Computer simulation of thermal convection in Rayleigh-Bénard cell ...

Hubert Jopek
Computer simulation of thermal convection in Rayleigh-Bénard cell
The next problem is the effect of the buoyant force on the pocket of fluid
which is proportional to the density different between the pocket and its
surroundings. On the other hand the density difference is proportional to the thermal
expansion coefficient α and the temperature difference
buoyant force may be calculated as:
∆ T
. Consequently the
δT
F = αρ
0
g∆T
= αρ
0g
∆T
, (4.5)
h
where: ρ0
- the original fluid density; g - strength of the local gravitational field.
Assuming that buoyant force balances the fluid viscous force the pocket
moves with the constant velocity υ
z
.Hence the displacement through a distance
takes for the pocket a time:
∆ z
τ
d
∆z
= . (4.6)
υ
z
As the viscous force is equal to the viscosity of the fluid multiplied by the
Laplacian of the velocity, the viscous force may be approximated as follows:
F
v
2
υz
= µ ∇ υz
≈ µ . (4.7)
2
h
Now, by equating buoyant and viscous force one can obtain the υz
expression:
αρ0 ghδT
υz
= ∆z
, (4.8)
µ
10