  A kinetic theory of classical particles serves as a unified basis for
developing a geometric $3+1$ spacetime perspective on fluid dynamics capable of
embracing both Minkowski and Galilei/Newton spacetimes. Parallel treatment of
these cases on as common a footing as possible reveals that the particle
four-momentum is better regarded as comprising momentum and \textit{inertia}
rather than momentum and energy; and consequently, that the object now known as
the stress-energy or energy-momentum tensor is more properly understood as a
stress-\textit{inertia} or \textit{inertia}-momentum tensor. In dealing with
both fiducial and comoving frames as fluid dynamics requires, tensor
decompositions in terms of the four-velocities of observers associated with
these frames render use of coordinate-free geometric notation not only fully
viable, but conceptually simplifying. A particle number four-vector,
three-momentum $(1,1)$ tensor, and kinetic energy four-vector characterize a
simple fluid and satisfy balance equations involving spacetime divergences on
both Minkowski and Galilei/Newton spacetimes. Reduced to a fully $3+1$ form,
these equations yield the familiar conservative formulations of special
relativistic and non-relativistic hydrodynamics as partial differential
equations in inertial coordinates, and in geometric form will provide a useful
conceptual bridge to arbitrary-Lagrange-Euler and general relativistic
formulations.
