牛顿黏度定律【Newton's Law of Viscosity】

本文介绍了牛顿黏度定律，详细阐述了在直角坐标系、圆柱坐标系和球坐标系中，黏度张量ττττ的定义，并引用了经典教材《Transport Phenomena》作为参考。

牛顿黏度定律

Newton’s Law of Viscosity

先定义矢量 $\pmb{\tau }$

τ τ = - μ (\nabla v v + (\nabla v v) †) + (2 3 μ - κ) (\nabla \cdot v v) δ

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

τyxτyx $\tau_{yx}$ 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

τ y x = - μ d v x d y

$\tau_{yx}=-\mu \frac {d v_x}{dy}$
其中

ττ $\tau$ 是流体所受的剪应力

[Pa][Pa] $[Pa]$

μμ $\mu$ 是流体的黏度

[Pa⋅s][Pa·s] $[Pa·s]$

dvxdydvxdy $\frac {dv_x}{dy}$ 是

xx $x$ 方向上速度的分量在

y

$y$ 方向上的梯度

[s−1][s−1] $[s^{−1}]$

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 v v = \partial v x \partial x + \partial v y \partial y + \partial v z \partial z

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

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 v v = 1 r \partial \partial r (r v r) + 1 r \partial v θ \partial θ + \partial v z \partial z

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

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 v v = 1 r 2 \partial \partial r (r 2 v r) + 1 r s i n θ \partial \partial θ (v θ s i n θ) + 1 r s i n θ \partial v θ \partial ϕ

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