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# The exact equations of motion

We consider a layer of fluid of depth have infinite horizontal dimensions. The temperature at the bottom and the top are maintained at and respectively. We use tensor notation in writing down the equations of motion. Thus we use cartesian tensor with and velocity for j=1,2,3. [Note: We frequently mix vector and tensor notation, e.g. the velocity vector can be written as ]

Equation of continuity

or in equaivalent form

 (1)

where is the density of the fluid.

Equations of momentum

 (2)

where

is the stress tensor where is the shear viscosity. Strictly speaking, the pressure term in the stress tensor is the mechanical tensor which is different than the thermodynamic pressure term. But their difference is so small that we assume they are same for all the practical case.
Equation of energy

 (3)

internal energy per unit mass of fluid [ ( specific heat at constant volume) for gases and ( specific heat) for liquid], thermal diffusivity, temperature and rate of viscous dissipation is

Next: The Boussinesq approximation Up: Rayleigh-Benard Convection Previous: Intrduction
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