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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View Full DocumentThis becomes an energetics proof, because total energy in perturbation in domain is being
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 Fall '08
 Staff
 Thermodynamics, Energy, Kinetic Energy

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