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The sign of the real part of
determines whether the flow
is stable or unstable. We show that for the Benard problem is real. To
show this multiply equation (30) by (* denotes complex conjugate)
and
integrate between to . Then
which on integrating by parts gives (using BCs)

(35) 
where
are clearly positive quantities. Similarly we multiply (31) by and
integrate between to .
which again on integration by parts using the BCs gives

(36) 
where
which are also clearly positive.
Taking the complex conjugate of (36) and subtracting from
(35) gives

(37) 
Taking real parts of (37) gives
which shows the obvious fact
for (i.e. flow is stable if
upper boundary is hotter than the lower). Taking the imaginary parts of
(37) we get
which implies that
for .
Next: A variation principle
Up: General stability characteristics
Previous: General stability characteristics
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