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free-free boundaries

The eigen value problem becomes

$\displaystyle (D^2-a^2)^3W=-a^2 R_a W$ (51)

subject to boundary condition

$\displaystyle W=D^2 W = D^4 W = 0$   at$\displaystyle \quad z=0,1$ (52)

In view of (52), using (51) we can shown that

$\displaystyle D^{2m}W=0$   for$\displaystyle \;z=0,1$   and$\displaystyle \;m=1,2,\cdots
$

From this it follows that the required solution must be

$\displaystyle W = A \sin(n \pi z) \qquad n=1,2,3,\cdots$ (53)

where $ A$ is a constant and $ n$ is an integer. Substitution of $ W$ in (51) leads to eigen value relation

$\displaystyle R_a = (n^2\pi^2+a^2)^3/a^2$ (54)

For a given $ a^2$, the lowest value of $ R_a$ occurs when $ n=1$ which is the lowest mode:

$\displaystyle R_a = (\pi^2+a^2)^3/a^2$ (55)

The critical Rayleigh number $ R_c$ is obtained by finding the minimum value of $ R_a$ when $ a^2$ is varied.

$\displaystyle \frac{dR_a}{da^2}=0 \rightarrow a_c = \pi/\sqrt{2}
$

and the corresponding $ R_c$ is given by

$\displaystyle R_c = \frac{27}{4}\pi^4
$


next up previous
Next: rigid-rigid boundaries Up: Exact solutions of characteristics Previous: Exact solutions of characteristics
A/C for Homepage of Dr. S Ghorai 2003-01-16