牛顿流体动量控制方程

流体动量控制方程

Equation of Motion for a Newtonian Fluid with Constant ρ ρ and μ μ

控制方程通式：

ρDvvDt=−∇p+μ∇2vv+ρgg ρ D v v D t = − ∇ p + μ ∇ 2 v v + ρ g g

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1.直角坐标系( x,y,z x , y , z )

直角坐标系Cartesian coordinates (  x,y,z   x,y,z  ):	NO.
ρ(∂vx∂t+vx∂vx∂x+vy∂vx∂y+vz∂vx∂z)=−∂p∂x+μ[∂2vx∂x+∂2vx∂y+∂2vx∂z]+ρgx ρ ( ∂ v x ∂ t + v x ∂ v x ∂ x + v y ∂ v x ∂ y + v z ∂ v x ∂ z ) = − ∂ p ∂ x + μ [ ∂ 2 v x ∂ x + ∂ 2 v x ∂ y + ∂ 2 v x ∂ z ] + ρ g x	1-1
ρ(∂vy∂t+vx∂vy∂x+vy∂vy∂y+vz∂vy∂z)=−∂p∂y+μ[∂2vy∂x+∂2vy∂y+∂2vy∂z]+ρgy ρ ( ∂ v y ∂ t + v x ∂ v y ∂ x + v y ∂ v y ∂ y + v z ∂ v y ∂ z ) = − ∂ p ∂ y + μ [ ∂ 2 v y ∂ x + ∂ 2 v y ∂ y + ∂ 2 v y ∂ z ] + ρ g y	1-2
ρ(∂vz∂t+vx∂vz∂x+vy∂vz∂y+vz∂vz∂z)=−∂p∂z+μ[∂2vz∂x+∂2vz∂y+∂2vz∂z]+ρgz ρ ( ∂ v z ∂ t + v x ∂ v z ∂ x + v y ∂ v z ∂ y + v z ∂ v z ∂ z ) = − ∂ p ∂ z + μ [ ∂ 2 v z ∂ x + ∂ 2 v z ∂ y + ∂ 2 v z ∂ z ] + ρ g z	1-3

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2.圆柱坐标系（ r,θ,z r , θ , z )

圆柱坐标系Cylindrical coordinates coordinates ( r, θ, z  r,  θ , z  ):	NO.
ρ(∂vr∂t+vr∂vr∂r+vθr∂vr∂θ+vz∂vr∂z−v2θr)=−∂p∂r+μ[∂∂r(1r∂∂r(rvr))+1r2∂2vr∂θ2+∂2vr∂z2−2r2∂vθ∂θ]+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 + μ [ ∂ ∂ r ( 1 r ∂ ∂ r ( r v r ) ) + 1 r 2 ∂ 2 v r ∂ θ 2 + ∂ 2 v r ∂ z 2 − 2 r 2 ∂ v θ ∂ θ ] + ρ g r	2-1
ρ(∂vθ∂t+vr∂vθ∂r+vθr∂vθ∂θ+vz∂vθ∂z+vrvθr)=−1r∂p∂θ+μ[∂∂r(1r∂∂r(rvθ))+1r2∂2vθ∂θ2+∂2vθ∂z2+2r2∂vr∂θ]+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 + μ [ ∂ ∂ r ( 1 r ∂ ∂ r ( r v θ ) ) + 1 r 2 ∂ 2 v θ ∂ θ 2 + ∂ 2 v θ ∂ z 2 + 2 r 2 ∂ v r ∂ θ ] + ρ g θ	2-2
ρ(∂vz∂t+vr∂vz∂r+vθr∂vz∂θ+vz∂vz∂z)=−∂p∂z+μ[1r∂∂r(r∂vz∂r)+1r2∂2vz∂θ2+∂2vz∂z2]+ρgz ρ ( ∂ v z ∂ t + v r ∂ v z ∂ r + v θ r ∂ v z ∂ θ + v z ∂ v z ∂ z ) = − ∂ p ∂ z + μ [ 1 r ∂ ∂ r ( r ∂ v z ∂ r ) + 1 r 2 ∂ 2 v z ∂ θ 2 + ∂ 2 v z ∂ z 2 ] + ρ g z	2-3

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3.球坐标系（ r,θ,ϕ r , θ , ϕ )

球坐标系Spherical coordinates（ r, θ, ϕ  r,  θ ,  ϕ   ):	NO.
ρ(∂vr∂t+vr∂vr∂r+vθr∂vr∂θ+vϕrsinθ∂vr∂ϕ−v2θ+v2ϕr)=−∂p∂r+μ[1r2∂2∂r2(r2vr)+1rsinθ∂∂θ(sinθ∂vr∂θ)+1r2sin2θ∂2vr∂ϕ2]+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 + μ [ 1 r 2 ∂ 2 ∂ r 2 ( r 2 v r ) + 1 r s i n θ ∂ ∂ θ ( s i n θ ∂ v r ∂ θ ) + 1 r 2 s i n 2 θ ∂ 2 v r ∂ ϕ 2 ] + ρ g r	3-1
ρ(∂vθ∂t+vr∂vθ∂r+vθr∂vθ∂θ+vϕrsinθ∂vθ∂ϕ+vrvθ−v2ϕcotθr)=−1r∂p∂θ+μ[1r2∂∂r(r2∂vθ∂r)+1r2∂∂θ(1sinθ∂∂θ(vθsinθ))+1r2sin2θ∂2vθ∂ϕ2+2r2∂vr∂θ−2cotθr2sinθ∂vϕ∂ϕ]+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 + μ [ 1 r 2 ∂ ∂ r ( r 2 ∂ v θ ∂ r ) + 1 r 2 ∂ ∂ θ ( 1 s i n θ ∂ ∂ θ ( v θ s i n θ ) ) + 1 r 2 s i n 2 θ ∂ 2 v θ ∂ ϕ 2 + 2 r 2 ∂ v r ∂ θ − 2 c o t θ r 2 s i n θ ∂ v ϕ ∂ ϕ ] + ρ g θ	3-2
ρ(∂vϕ∂t+vr∂vϕ∂r+vθr∂vϕ∂θ+vϕrsinθ∂vϕ∂ϕ+vϕvr+vθvϕcotθr)=−1rsinθ∂p∂ϕ+μ[1r2∂∂r(r2∂vϕ∂r)+1r2∂∂θ(1sinθ∂∂θ(vϕsinθ))+1r2sin2θ∂2vϕ∂ϕ2+2r2sinθ∂vr∂ϕ+2cotθr2sinθ∂vθ∂ϕ]+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 + μ [ 1 r 2 ∂ ∂ r ( r 2 ∂ v ϕ ∂ r ) + 1 r 2 ∂ ∂ θ ( 1 s i n θ ∂ ∂ θ ( v ϕ s i n θ ) ) + 1 r 2 s i n 2 θ ∂ 2 v ϕ ∂ ϕ 2 + 2 r 2 s i n θ ∂ v r ∂ ϕ + 2 c o t θ r 2 s i n θ ∂ v θ ∂ ϕ ] + ρ g ϕ	3-3

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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.