Past exam of the astrophysical fluid dynamics course of the University of Cambridge / 2019 / Section A / 3

For an inverted free surface, gravity promotes a displacement instead of restoring it. The surface tension term still resists curvature. Hence the gravity-capillary wave relation becomes, with $q=|k|$,For $0<q<k_c$, where $k_c=\sqrt{\rho g/\sigma }$, ${\omega }^2<0$. Writing the displacement as $\eta \propto e^{st+ikx}$ gives the Rayleigh-Taylor instability growth rateShorter wavelengths, $q>k_c$, remain oscillatory because surface tension dominates. The marginal scale is determined by the capillary length,A sufficiently small aperture excludes the long unstable modes: this is finite-aperture stabilization of an inverted liquid surface. The estimate requested is therefore a container opening of order one centimetre, with $1.7\mathrm{cm}$ the marginal full wavelength rather than a geometry-independent maximum diameter.

The exact largest opening cannot be fixed from the stated infinite-surface dispersion relation alone. It requires the aperture shape and contact-line boundary conditions. For example, a one-dimensional span of width $L$ whose contact line is pinned at both ends has a half-sine lowest mode with $q_{\min }=\pi /L$. In that simple model,Different geometries change the numerical factor. This is a small-disturbance stability estimate with negligible air inertia and no leakage or contact-line failure; a guarantee that a real inverted container retains its water needs those additional boundary details.