NO.
$\tau_{rr}=-\mu[2\dfrac{\partial v_r}{\partial r}]+(\dfrac{2}{3}\mu-\kappa)(\nabla\cdot \pmb v)$	2-1
$\tau_{\theta \theta}=-\mu[2(\dfrac {1}{r}\frac{\partial v_\theta}{\partial \theta}+\frac {v_r}{r})]+(\dfrac{2}{3}\mu-\kappa)(\nabla\cdot \pmb v)$	2-2
$\tau_{zz}=-\mu[2\dfrac{\partial v_z}{\partial z}]+(\dfrac{2}{3}\mu-\kappa)(\nabla\cdot \pmb v)$	2-3
$\tau_{r \theta}=\tau_{\theta r}=-\mu[r \dfrac {\partial}{\partial r}(\dfrac {v_\theta}{r})+\dfrac {1}{r} \dfrac {\partial v_r}{\partial \theta}]$	2-4
$\tau_{\theta z}=\tau_{z \theta}=-\mu[\dfrac {1}{r} \dfrac{\partial v_z}{\partial \theta}+\dfrac {\partial v_\theta}{\partial z}]$	2-5
$\tau_{z r}=\tau_{r z}=-\mu[\dfrac{\partial v_r}{\partial z} + \dfrac{\partial v_z}{\partial r}]$	2-6

其中

$\nabla \cdot \pmb v=\frac {1}{r} \frac {\partial }{\partial r}(r v_r) +\frac {1}{r} \frac {\partial v_\theta}{\partial \theta}+ \frac {\partial v_z}{\partial z}$

3.球坐标系（$r, \theta, \phi $)

球坐标系Spherical coordinates（$\textit {r, $\theta$, $\phi$ }$):	NO.
$\tau_{rr}=-\mu[2\dfrac{\partial v_r}{\partial r}]+(\dfrac{2}{3}\mu-\kappa)(\nabla\cdot \pmb v)$	3-1
$\tau_{\theta \theta}=-\mu[2(\frac {1}{r}\dfrac{\partial v_\theta}{\partial \theta}+\dfrac {v_r}{r})]+(\dfrac{2}{3}\mu-\kappa)(\nabla\cdot \pmb v)$	3-2
$\tau_{zz}=-\mu[2(\dfrac{1}{r sin \theta} \dfrac {\partial v_\phi}{\partial \phi}+\dfrac {v_r+v_\theta cot \theta }{r})]+(\dfrac{2}{3}\mu-\kappa)(\nabla\cdot \pmb v)$	3-3
$\tau_{r \theta}=\tau_{\theta r}=-\mu[r \dfrac {\partial}{\partial r}(\dfrac {v_\theta}{r})+\dfrac {1}{r} \dfrac {\partial v_r}{\partial \theta}]$	3-4
$\tau_{\theta \phi}=\tau_{\phi \theta}=-\mu[\dfrac {sin \theta}{r} \dfrac{\partial }{\partial \theta}(\dfrac {v_\phi}{sin \theta})+\dfrac {1}{r sin \theta} \dfrac {\partial v_\theta}{\partial \phi}]$	3-5
$\tau_{\phi r}=\tau_{r \phi}=-\mu[\dfrac {1}{r sin \theta} \dfrac {\partial v_r}{\partial \phi}+r \dfrac {\partial}{\partial r}(\dfrac {v_\phi}{r})]$	3-6

其中

$\nabla \cdot \pmb v=\frac {1}{r^2} \dfrac {\partial }{\partial r}(r^2 v_r) +\dfrac {1}{r sin \theta} \dfrac {\partial }{\partial \theta} (v_\theta sin \theta)+ \dfrac {1}{r sin\theta }\dfrac{\partial v_\theta}{\partial \phi}$

参考文献

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