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

Minkowski and Galilei/Newton Fluid Dynamics: A Geometric 3+1 Spacetime Perspective