free-free boundaries

The eigen value problem becomes

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

subject to boundary condition

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

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

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

From this it follows that the required solution must be

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

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

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

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

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

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

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

and the corresponding $R_c$ is given by

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