Thermal Convection in a Cylindrical Enclosure withWavy Sidewall - Selected Topics of Computational and Experimental Fluid Mechanics - page 382

Fig. 1 a Physical model, b coordinates and boundary conditions

3 Mathematical Formulation

The problem can be solved as an axisymmetric flow using cylindrical coordinates.
The axial direction z ∗ coincides with the axis of the cylinder and the wavy sidewall
is located at r ∗ = z ∗ /ʻ) R + a cos ( 2 ˀ . The continuity and steady-state momentum

and energy equations are respectively:

+ ∂v z
∂

∂v r
∂

+ v r
r ∗

= 0 (1)

r ∗ z ∗

∂

v r ∂v r

r ∗ + v z ∂v r

P ∗

2

v r ∂v r

∂

v r

2

v r 1

ˁ

∂ 1

r ∗ r ∗ 2 + ∂ z ∗ =− r ∗ + ʽ

+

r ∗ −

(2)

r ∗ 2 z ∗ 2 ∂ ∂ ∂ ∂ ∂

v r ∂v z

r ∗ + v z ∂v z

2

v z ∂v z

∂

2

v z P ∗ 1

ˁ

∂ ∂ 1

r ∗ r ∗ + ∂ z ∗ =− z ∗ + ʽ

+ + g ʲ ( T − T m ) (3)

r ∗ 2 z ∗ 2

∂ ∂ ∂ ∂ ∂

1
r ∗

r ∗ ∂

T ∗ T ∗

∂

T ∗
∂

2 T ∗
∂

v r ∂

r ∗ + v z ∂

∂
∂

+ ∂

z ∗ = ʱ

, (4)

r ∗ r ∗ z 2 ∗ ∂

subject to the following boundary conditions:

v r = v z T 2 at z ∗ = r ∗ ≤ = 0 , T = 0 , 0 ≤ R + a , (5)

v r = v z T 1 at z ∗ = r ∗ ≤ = 0 , T = L , 0 ≤ R + a , (6)