Page 256 - Academic Press Encyclopedia of Physical Science and Technology 3rd Chemical Engineering

P1: GLM/GLT P2: GLM Final
Encyclopedia of Physical Science and Technology En006G-249 June 27, 2001 14:7

48 Fluid Dynamics (Chemical Engineering)

Cartesian: Spherical Polar:
x Component r Component

∂p
∂v x ∂v x ∂v x ∂v x 2 2
ρ + v x + v y + v z =− ∂v r ∂v r v θ ∂v r v φ ∂v r v + v φ
θ
∂t ∂x ∂y ∂z ∂x ρ + v r + + −
∂t ∂r r ∂θ r sin θ ∂φ r

∂τ xx ∂τ yx ∂τ zx
− + + + ρg x (14) ∂p 1 ∂ 2 1 ∂
∂x ∂y ∂z =− − 2 r τ rr + (τ rθ sin θ)
∂r r ∂r r sin θ ∂θ
y Component

1
∂τ rφ τ θθ + τ φφ
∂p + − + ρg r (20)
∂v y ∂v y ∂v y ∂v y
ρ + v x + v y + v z =− r sin θ ∂φ r
∂t ∂x ∂y ∂z ∂y
θ Component

∂τ xy ∂τ yy ∂τ zy
− + + + ρg y (15)
∂x ∂y ∂z ∂v θ ∂v θ v θ ∂v θ v φ ∂v θ v r v θ
ρ + v r + + +
z Component ∂t ∂r r ∂θ r sin θ ∂φ r

2
∂v z ∂v z ∂v z ∂v z ∂p v cot θ 1 ∂p 1 ∂
ρ + v x + v y + v z =− − φ =− − 1 ∂ r τ rθ +
2

∂t ∂z ∂y ∂z ∂z r r ∂θ r ∂r r sin θ ∂θ
2

∂τ xz ∂τ yz ∂τ zz 1 ∂τ θφ τ rθ cot θ
− + + + ρg z (16) × (τ θθ sin θ) + + − τ φφ + ρg θ
∂x ∂y ∂z r sin θ ∂φ r r
Cylindrical Polar: (21)
r Component φ Component
2
∂v r ∂v r v θ ∂v r v θ ∂v r ∂p
ρ + v r + − + v z =− ∂v φ ∂v φ v θ ∂v φ v φ ∂v φ v φ v r
∂t ∂r r ∂θ r ∂z ∂r ρ + v r + + +
∂t ∂r r ∂θ r sin θ ∂φ r

1 ∂ 1 ∂τ rθ τ θθ ∂τ rz 1 ∂p
− (rτ rr ) + − + + ρg r (17) + v θ v φ cot θ =− − 1 ∂ 2
r ∂r r ∂θ r ∂z r r sin θ ∂φ r ∂r r τ rφ
2
θ Component

1 ∂τ θφ 1 ∂τ φφ τ rφ 2 cot θ

1 ∂p + + + + τ θφ + ρg φ
∂v θ ∂v θ v θ ∂v θ v r v θ ∂v θ r ∂θ r sin θ ∂φ r r
ρ + v r + + + v z =−
∂t ∂r r ∂θ r ∂z r ∂θ
(22)

1 ∂ 2 1 ∂τ θθ ∂τ θz
− (r τ rθ ) + + + ρg θ (18) Two terms in Eqs. (17) and (18) are worthy of special
2
r ∂r r ∂θ ∂z 2
note. In Eq. (17) the term ρv /r is the centrifugal “force.”
θ
That is, it is the effective force in the r direction arising
z Component
from fluid motion in the θ direction. Similarly, in Eq. (18)

∂v z ∂v z v θ ∂v z ∂v z ∂p ρv r v θ /r is the Coriolis force, or effective force in the θ
ρ + v r + + v z =−
∂t ∂r r ∂θ ∂z ∂z direction due to motion in both the r and θ directions.
Both of these forces arise naturally in the transformation
of coordinates from the Cartesian frame to the cylindrical

1 ∂ 1 ∂τ θz ∂τ zz polar frame. They are properly part of the acceleration
− (rτ rz ) + + + ρg z (19)
r ∂r r ∂θ ∂z vector and do not need to be added on physical grounds.