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Exchange of stabilities

The sign of the real part of $ \sigma=\sigma_r + i \sigma_i$ determines whether the flow is stable or unstable. We show that for the Benard problem $ \sigma$ is real. To show this multiply equation (30) by $ \Theta^*$ (* denotes complex conjugate) and integrate between $ z=0$ to $ 1$. Then

$\displaystyle \int_0^1 \Theta^*(D^2-a^2-\sigma)\Theta dz = -\int_0^1 \Theta^*W dz
$

which on integrating by parts gives (using BCs)

$\displaystyle \sigma I_0 + I_1 = \int_0^1 \Theta^* W dz$ (35)

where

$\displaystyle I_0 = \int_0^1\vert\Theta\vert^2 dz\qquad I_1=\int_0^1\left(\vert D\Theta\vert^2+a^2
\vert\Theta\vert^2\right) dz
$

are clearly positive quantities. Similarly we multiply (31) by $ W^*$ and integrate between $ z=0$ to $ 1$.

$\displaystyle \int_0^1 W^*(D^2-a^2)(D^2-a^2-\frac{\sigma}{P_r})W dz = a^2 R_a \int_0^1 W*\Theta dz
$

which again on integration by parts using the BCs gives

$\displaystyle J_1 + \sigma J_0/P_r = a^2 R_a \int_0^1 W^* \Theta dz$ (36)

where

$\displaystyle J_0=\int_0^1\left(\vert DW\vert^2+a^2\vert W\vert^2\right)dz\quad...
...\int_0^1\left(\vert D^2W\vert^2+2a^2\vert DW\vert^2+a^4\vert W\vert^2\right)dz
$

which are also clearly positive. Taking the complex conjugate of (36) and subtracting from $ a^2 R_a
\times$(35) gives

$\displaystyle a^2 R_a I_1 - J_1 + \sigma a^2 R_a I_0 - \sigma^* J_0/P_r =0$ (37)

Taking real parts of (37) gives

$\displaystyle \sigma_r (a^2 R_a I_0-J_0/P_r) + a^2 R_a I_1 - J_1 = 0
$

which shows the obvious fact $ \sigma_r < 0$ for $ R_a < 0$ (i.e. flow is stable if upper boundary is hotter than the lower). Taking the imaginary parts of (37) we get

$\displaystyle \sigma_i (a^2 R_a I_0 + J_0/P_r) = 0
$

which implies that $ \sigma_i=0$ for $ R_a > 0$.
next up previous
Next: A variation principle Up: General stability characteristics Previous: General stability characteristics
A/C for Homepage of Dr. S Ghorai 2003-01-16