Stability lecture 12

# Stability lecture 12 - Hydrodynamic Stability Lecture 12...

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Hydrodynamic Stability Lecture 12 Principle of exchange of stabilities. As Ra is varied the unstable modes are separated from the stable modes () by modes for which ( 0 R σ > ) 0 R > 0, 0 RI == . N.B. This isn’t usually the case in other physical problems. (When IR = doesn’t have to be zero). Expect a steady cellular solution near R instead of a growing wavelike (oscillatory) solution. cRt a This principle simplifies the math. It is misnamed [Exchange of Stabilities] meaning if 0, then 0 = = . It is easy to look for the neutral curve, set 0 = , this greatly simplifies the math. Proof – for this problem. Energy like eqn. Let ‘ d dz 2 21 22 and Pr Governing equation becomes Ra with temperature B.C. 0 at 0,1. Gwkw FG k G Fk F k w Fz ′′ =− =− + −+ = Multiply governing equation by conjugate of ( ) FF and integrate from at 0,1. z = N 2 11 00 Ra F FF dz k FFdz k Fwdz = ∫∫ 1 0 1

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This becomes an energetics proof, because total energy in perturbation in domain is being considered.
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Stability lecture 12 - Hydrodynamic Stability Lecture 12...

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