  To an outside observer, a black hole's event horizon appears to behave
exactly like a dynamical fluid membrane. We extend this membrane paradigm to
black holes in general $f(R)$ theories of gravity. We derive the stress tensor
and various transport coefficients of the fluid and find that the membrane
behaves as a non-Newtonian fluid except for the special case of Einstein
gravity. Using Euclidean methods, we study the thermodynamics of the membrane.
We speculate on what theories of gravity admit horizons with fluid properties.
