At stationary onset, $\mu=0$. The lowest stress-free vertical mode is $\widehat W=\sin\pi z$, for which $D^2=-\pi^2$ and $\widehat D^2=-(\pi^2+k^2)$. Substitution gives
$$
\boxed{
Ra(k)=\frac{(\pi^2+k^2)^3+\pi^2(\lambda/\sigma)^2}{k^2}}.
$$
The rotation term is positive, so rotation raises the critical Rayleigh number and is stabilizing.


Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 331 / 1 / e / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-331/1/e/solution