  Analyzing a capillary minimizing problem for a higher-dimensional extended
fluid, we find that there exist startling similarities between the black
hole-black string system (the Gregory-Laflamme instability) and the liquid
drop-liquid bridge system (the Rayleigh-Plateau instability), which were first
suggested by a perturbative approach. In the extended fluid system, we confirm
the existence of the critical dimension above which the non-uniform bridge
(NUB, i.e., {\it Delaunay unduloid}) serves as the global minimizer of surface
area. We also find a variety of phase structures (one or two cusps in the
volume-area phase diagram) near the critical dimension. Applying a catastrophe
theory, we predict that in the 9 dimensional (9D) space and below, we have the
first order transition from a uniform bridge (UB) to a spherical drop (SD),
while in the 10D space and above, we expect the transition such that UB $\to$
NUB $\to$ SD. This gives an important indication for a transition in the black
hole-black string system.
