  We provide the set of equations for non-relativistic fluid dynamics on
arbitrary, possibly time-dependent spaces, in general coordinates. These
equations are fully covariant under either local Galilean or local Carrollian
transformations, and are obtained from standard relativistic hydrodynamics in
the limit of infinite or vanishing velocity of light. All dissipative phenomena
such as friction and heat conduction are included in our description. Part of
our work consists in designing the appropriate coordinate frames for
relativistic spacetimes, invariant under Galilean or Carrollian
diffeomorphisms. The guide for the former is the dynamics of relativistic point
particles, and leads to the Zermelo frame. For the latter, the relevant objects
are relativistic instantonic space-filling branes in Randers-Papapetrou
backgrounds. We apply our results for obtaining the general
first-derivative-order Galilean fluid equations, in particular for
incompressible fluids (Navier-Stokes equations) and further illustrate our
findings with two applications: Galilean fluids in rotating frames or inflating
surfaces and Carrollian conformal fluids on two-dimensional time-dependent
geometries. The first is useful in atmospheric physics, while the dynamics
emerging in the second is governed by the Robinson-Trautman equation,
describing a Calabi flow on the surface, and known to appear when solving
Einstein's equations for algebraically special Ricci-flat or Einstein
spacetimes.
