Neutral curve (1) Convection in a rotating cylindrical annulus with fixed heat flux boundaries Governing equations Neutral curve (2)

We linearize the governing equations by assuming small amplitudes for y and q. Further, we expand y and q by a series of functions satisfying the boundary conditions and truncate up to the vertical wavenumber 1 (see Appendix B). As a result, we can easily obtain the approximate expression of the neutral curve of the system.

 \begin{displaymath}

 R = \frac{\pi^2}{8}

 \left[K^4 + \frac{P^2\eta^2k^2}{(PK^2 ...

 ...ight)^2

 \frac{k^2}{(k^2 + \frac{P\pi^2}{P+1})^2} 

 \right]. 

 \end{displaymath} (11)

The first term on the right hand side   (p2/8)(k2+p2)2 describes the neutral curve for the case of the non-rotating system (center graph in figure). This neutral curve has a minimum at k=0. As the rotation rate increases, the effect of the second term (right graph in figure) appears  and the neutral curve goes up in the k-R plane. However, since the effect of the second term vanishes at k=0, the critical horizontal wave number is zero even with the effect of rotation. Therefore, the critical values are

\begin{displaymath}

k_c = 0, \quad \omega_c=0, \quad

 R_c= \frac{\pi^6}{8}\sim 120.17\DselJ{,}{.}

 \end{displaymath} (12)

Neutral curve (1) Convection in a rotating cylindrical annulus with fixed heat flux boundaries Governing equations Neutral cureve (2)