牛顿黏度定律【Newton's Law of Viscosity】

　

牛顿定律

牛顿黏度定律

Newton’s Law of Viscosity

先定义矢量 ττ" role="presentation" style="position: relative;">ττ$\tau \tau$

ττ=−μ(∇vv+(∇vv)†)+(23μ−κ)(∇⋅vv)δ" role="presentation">ττ=−μ(∇vv+(∇vv)†)+(23μ−κ)(∇⋅vv)δ$$\tau \tau =-\mu (\mathrm{\nabla }vv+(\mathrm{\nabla }vv{)}^{†})+(\frac{2}{3}\mu -\kappa )(\mathrm{\nabla }\cdot vv)\delta$$

τyx" role="presentation" style="position: relative;">τyx${\tau }_{yx}$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" role="presentation">τyx=−μdvxdy$${\tau }_{yx}=-\mu \frac{d{v}_x}{dy}$$

其中

τ" role="presentation" style="position: relative;">τ$\tau$是流体所受的剪应力[Pa]" role="presentation" style="position: relative;">[Pa]$[Pa]$

μ" role="presentation" style="position: relative;">μ$\mu$是流体的黏度 [Pa·s]" role="presentation" style="position: relative;">[Pa⋅s]$[Pa·s]$

dvxdy" role="presentation" style="position: relative;">dvxdy$\frac{d{v}_x}{dy}$是x" role="presentation" style="position: relative;">x$x$方向上速度的分量在y" role="presentation" style="position: relative;">y$y$方向上的梯度[s−1]" role="presentation" style="position: relative;">[s−1]$[s^{-1}]$

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1.直角坐标系(x,y,z" role="presentation" style="position: relative;">x,y,z$x,y,z$)

直角坐标系Cartesian coordinates ( x,y,z " role="presentation" style="position: relative;">x,y,z$\mathit{\text{x,y,z}}$):	NO.
τxx=−μ[2∂vx∂x]+(23μ−κ)(∇⋅vv)" role="presentation" style="position: relative;">τxx=−μ[2∂vx∂x]+(23μ−κ)(∇⋅vv)${\tau }_{xx}=-\mu [2{\displaystyle \frac{\mathrm{\partial }{v}_x}{\mathrm{\partial }x}}]+({\displaystyle \frac{2}{3}}\mu -\kappa )(\mathrm{\nabla }\cdot vv)$	1-1
τyy=−μ[2∂vy∂y]+(23μ−κ)(∇⋅vv)" role="presentation" style="position: relative;">τyy=−μ[2∂vy∂y]+(23μ−κ)(∇⋅vv)${\tau }_{yy}=-\mu [2{\displaystyle \frac{\mathrm{\partial }{v}_y}{\mathrm{\partial }y}}]+({\displaystyle \frac{2}{3}}\mu -\kappa )(\mathrm{\nabla }\cdot vv)$	1-2
τzz=−μ[2∂vz∂z]+(23μ−κ)(∇⋅vv)" role="presentation" style="position: relative;">τzz=−μ[2∂vz∂z]+(23μ−κ)(∇⋅vv)${\tau }_{zz}=-\mu [2{\displaystyle \frac{\mathrm{\partial }{v}_z}{\mathrm{\partial }z}}]+({\displaystyle \frac{2}{3}}\mu -\kappa )(\mathrm{\nabla }\cdot vv)$	1-3
τxy=τyx=−μ[∂vy∂x+∂vx∂y]" role="presentation" style="position: relative;">τxy=τyx=−μ[∂vy∂x+∂vx∂y]${\tau }_{xy}={\tau }_{yx}=-\mu [{\displaystyle \frac{\mathrm{\partial }{v}_y}{\mathrm{\partial }x}}+{\displaystyle \frac{\mathrm{\partial }{v}_x}{\mathrm{\partial }y}}]$	1-4
τyz=τzy=−μ[∂vz∂y+∂vy∂z]" role="presentation" style="position: relative;">τyz=τzy=−μ[∂vz∂y+∂vy∂z]${\tau }_{yz}={\tau }_{zy}=-\mu [{\displaystyle \frac{\mathrm{\partial }{v}_z}{\mathrm{\partial }y}}+{\displaystyle \frac{\mathrm{\partial }{v}_y}{\mathrm{\partial }z}}]$	1-5
τzx=τxz=−μ[∂vx∂z+∂vz∂x]" role="presentation" style="position: relative;">τzx=τxz=−μ[∂vx∂z+∂vz∂x]${\tau }_{zx}={\tau }_{xz}=-\mu [{\displaystyle \frac{\mathrm{\partial }{v}_x}{\mathrm{\partial }z}}+{\displaystyle \frac{\mathrm{\partial }{v}_z}{\mathrm{\partial }x}}]$	1-6

其中

∇⋅vv=∂vx∂x+∂vy∂y+∂vz∂z" role="presentation">∇⋅vv=∂vx∂x+∂vy∂y+∂vz∂z$$\mathrm{\nabla }\cdot vv=\frac{\mathrm{\partial }{v}_x}{\mathrm{\partial }x}+\frac{\mathrm{\partial }{v}_y}{\mathrm{\partial }y}+\frac{\mathrm{\partial }{v}_z}{\mathrm{\partial }z}$$

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2.圆柱坐标系（r,θ,z" role="presentation" style="position: relative;">r,θ,z$r,\theta ,z$)

圆柱坐标系Cylindrical coordinates coordinates (r, θ, z " role="presentation" style="position: relative;">r,θ, z$\mathit{\text{r,}}\theta \mathit{\text{, z}}$):	NO.
τrr=−μ[2∂vr∂r]+(23μ−κ)(∇⋅vv)" role="presentation" style="position: relative;">τrr=−μ[2∂vr∂r]+(23μ−κ)(∇⋅vv)${\tau }_{rr}=-\mu [2{\displaystyle \frac{\mathrm{\partial }{v}_r}{\mathrm{\partial }r}}]+({\displaystyle \frac{2}{3}}\mu -\kappa )(\mathrm{\nabla }\cdot vv)$	2-1
τθθ=−μ[2(1r∂vθ∂θ+vrr)]+(23μ−κ)(∇⋅vv)" role="presentation" style="position: relative;">τθθ=−μ[2(1r∂vθ∂θ+vrr)]+(23μ−κ)(∇⋅vv)${\tau }_{\theta \theta }=-\mu [2({\displaystyle \frac{1}{r}}\frac{\mathrm{\partial }{v}_{\theta }}{\mathrm{\partial }\theta }+\frac{v_r}{r})]+({\displaystyle \frac{2}{3}}\mu -\kappa )(\mathrm{\nabla }\cdot vv)$	2-2
τzz=−μ[2∂vz∂z]+(23μ−κ)(∇⋅vv)" role="presentation" style="position: relative;">τzz=−μ[2∂vz∂z]+(23μ−κ)(∇⋅vv)${\tau }_{zz}=-\mu [2{\displaystyle \frac{\mathrm{\partial }{v}_z}{\mathrm{\partial }z}}]+({\displaystyle \frac{2}{3}}\mu -\kappa )(\mathrm{\nabla }\cdot vv)$	2-3
τrθ=τθr=−μ[r∂∂r(vθr)+1r∂vr∂θ]" role="presentation" style="position: relative;">τrθ=τθr=−μ[r∂∂r(vθr)+1r∂vr∂θ]${\tau }_{r\theta }={\tau }_{\theta r}=-\mu [r{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }r}}({\displaystyle \frac{v_{\theta }}{r}})+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\mathrm{\partial }{v}_r}{\mathrm{\partial }\theta }}]$	2-4
τθz=τzθ=−μ[1r∂vz∂θ+∂vθ∂z]" role="presentation" style="position: relative;">τθz=τzθ=−μ[1r∂vz∂θ+∂vθ∂z]${\tau }_{\theta z}={\tau }_{z\theta }=-\mu [{\displaystyle \frac{1}{r}}{\displaystyle \frac{\mathrm{\partial }{v}_z}{\mathrm{\partial }\theta }}+{\displaystyle \frac{\mathrm{\partial }{v}_{\theta }}{\mathrm{\partial }z}}]$	2-5
τzr=τrz=−μ[∂vr∂z+∂vz∂r]" role="presentation" style="position: relative;">τzr=τrz=−μ[∂vr∂z+∂vz∂r]${\tau }_{zr}={\tau }_{rz}=-\mu [{\displaystyle \frac{\mathrm{\partial }{v}_r}{\mathrm{\partial }z}}+{\displaystyle \frac{\mathrm{\partial }{v}_z}{\mathrm{\partial }r}}]$	2-6

其中

∇⋅vv=1r∂∂r(rvr)+1r∂vθ∂θ+∂vz∂z" role="presentation">∇⋅vv=1r∂∂r(rvr)+1r∂vθ∂θ+∂vz∂z$$\mathrm{\nabla }\cdot vv=\frac{1}{r}\frac{\mathrm{\partial }}{\mathrm{\partial }r}(r{v}_r)+\frac{1}{r}\frac{\mathrm{\partial }{v}_{\theta }}{\mathrm{\partial }\theta }+\frac{\mathrm{\partial }{v}_z}{\mathrm{\partial }z}$$

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3.球坐标系（r,θ,ϕ" role="presentation" style="position: relative;">r,θ,ϕ$r,\theta ,\varphi$)

球坐标系Spherical coordinates（r, θ, ϕ " role="presentation" style="position: relative;">r,θ,ϕ$\mathit{\text{r,}}\theta \mathit{\text{,}}\varphi$):	NO.
τrr=−μ[2∂vr∂r]+(23μ−κ)(∇⋅vv)" role="presentation" style="position: relative;">τrr=−μ[2∂vr∂r]+(23μ−κ)(∇⋅vv)${\tau }_{rr}=-\mu [2{\displaystyle \frac{\mathrm{\partial }{v}_r}{\mathrm{\partial }r}}]+({\displaystyle \frac{2}{3}}\mu -\kappa )(\mathrm{\nabla }\cdot vv)$	3-1
τθθ=−μ[2(1r∂vθ∂θ+vrr)]+(23μ−κ)(∇⋅vv)" role="presentation" style="position: relative;">τθθ=−μ[2(1r∂vθ∂θ+vrr)]+(23μ−κ)(∇⋅vv)${\tau }_{\theta \theta }=-\mu [2(\frac{1}{r}{\displaystyle \frac{\mathrm{\partial }{v}_{\theta }}{\mathrm{\partial }\theta }}+{\displaystyle \frac{v_r}{r}})]+({\displaystyle \frac{2}{3}}\mu -\kappa )(\mathrm{\nabla }\cdot vv)$	3-2
τzz=−μ[2(1rsinθ∂vϕ∂ϕ+vr+vθcotθr)]+(23μ−κ)(∇⋅vv)" role="presentation" style="position: relative;">τzz=−μ[2(1rsinθ∂vϕ∂ϕ+vr+vθcotθr)]+(23μ−κ)(∇⋅vv)${\tau }_{zz}=-\mu [2({\displaystyle \frac{1}{rsin\theta }}{\displaystyle \frac{\mathrm{\partial }{v}_{\varphi }}{\mathrm{\partial }\varphi }}+{\displaystyle \frac{v_r+v_{\theta }cot\theta }{r}})]+({\displaystyle \frac{2}{3}}\mu -\kappa )(\mathrm{\nabla }\cdot vv)$	3-3
τrθ=τθr=−μ[r∂∂r(vθr)+1r∂vr∂θ]" role="presentation" style="position: relative;">τrθ=τθr=−μ[r∂∂r(vθr)+1r∂vr∂θ]${\tau }_{r\theta }={\tau }_{\theta r}=-\mu [r{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }r}}({\displaystyle \frac{v_{\theta }}{r}})+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\mathrm{\partial }{v}_r}{\mathrm{\partial }\theta }}]$	3-4
τθϕ=τϕθ=−μ[sinθr∂∂θ(vϕsinθ)+1rsinθ∂vθ∂ϕ]" role="presentation" style="position: relative;">τθϕ=τϕθ=−μ[sinθr∂∂θ(vϕsinθ)+1rsinθ∂vθ∂ϕ]${\tau }_{\theta \varphi }={\tau }_{\varphi \theta }=-\mu [{\displaystyle \frac{sin\theta }{r}}{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }\theta }}({\displaystyle \frac{v_{\varphi }}{sin\theta }})+{\displaystyle \frac{1}{rsin\theta }}{\displaystyle \frac{\mathrm{\partial }{v}_{\theta }}{\mathrm{\partial }\varphi }}]$	3-5
τϕr=τrϕ=−μ[1rsinθ∂vr∂ϕ+r∂∂r(vϕr)]" role="presentation" style="position: relative;">τϕr=τrϕ=−μ[1rsinθ∂vr∂ϕ+r∂∂r(vϕr)]${\tau }_{\varphi r}={\tau }_{r\varphi }=-\mu [{\displaystyle \frac{1}{rsin\theta }}{\displaystyle \frac{\mathrm{\partial }{v}_r}{\mathrm{\partial }\varphi }}+r{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }r}}({\displaystyle \frac{v_{\varphi }}{r}})]$	3-6

其中

∇⋅vv=1r2∂∂r(r2vr)+1rsinθ∂∂θ(vθsinθ)+1rsinθ∂vθ∂ϕ" role="presentation">∇⋅vv=1r2∂∂r(r2vr)+1rsinθ∂∂θ(vθsinθ)+1rsinθ∂vθ∂ϕ$$\mathrm{\nabla }\cdot vv=\frac{1}{r^2}{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }r}}(r^2{v}_r)+{\displaystyle \frac{1}{rsin\theta }}{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }\theta }}(v_{\theta }sin\theta )+{\displaystyle \frac{1}{rsin\theta }}{\displaystyle \frac{\mathrm{\partial }{v}_{\theta }}{\mathrm{\partial }\varphi }}$$

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参考文献

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

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