  Fluid dynamics in intrinsically curved geometries is encountered in many
physical systems in nature, ranging from microscopic bio-membranes all the way
up to general relativity at cosmological scales. Despite the diversity of
applications, all of these systems share a common feature: the free motion of
particles is affected by inertial forces originating from the curvature of the
embedding space. Here we reveal a fundamental process underlying fluid dynamics
in curved space: the free motion of fluids, in the complete absence of solid
walls or obstacles, exhibits loss of energy due exclusively to the intrinsic
curvature of space. We find that local sources of curvature generate viscous
stresses as a result of the inertial forces. The curvature-induced viscous
forces are shown to cause hitherto unnoticed and yet appreciable energy
dissipation, which might play a significant role for a variety of physical
systems involving fluid dynamics in curved spaces.
