Navier-Stokes equations for an incompressible fluid
∂u ∂v ∂w
+
+
=0
∂x ∂y
∂z
Cartesian Coordinates
⎡∂ 2 u ∂ 2 u ∂ 2 u ⎤
∂u
∂u
∂u
∂u
1∂p
+u
+v
+w
=−
+ν⎢ 2 +
+
+g
∂t
∂x
∂y
∂z
ρ ∂x
∂ y 2 ∂ z2 ⎥⎦ x
⎣∂x
⎡∂ 2 v ∂ 2 v ∂ 2 v ⎤
∂v
∂v
∂v
∂v
1 ∂p
+u
+v
+w
=−
+ ν⎢ 2 +
+
+g
∂t
∂x
∂y
∂z
ρ ∂y
∂ y2
∂ z2 ⎥⎦ y
⎣∂x
⎡∂2 w ∂2 w ∂ 2 w ⎤
∂w
∂w
∂w
∂w
1 ∂p
+u
+v
+w
=−
+ ν⎢ 2 +
+
+g
∂t
∂x
∂y
∂z
ρ ∂z
∂ y2
∂ z2 ⎥⎦ z
⎣ ∂x
Cylindrical Coordinates
∂u
1 ∂ ( rur ) 1 ∂ uθ
+
+ z =0
r ∂r
r ∂θ
∂z
⎡ ∂ ⎛ 1 ∂ ( ru ) ⎞ 1 ∂ 2 u
∂ ur
∂u
u ∂ ur uθ2
∂u
1 ∂p
2 ∂ uθ ∂ 2 ur ⎤
r
r
+ ur r + θ
−
+ uz r = −
+ ν⎢ ⎜
−
+
⎥+g
⎟+ 2
∂t
∂r
r ∂θ
r
∂z
ρ ∂r
r2 ∂θ
∂ z2 ⎥⎦ r
⎢⎣ ∂ r ⎝ r ∂ r ⎠ r ∂θ 2
⎡ ∂ ⎛ 1 ∂ ( ruθ ) ⎞ 1 ∂ 2 u
∂ uθ
∂u
u ∂u
uu
∂u
1 1 ∂p
2 ∂ ur ∂ 2 uθ ⎤
θ
+ ur θ + θ θ + r θ + uz θ = −
+ ν⎢ ⎜
+
+
+
⎥+ g
⎟
2
2
∂t
∂r
r ∂θ
r
∂z
ρ r ∂θ
r 2 ∂θ
∂ z2 ⎥⎦ θ
⎢⎣ ∂ r ⎝ r ∂ r ⎠ r ∂θ
⎡ 1 ∂ ⎛ ∂ uz ⎞ 1 ∂ 2 uz
∂ uz
∂u
u ∂ uz
∂u
∂ 2 uz ⎤
1 ∂p
+ ur z + θ
+ uz z = −
+ ν⎢
r
+
+
⎜
⎟
⎥ + gz
2
2
∂t
∂r
r ∂θ
∂z
ρ ∂z
∂z2 ⎦
⎣ r ∂ r ⎝ ∂ r ⎠ r ∂θ
LIST OF USEFUL FORMULAE
z
in Cartesian coordinates -> Flow rate: V = b ∫ u dz
0
2π
ro
0
ri
in cylindrical coordinates -> Flow rate: V = ∫ dθ ∫ v z r dr