牛顿流体耗散方程【Dissipation Function】

牛顿流体耗散方程

Dissipation Function for Newtonian Fluids

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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.
ΦΦv=2[(∂vx∂x)2+(∂vy∂y)2+(∂vz∂z)2]+[∂vy∂x+∂vx∂y]2+[∂vz∂y+∂vy∂z]2+[∂vx∂z+∂vz∂x]2−23[∂vx∂x+∂vy∂y++∂vz∂z]2 Φ Φ v = 2 [ ( ∂ v x ∂ x ) 2 + ( ∂ v y ∂ y ) 2 + ( ∂ v z ∂ z ) 2 ] + [ ∂ v y ∂ x + ∂ v x ∂ y ] 2 + [ ∂ v z ∂ y + ∂ v y ∂ z ] 2 + [ ∂ v x ∂ z + ∂ v z ∂ x ] 2 − 2 3 [ ∂ v x ∂ x + ∂ v y ∂ y + + ∂ v z ∂ z ] 2	1-1

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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.
ΦΦv=2[(∂vr∂r)2+(1r∂vθ∂θ+vrr)2+(∂vz∂z)2]+[r∂∂r(vθr)+1r∂vr∂θ]2+[1r∂vz∂θ+∂vθ∂z]2+[∂vr∂z+∂vz∂r]2−23[1r∂∂r(rvr)+1r∂vθ∂θ+∂vz∂z]2 Φ Φ v = 2 [ ( ∂ v r ∂ r ) 2 + ( 1 r ∂ v θ ∂ θ + v r r ) 2 + ( ∂ v z ∂ z ) 2 ] + [ r ∂ ∂ r ( v θ r ) + 1 r ∂ v r ∂ θ ] 2 + [ 1 r ∂ v z ∂ θ + ∂ v θ ∂ z ] 2 + [ ∂ v r ∂ z + ∂ v z ∂ r ] 2 − 2 3 [ 1 r ∂ ∂ r ( r v r ) + 1 r ∂ v θ ∂ θ + ∂ v z ∂ z ] 2	2-1

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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.

ΦΦv=2[(∂vr∂r)2+(1r∂vθ∂θ+vrr)2+(1rsinθ∂vϕ∂ϕ+vr+vθcotθr)2]+[r∂∂r(vθr)+1r∂vr∂θ]2+[sinθr∂∂θ(vϕsinθ)+1rsinθ∂vθ∂ϕ]2+[1rsinθ∂vr∂ϕ+r∂∂r(vϕr)]2−23[1r2∂∂r(r2vr)+1rsinθ∂∂θ(vθsinθ)+1rsinθ∂vϕ∂ϕ]2 Φ Φ v = 2 [ ( ∂ v r ∂ r ) 2 + ( 1 r ∂ v θ ∂ θ + v r r ) 2 + ( 1 r s i n θ ∂ v ϕ ∂ ϕ + v r + v θ c o t θ r ) 2 ] + [ r ∂ ∂ r ( v θ r ) + 1 r ∂ v r ∂ θ ] 2 + [ s i n θ r ∂ ∂ θ ( v ϕ s i n θ ) + 1 r s i n θ ∂ v θ ∂ ϕ ] 2 + [ 1 r s i n θ ∂ v r ∂ ϕ + r ∂ ∂ r ( v ϕ r ) ] 2 − 2 3 [ 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 ϕ ∂ ϕ ] 2

3-1

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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.