Hydrodynamic Stability
Lecture 10
Convective (thermal) instability.
Instability of a fluid layer heated from below.
Basic Problem.
1. Assume initial instability is isentropic – heat transfer by conduction is relatively slow
(steady state).
2. Temperature and density fluctuations are small.
0
0
1
TT
T
−
±
() ( )
00
1
ρρ
ρ
α
==−−
T
Taylor series expansion using these assumptions.
0
density at
0
z
−=
0
0
const
0
T
ρα
∂
=−
=
>
∂
 coefficient of volumetric expansion.
Basic Backgroud state.
()
[ ]
0
,1
x
tz
T
B
z
ραβ
=+
∂
∂
²
Only a small
loss of entropy due to viscous effects.
1
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View Full DocumentThere is an entropy flux at boundaries, but the whole fluid doesn’t change state at all.
Flow is incompressible.
Incompressibility means
0
leads to
0
D
Dt
u
ρ
=
∇=
i
±
In the Boussinesq approximation we use
u
g
t
∂
∂
²
±
so that to a good approximation, for variable
density flows, density differences only enter the momentum equation through buoyancy terms.
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 Fall '08
 Staff
 Fluid Dynamics, Thermodynamics, Boundary value problem, Benard, Boussinesq approximation, Assume initial instability

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