  The Bondi-van der Burg-Metzner-Sachs (BMS) group is the asymptotic symmetry
group of asymptotically flat gravity. Recently, Donnay et al. have derived an
analogous symmetry group acting on black hole event horizons. For a certain
choice of boundary conditions, it is a semidirect product of ${\rm Diff}(S^2)$,
the smooth diffeomorphisms of the two-sphere, acting on $C^\infty(S^2)$, the
smooth functions on the two-sphere. We observe that the same group appears in
fluid dynamics as symmetries of the compressible Euler equations. We relate
these two realizations of ${\rm Diff}(S^2)\ltimes C^\infty(S^2)$ using the
black hole membrane paradigm. We show that the Lie-Poisson brackets of membrane
paradigm fluid charges reproduce the near-horizon BMS algebra. The perspective
presented here may be useful for understanding the BMS algebra at null
infinity.
