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## Scaling of the equations

We use as length scale, as time scale and as the scale for temperature. Let are the nondimensional variables. The linear stability equations become

 (10)

 (11)

 (12)

Here the nondimensional Rayleigh number and Prandtl number are given by

 (13)

The boundary conditions are to be applied on the boundary at and . Note that Rayleigh number is positive when and is the ratio of buoyancy force and viscous force. On the other hand, Prandtl number depends on the properties of the fluid: ratio of molecular diffusion due to momentum and heat.

We now eliminate all the dependent variables except . The curl of equation (11) gives

 (14)

where we is the vorticity of the flow. In the above equation we have used the vector identity

Taking the curl of equation (14) again and using the vector identities

we get

 (15)

In particular for the vertical velocity we have

 (16)

where is the horizontal laplacian. This equation can be written is

Similarly the heat equation (12) can be written as

If we eliminate then we get

 (17)

If we eliminate then we get same equation (17) satisfied by .

From the vertical component of we have

 (18)

from which we can determine the vertical vorticity component. From the definition of and the equation of continuity, we get

 (19)

 (20)

and these equation can be used to find and .

Next: Boundary conditions Up: The stability problem Previous: The linearized equations
A/C for Homepage of Dr. S Ghorai 2003-01-16