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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.
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