流体动量控制方程

The Equation of Motion in terms of τ τ \tau

控制方程通式：

ρDvvDt=−∇p−∇⋅ττ+ρgg ρ D v v D t = − ∇ p − ∇ ⋅ τ τ + ρ g g

\rho \dfrac {D \pmb v}{D t}=- \nabla p-\nabla \cdot \pmb \tau+\rho \pmb g

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.
ρ(∂vx∂t+vx∂vx∂x+vy∂vx∂y+vz∂vx∂z)=−∂p∂x−[∂∂xτxx+∂∂yτyx+∂∂zτzx]+ρgx ρ ( ∂ v x ∂ t + v x ∂ v x ∂ x + v y ∂ v x ∂ y + v z ∂ v x ∂ z ) = − ∂ p ∂ x − [ ∂ ∂ x τ x x + ∂ ∂ y τ y x + ∂ ∂ z τ z x ] + ρ g x \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
ρ(∂vy∂t+vx∂vy∂x+vy∂vy∂y+vz∂vy∂z)=−∂p∂y−[∂∂xτxy+∂∂yτyy+∂∂zτzy]+ρgy ρ ( ∂ v y ∂ t + v x ∂ v y ∂ x + v y ∂ v y ∂ y + v z ∂ v y ∂ z ) = − ∂ p ∂ y − [ ∂ ∂ x τ x y + ∂ ∂ y τ y y + ∂ ∂ z τ z y ] + ρ g y \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
ρ(∂vz∂t+vx∂vz∂x+vy∂vz∂y+vz∂vz∂z)=−∂p∂z−[∂∂xτxz+∂∂yτyz+∂∂zτzz]+ρgz ρ ( ∂ v z ∂ t + v x ∂ v z ∂ x + v y ∂ v z ∂ y + v z ∂ v z ∂ z ) = − ∂ p ∂ z − [ ∂ ∂ x τ x z + ∂ ∂ y τ y z + ∂ ∂ z τ z z ] + ρ g z \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,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.
ρ(∂vr∂t+vr∂vr∂r+vθr∂vr∂θ+vz∂vr∂z−v2θr)=−∂p∂r−[1r∂∂r(rτrr)+1r∂∂θτθr+∂∂zτzr−τθθr]+ρgr ρ ( ∂ v r ∂ t + v r ∂ v r ∂ r + v θ r ∂ v r ∂ θ + v z ∂ v r ∂ z − v θ 2 r ) = − ∂ p ∂ r − [ 1 r ∂ ∂ r ( r τ r r ) + 1 r ∂ ∂ θ τ θ r + ∂ ∂ z τ z r − τ θ θ r ] + ρ g r \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
ρ(∂vθ∂t+vr∂vθ∂r+vθr∂vθ∂θ+vz∂vθ∂z+vrvθr)=−1r∂p∂θ−[1r2∂∂r(r2τrθ)+1r∂∂θτθθ+∂∂zτzθ+τθr−τrθr]+ρgθ ρ ( ∂ v θ ∂ t + v r ∂ v θ ∂ r + v θ r ∂ v θ ∂ θ + v z ∂ v θ ∂ z + v r v θ r ) = − 1 r ∂ p ∂ θ − [ 1 r 2 ∂ ∂ r ( r 2 τ r θ ) + 1 r ∂ ∂ θ τ θ θ + ∂ ∂ z τ z θ + τ θ r − τ r θ r ] + ρ g θ \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
ρ(∂vz∂t+vr∂vz∂r+vθr∂vz∂θ+vz∂vz∂z)=−∂p∂z−[1r∂∂r(rτzz)+1r∂∂θτθz+∂∂zτzz]+ρgz ρ ( ∂ v z ∂ t + v r ∂ v z ∂ r + v θ r ∂ v z ∂ θ + v z ∂ v z ∂ z ) = − ∂ p ∂ z − [ 1 r ∂ ∂ r ( r τ z z ) + 1 r ∂ ∂ θ τ θ z + ∂ ∂ z τ z z ] + ρ g z \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,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.
ρ(∂vr∂t+vr∂vr∂r+vθr∂vr∂θ+vϕrsinθ∂vr∂ϕ−v2θ+v2ϕr)=−∂p∂r−[1r2∂∂r(r2τrr)+1rsinθ∂∂θ(τθrsinθ)+1rsinθ∂∂ϕτϕr−τθθ+τϕϕr]+ρgr ρ ( ∂ v r ∂ t + v r ∂ v r ∂ r + v θ r ∂ v r ∂ θ + v ϕ r s i n θ ∂ v r ∂ ϕ − v θ 2 + v ϕ 2 r ) = − ∂ p ∂ r − [ 1 r 2 ∂ ∂ r ( r 2 τ r r ) + 1 r s i n θ ∂ ∂ θ ( τ θ r s i n θ ) + 1 r s i n θ ∂ ∂ ϕ τ ϕ r − τ θ θ + τ ϕ ϕ r ] + ρ g r \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
ρ(∂vθ∂t+vr∂vθ∂r+vθr∂vθ∂θ+vϕrsinθ∂vθ∂ϕ+vrvθ−v2ϕcotθr)=−1r∂p∂θ−[1r3∂∂r(r3τrθ)+1rsinθ∂∂θ(τθθsinθ)+1rsinθ∂∂ϕτϕθ+(τθr−τrθ)−τϕϕcotθr]+ρgθ ρ ( ∂ v θ ∂ t + v r ∂ v θ ∂ r + v θ r ∂ v θ ∂ θ + v ϕ r s i n θ ∂ v θ ∂ ϕ + v r v θ − v ϕ 2 c o t θ r ) = − 1 r ∂ p ∂ θ − [ 1 r 3 ∂ ∂ r ( r 3 τ r θ ) + 1 r s i n θ ∂ ∂ θ ( τ θ θ s i n θ ) + 1 r s i n θ ∂ ∂ ϕ τ ϕ θ + ( τ θ r − τ r θ ) − τ ϕ ϕ c o t θ r ] + ρ g θ \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
ρ(∂vϕ∂t+vr∂vϕ∂r+vθr∂vϕ∂θ+vϕrsinθ∂vϕ∂ϕ+vϕvr+vθvϕcotθr)=−1rsinθ∂p∂ϕ−[1r3∂∂r(r3τrϕ)+1rsinθ∂∂θ(τθϕsinθ)+1rsinθ∂∂ϕτϕϕ+(τϕr−τrϕ)+τϕθcotθr]+ρgϕ ρ ( ∂ v ϕ ∂ t + v r ∂ v ϕ ∂ r + v θ r ∂ v ϕ ∂ θ + v ϕ r s i n θ ∂ v ϕ ∂ ϕ + v ϕ v r + v θ v ϕ c o t θ r ) = − 1 r s i n θ ∂ p ∂ ϕ − [ 1 r 3 ∂ ∂ r ( r 3 τ r ϕ ) + 1 r s i n θ ∂ ∂ θ ( τ θ ϕ s i n θ ) + 1 r s i n θ ∂ ∂ ϕ τ ϕ ϕ + ( τ ϕ r − τ r ϕ ) + τ ϕ θ c o t θ r ] + ρ g ϕ \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 ττ τ τ \pmb \tau 具有对称性，那么 τrθ−τθr=0 τ r θ − τ θ r = 0 \tau_{r \theta}-\tau_{\theta r}=0

参考文献

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

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