牛顿黏度定律

Newton’s Law of Viscosity

先定义矢量 ττ τ τ \pmb{\tau }

ττ=−μ(∇vv+(∇vv)†)+(23μ−κ)(∇⋅vv)δ τ τ = − μ ( ∇ v v + ( ∇ v v ) † ) + ( 2 3 μ − κ ) ( ∇ ⋅ v v ) δ

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

τyx=−μdvxdy τ y x = − μ d v x d y

\tau_{yx}=-\mu \frac {d v_x}{dy}
其中
τ τ \tau是流体所受的剪应力 [Pa] [ P a ] [Pa]
μ μ \mu 是流体的黏度 [Pa⋅s] [ P a · s ] [Pa·s]
dvxdy d v x d y \frac {dv_x}{dy} 是 x x x方向上速度的分量在y" role="presentation" style="position: relative;">yyy方向上的梯度 [s−1] [ s − 1 ] [s^{−1}]

1.直角坐标系( x,y,z x , y , z x, y, z )

直角坐标系Cartesian coordinates ( x,y,z x,y,z \textit { x,y,z }):	NO.
τxx=−μ[2∂vx∂x]+(23μ−κ)(∇⋅vv) τ x x = − μ [ 2 ∂ v x ∂ x ] + ( 2 3 μ − κ ) ( ∇ ⋅ v v ) \tau_{xx}=-\mu[2\dfrac{\partial v_x}{\partial x}]+(\dfrac{2}{3}\mu-\kappa)(\nabla\cdot \pmb v)	1-1
τyy=−μ[2∂vy∂y]+(23μ−κ)(∇⋅vv) τ y y = − μ [ 2 ∂ v y ∂ y ] + ( 2 3 μ − κ ) ( ∇ ⋅ v v ) \tau_{yy}=-\mu[2\dfrac{\partial v_y}{\partial y}]+(\dfrac{2}{3}\mu-\kappa)(\nabla\cdot \pmb v)	1-2
τzz=−μ[2∂vz∂z]+(23μ−κ)(∇⋅vv) τ z z = − μ [ 2 ∂ v z ∂ z ] + ( 2 3 μ − κ ) ( ∇ ⋅ v v ) \tau_{zz}=-\mu[2\dfrac{\partial v_z}{\partial z}]+(\dfrac{2}{3}\mu-\kappa)(\nabla\cdot \pmb v)	1-3
τxy=τyx=−μ[∂vy∂x+∂vx∂y] τ x y = τ y x = − μ [ ∂ v y ∂ x + ∂ v x ∂ y ] \tau_{xy}=\tau_{yx}=-\mu[\dfrac{\partial v_y}{\partial x}+\dfrac{\partial v_x}{\partial y}]	1-4
τyz=τzy=−μ[∂vz∂y+∂vy∂z] τ y z = τ z y = − μ [ ∂ v z ∂ y + ∂ v y ∂ z ] \tau_{yz}=\tau_{zy}=-\mu[\dfrac{\partial v_z}{\partial y}+\dfrac{\partial v_y}{\partial z}]	1-5
τzx=τxz=−μ[∂vx∂z+∂vz∂x] τ z x = τ x z = − μ [ ∂ v x ∂ z + ∂ v z ∂ x ] \tau_{zx}=\tau_{xz}=-\mu[\dfrac{\partial v_x}{\partial z}+\dfrac{\partial v_z}{\partial x}]	1-6

其中

∇⋅vv=∂vx∂x+∂vy∂y+∂vz∂z ∇ ⋅ v v = ∂ v x ∂ x + ∂ v y ∂ y + ∂ v z ∂ 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,θ,z r , θ , z r,\theta, z)

圆柱坐标系Cylindrical coordinates coordinates ( r, θ, z r, θ , z \textit {r, $\theta$, z }):	NO.
τrr=−μ[2∂vr∂r]+(23μ−κ)(∇⋅vv) τ r r = − μ [ 2 ∂ v r ∂ r ] + ( 2 3 μ − κ ) ( ∇ ⋅ v v ) \tau_{rr}=-\mu[2\dfrac{\partial v_r}{\partial r}]+(\dfrac{2}{3}\mu-\kappa)(\nabla\cdot \pmb v)	2-1
τθθ=−μ[2(1r∂vθ∂θ+vrr)]+(23μ−κ)(∇⋅vv) τ θ θ = − μ [ 2 ( 1 r ∂ v θ ∂ θ + v r r ) ] + ( 2 3 μ − κ ) ( ∇ ⋅ v v ) \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
τzz=−μ[2∂vz∂z]+(23μ−κ)(∇⋅vv) τ z z = − μ [ 2 ∂ v z ∂ z ] + ( 2 3 μ − κ ) ( ∇ ⋅ v v ) \tau_{zz}=-\mu[2\dfrac{\partial v_z}{\partial z}]+(\dfrac{2}{3}\mu-\kappa)(\nabla\cdot \pmb v)	2-3
τrθ=τθr=−μ[r∂∂r(vθr)+1r∂vr∂θ] τ r θ = τ θ r = − μ [ r ∂ ∂ r ( v θ r ) + 1 r ∂ v r ∂ θ ] \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
τθz=τzθ=−μ[1r∂vz∂θ+∂vθ∂z] τ θ z = τ z θ = − μ [ 1 r ∂ v z ∂ θ + ∂ v θ ∂ z ] \tau_{\theta z}=\tau_{z \theta}=-\mu[\dfrac {1}{r} \dfrac{\partial v_z}{\partial \theta}+\dfrac {\partial v_\theta}{\partial z}]	2-5
τzr=τrz=−μ[∂vr∂z+∂vz∂r] τ z r = τ r z = − μ [ ∂ v r ∂ z + ∂ v z ∂ r ] \tau_{z r}=\tau_{r z}=-\mu[\dfrac{\partial v_r}{\partial z} + \dfrac{\partial v_z}{\partial r}]	2-6

其中

∇⋅vv=1r∂∂r(rvr)+1r∂vθ∂θ+∂vz∂z ∇ ⋅ v v = 1 r ∂ ∂ r ( r v r ) + 1 r ∂ v θ ∂ θ + ∂ v z ∂ 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,θ,ϕ r , θ , ϕ r, \theta, \phi )

球坐标系Spherical coordinates（ r, θ, ϕ r, θ , ϕ \textit {r, $\theta$, $\phi$ }):	NO.
τrr=−μ[2∂vr∂r]+(23μ−κ)(∇⋅vv) τ r r = − μ [ 2 ∂ v r ∂ r ] + ( 2 3 μ − κ ) ( ∇ ⋅ v v ) \tau_{rr}=-\mu[2\dfrac{\partial v_r}{\partial r}]+(\dfrac{2}{3}\mu-\kappa)(\nabla\cdot \pmb v)	3-1
τθθ=−μ[2(1r∂vθ∂θ+vrr)]+(23μ−κ)(∇⋅vv) τ θ θ = − μ [ 2 ( 1 r ∂ v θ ∂ θ + v r r ) ] + ( 2 3 μ − κ ) ( ∇ ⋅ v v ) \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
τzz=−μ[2(1rsinθ∂vϕ∂ϕ+vr+vθcotθr)]+(23μ−κ)(∇⋅vv) τ z z = − μ [ 2 ( 1 r s i n θ ∂ v ϕ ∂ ϕ + v r + v θ c o t θ r ) ] + ( 2 3 μ − κ ) ( ∇ ⋅ v v ) \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
τrθ=τθr=−μ[r∂∂r(vθr)+1r∂vr∂θ] τ r θ = τ θ r = − μ [ r ∂ ∂ r ( v θ r ) + 1 r ∂ v r ∂ θ ] \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
τθϕ=τϕθ=−μ[sinθr∂∂θ(vϕsinθ)+1rsinθ∂vθ∂ϕ] τ θ ϕ = τ ϕ θ = − μ [ s i n θ r ∂ ∂ θ ( v ϕ s i n θ ) + 1 r s i n θ ∂ v θ ∂ ϕ ] \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
τϕr=τrϕ=−μ[1rsinθ∂vr∂ϕ+r∂∂r(vϕr)] τ ϕ r = τ r ϕ = − μ [ 1 r s i n θ ∂ v r ∂ ϕ + r ∂ ∂ r ( v ϕ r ) ] \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

∇⋅vv=1r2∂∂r(r2vr)+1rsinθ∂∂θ(vθsinθ)+1rsinθ∂vθ∂ϕ ∇ ⋅ v v = 1 r 2 ∂ ∂ r ( r 2 v r ) + 1 r s i n θ ∂ ∂ θ ( v θ s i n θ ) + 1 r s i n θ ∂ v θ ∂ 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}

参考文献

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