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# The Boussinesq approximation

The basis of this approximation is that there are flows in which the temperature varies little, and therefore the density varies little, yet in which the buoyancy drives the motion. Thus the variation in density is neglected everywhere except in the buoyancy term. Let denotes the density at the bottom where temperature is . For small temperature difference between the top and bottom layer we can write where coefficient of volume expansion. For liquid is in the range to . For a temperature variation of moderate amount we have but the buoyancy term is the same order of magnitude as the inertia, acceleration or the viscous stress so is not negligible.

The differential in density in the continuity equation (1) are of the order and hence neglected to give (4)

as for an incompressible fluid. Then the stress tensor becomes Again the momentum equation becomes which after using the Boussinesq approximation becomes (after a little manipulation) (5)

where is the Laplacian operator.

Now we consider the energy equation. The velocity is of the order and thus the ratio of term to the term due to conduction of heat is of the order of and this is in the range from to for gases and liquids and the viscous dissipation is neglected. The term Using perfect gas relations and we get Thus though holds in the equation of continuity, we should not use this relation in the heating due to compression term for gases. For liquids however, this heating is negligible. Thus the final form of the energy equation is (6)

where for a perfect gas and for liquids. Equations (4), (5) and (6) are called Boussinesq equations and describe the motion of a Boussinesq fluid.   Next: The stability problem Up: Rayleigh-Benard Convection Previous: The exact equations of
A/C for Homepage of Dr. S Ghorai 2003-01-16