流体动量控制方程
The Equation of Motion in terms of
τ
τ
控制方程通式：
ρ
D
v
v
D
t
=
−
∇
p
−
∇
⋅
τ
τ
+
ρ
g
g
ρ
D
v
v
D
t
=
−
∇
p
−
∇
⋅
τ
τ
+
ρ
g
g
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 ( ∂ 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 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 |
1-1 |
|
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 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 |
1-2 |
|
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 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 |
1-3 |
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 ( ∂ v r ∂ t + v r ∂ v r ∂ r + v θ r ∂ v r ∂ θ + v z ∂ v r ∂ z − v 2 θ 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 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 |
2-1 |
|
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 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 |
2-2 |
|
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 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 |
2-3 |
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 ( ∂ 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 2 θ + v 2 ϕ 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 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 |
3-1 |
|
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 2 ϕ 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 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 |
3-2 |
|
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 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 |
3-3 |
注：如果
τ
τ
τ
τ
具有对称性，那么
τ
r
θ
−
τ
θ
r
=
0
τ
r
θ
−
τ
θ
r
=
0
参考文献
- R. Byron Bird, Warren E. stewart, Edwin N. Lightfoot.* Transport phenomena:Revised second edition* John Wiely &Sons, Inc.