Bridge between particle and fluid kinematics

Fluid kinematics is developed in a natural way, like particle kinematics, through differentiation rather than by applying the divergence theorem. Dynamical equations are obtained for densities and current densities defined as sums of delta functions. The delta function’s operational properties provide a simple, unified conceptual framework. For a charged fluid interacting with the electromagnetic field, the Poynting theorem and Maxwell stress relation are obtained as fundamental energy and momentum transport equations. Fluid and field are treated symmetrically in the development, account being taken of the stresses and fluxes for each. Then, a microscopic energy transport equation is derived in a similar way for a fluid whose particles interact pairwise through instantaneous forces. The requirement of form invariance under a Galilean transformation yields both the continuity equation and the momentum transport equation. The macroscopic continuum equations are derived by way of simple, spatial averaging, without reference to ensembles, virial theorems, or phase space. The development is tailored for presentation to advanced undergraduate students and assumes no previous exposure to continuum mechanics.