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## Exchange of stabilities

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
A/C for Homepage of Dr. S Ghorai 2003-01-16