4H FLUID DYNAMICS Vector Formulae and Theorems of Vector ...

4H FLUID DYNAMICS
Vector Formulae and Theorems of Vector Calculus
Vector Formulae
a × (b × c) = (a · c)b − (a · b)c
∇×∇f = 0
∇·(∇×F) = 0
∇·(fF) = F · ∇f + f∇·F
∇×(fF) = ∇f × F + f∇×F
∇×(∇×F) = ∇(∇·F) − ∇ 2 F
∇(F · G) = (F · ∇)G + (G · ∇)F + F × (∇×G) + G × (∇×F)
∇·(F × G) = G · (∇×F) − F · (∇×G)
∇×(F × G) = F(∇·G) − G(∇·F) + (G · ∇)F − (F · ∇)G
If x = re r is the position vector with respect to some origin, then
∇·x = 3 , ∇×x = 0 , ∇·e r = 2 r , ∇×e r = 0
The Divergence Theorem and Corollaries
In the following V is a volume bounded by a closed surface S,
∫
∫
∇·F dV = F · dS
V
S
∫
∫
∇f dV = f dS
V
S
∫
∫
∇×F dV = dS × F
V
S
∫
(
f∇ 2 g + ∇f · ∇g ) ∫
dV = f∇g · dS
V
S
∫
(
f∇ 2 g − g∇ 2 f ) ∫
dV = (f∇g − g∇f) · dS
V
Stoke’s Theorem and a Corollary
In the following S is an open surface bounded by the contour C,
∫
∮
(∇×F) · dS = F · dx
S
C
∫
∮
dS × ∇f = f dx
S
S
C
(Divergence theorem)
(Green’s first identity)
(Green’s theorem)
(Stoke’s theorem)
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Common Vector Differential Quantities in Polar Coordinates
Cylindrical Polars (s, φ, z)
∇f = ∂f
∂s e s + 1 ∂f
s ∂φ e φ + ∂f
∂z e z
∇·F = 1 ∂
s ∂s (sF s) + 1 ∂F φ
s ∂φ + ∂F z
∂z
( 1 ∂F z
∇×F =
s ∂φ − ∂F ) (
φ ∂Fs
e s +
∂z ∂z − ∂F z
∂s
( 1 ∂
+
s ∂s (sF φ) − 1 )
∂F s
e z
s ∂φ
∇ 2 f = 1 (
∂
s ∂f )
+ 1 ∂ 2 f
s ∂s ∂s s 2 ∂φ 2 + ∂2 f
∂z
(
2
∇ 2 F = ∇ 2 F s − F s
+
)
e s
)
e φ
s 2 − 2 ∂F φ
s 2 ∂φ
(
∇ 2 F φ + 2 ∂F s
s 2 ∂φ − F )
φ
s 2 e φ + ( ∇ 2 )
F z ez
Spherical Polars (r, θ, φ)
∇f = ∂f
∂r e r + 1 ∂f
r ∂θ e θ + 1 ∂f
r sin θ ∂φ e φ
∇·F = 1 ∂
r 2 ∂r (r2 F r ) + 1 ∂
r sin θ ∂θ (sin θF θ) + 1 ∂F φ
r sin θ ∂φ
∇×F = 1 ( ∂
r sin θ ∂θ (sin θF φ) − ∂F )
θ
e r
∂φ
( 1 ∂F r
+
r sin θ ∂φ − 1 ) (
∂
1
r ∂r (rF ∂
φ) e θ +
r ∂r (rF θ) − 1 r
∇ 2 f = 1 (
∂
r 2 r 2 ∂f )
(
1 ∂
+
∂r ∂r r 2 sin θ ∂f )
1 ∂ 2 f
+
sin θ ∂θ ∂θ r 2 sin 2 θ ∂φ
(
2
∇ 2 F = ∇ 2 F r − 2F )
r
r 2 − 2 ∂
r 2 sin θ ∂θ (sin θF 2 ∂F φ
θ) −
r 2 e r
sin θ ∂φ
(
+ ∇ 2 F θ + 2 ∂F r
r 2 ∂θ − F θ
r 2 sin 2 θ − 2 cos θ )
∂F φ
r 2 sin 2 e θ
θ ∂φ
(
)
+ ∇ 2 F φ +
e φ
2
r 2 sin θ
∂F r
∂φ + 2 cos θ
r 2 sin 2 θ
∂F θ
∂φ −
F φ
r 2 sin 2 θ
)
∂F r
e φ
∂θ
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The Navier-Stokes Equations in Cylindrical Polar Coordinates
The Navier-Stokes equations in cylindrical polars (s, φ, z) are:
∂u s
∂t + (u·∇)u s − u2 φ
s
∂u φ
∂t + (u·∇)u φ + u su φ
s
= − 1 ρ
= − 1 ρs
∂u z
∂t + (u·∇)u z
(
∂p
∂s + ν ∇ 2 u s − u s
s 2 − 2 )
∂u φ
s 2 ,
∂φ
(
∂p
∂φ + ν ∇ 2 u φ − u φ
s 2 + 2 )
∂u s
s 2 ,
∂φ
= − 1 ρ
1 ∂
s ∂s (su s) + 1 ∂u φ
s ∂φ + ∂u z
∂z
∂p
∂z + ν∇2 u z ,
= 0 .
where
∂f
u · ∇f = u s
∂s + u φ ∂f
s ∂φ + u ∂f
z
∂z .
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