  We construct the theory of dissipative hydrodynamics of uncharged fluids
living on embedded space-time surfaces to first order in a derivative expansion
in the case of codimension-1 surfaces (including fluid membranes) and the
theory of non-dissipative hydrodynamics to second order in a derivative
expansion in the case of codimension higher than one under the assumption of no
angular momenta in transverse directions to the surface. This construction
includes the elastic degrees of freedom, and hence the corresponding transport
coefficients, that take into account transverse fluctuations of the geometry
where the fluid lives. Requiring the second law of thermodynamics to be
satisfied leads us to conclude that in the case of codimension-1 surfaces the
stress-energy tensor is characterized by 2 hydrodynamic and 1 elastic
independent transport coefficient to first order in the expansion while for
codimension higher than one, and for non-dissipative flows, the stress-energy
tensor is characterized by 7 hydrodynamic and 3 elastic independent transport
coefficients to second order in the expansion. Furthermore, the constraints
imposed between the stress-energy tensor, the bending moment and the entropy
current of the fluid by these extra non-dissipative contributions are fully
captured by equilibrium partition functions. This analysis constrains the Young
modulus which can be measured from gravity by elastically perturbing black
branes.
