Stability of a two component fluid layer heated from below
We shall now be concerned with the so-called Bénard problem in a two component fluid layer. This problem has been studied recently by J.C. Legros [1, 2, 3] from the experimental point of view and theoretically by J.C. Legros, J.K. Platten, and P. Poty [4]
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11. Stability of a two component fluid layer heated from below.
11.1. Introduction.
We shall now be concerned with the so-called Benard problem in a two component fluid layer. This problem has been studied recently by j.C. Legros[l, 2,
3]
from the experi-
mental point of view and theoretically by j.e. Legros, j.K. Platten, and P. Poty
[4] (see also [9J ).
The kinetic theory of the Benard problem in a one component fluid layer has been solved in details by Chandrasekhar
[5,
chap. I~
using the usual normal mode analysis. The problem is to find out when free convection is generated due to the instability of the state at rest. As quoted in Section 10, the corresponding transition point is characterized by the cri tical value of the dimensionless parameter i. e. the Rayleigh number: (11.1)
(ft,a
=
gex ,Pd
...
,,'"
together with the critical value of the wave number k included in the expression of the vertical velo city perturbation: (11.2)
for a normal mode, with
P. Glansdorff, Thermodynamics in Contemporary Dynamics © Springer-Verlag Wien 1972
Stability of a Two-Component Layer
57
Finally starting from the linearized perturbation equations, Chandrasekhar obtains the dispersion equation (for
[5]):
more details, see (
2
o - k ) (0
2222
2
22
- k - 0') (0 - k -
0" Pr) W: - ~
2
where 0: d / d z , k: the wave number, coefficient, and
(Y' :
2
kW
(11.3)
the amplification
Pr: v/x the Prandtl number. The above equation
gives rise for suitable boundary conditions to the critical values: (la
CI".
657.5
and
CI'.
k :
2
~ ~ree surtclces)
or (!a
CI'.
:
er.
1707.762 and k : 3 .117~l.gl.d surfaces)
(11.5)
By comparison, using the local potential variational technique, Schechter and Himmelblau [6] found for two rigid surfaces and with only one variational parameter the excellent approximation CI'.
~a.
:
1750
and
CI'.
k : 3.1
Accordingly, we intend to follow the same way, in order to study the similar problem of stability for a two component fluid layer. In this respect let us emphasize that, already for such an apparently slight generalization to a mixture, an exact solution is excluded and only approximate solutions are to be expected. This remark enhances the interest of the local
Stability of a Two-Component Layer
58
potential method for a large class of problem in macroscopic physics. 11.2. The experimental problem.
11.2.1. Description of the apparatus.
As it can be seen on fig. I, the cell is constituted by two circular brass plates (L 20 cm; thickness
= 20 mm)
1
and L
2
=
coated with gold to avoid corrosion.
The plates are plane within a tolerance of 0.01 The
; diameter
L plates
mm.
are fitted in thick stainless steel
disks (A 1 and A2) to ensure the cell a high mechanical rigidi ty. The
Li plate is provided with a heating coil of
3000 (in Kanthal type 0, wire diameter = 0.5 mm). The latter thermally and electrically insulated on the of compressed asbestos (thickness
A~
side by a disk
5mm) and the Li side, is elec-
trically insulated by a thin
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