= Finite-aperture stabilization of an inverted liquid surface
{title2=$k_{min}\ge\sqrt{\rho g/\sigma}$}
For a deep inviscid liquid above a gas of negligible inertia, an inverted <free surface> has <dispersion relation> $\omega^2=q(\sigma q^2/\rho-g)$, $q=|k|$. The <Rayleigh-Taylor instability> affects $0<q<k_c=\sqrt{\rho g/\sigma}$. A bounded aperture can suppress it if its smallest allowed <wavenumber> is at least $k_c$. The <capillary length> is $\ell_c=k_c^{-1}$ and the marginal <wavelength> is $2\pi\ell_c$. A pinned one-dimensional span of width $L$ with a half-sine lowest mode instead has $k_{min}=\pi/L$, giving $L\le\pi\ell_c$. A numerical maximum container size therefore also requires aperture geometry and contact-line <boundary conditions>; the marginal wavelength alone is not a universal diameter.
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