Irreversible thermodynamic interpretations of experimental data involving molecular diffusion are usually based upon the assumption that in closed containers the local velocity v(o) of the so-called center of volume (relative to the fixed container walls) vanishes at every point of the system at every instant of time. This assumption greatly simplifies the interpretation of diffusion data, since by referring the convective-diffusive species flux vector to a local reference frame in which the convective component of the flux is v(o) and hence vanishes, the transport process occurs by molecular diffusion alone. In turn, this furnishes a straightforward, classical linear scheme for determining diffusion coefficients from experimental measurements of transient species concentrations in the closed diffusion cell. Were this not the case, one would have to determine the transient hydrodynamic velocity field v induced by the diffusional process, simultaneous with the solution of the transient species concentration field-a highly nonlinear analysis owing to the coupling between these fields, similar to that occurring in natural convection problems. In this paper, we first give a physical argument proving that v(o) does indeed describe the volume flux in a mixture. Subsequently, we derive a simple expression-valid for isothermal incompressible binary mixtures-connecting the barycentric or mass-average velocity field v to. the volume-average velocity field v(o), i.e., relating the mass and volume flows. Later on, after showing that the generic kinematic argument found in the literature ’proving’ that v(o) vanishes in closed containers is incompatible with hydrodynamics and even internally inconsistent, we expose an alternative, more general development incorporating hydrodynamic effects, one that supplies a (necessary but insufficient) compatibility condition based upon the Navier-Stokes equation. This criterion permits one to identify a priori those classes of systems for which the possibility exists that v(o) = 0. These circumstances are shown to include all laterally unbounded one-dimensional transport processes as well as all unbounded three-dimensional Navier-Stokes flows for which inertial effects are small compared with viscous effects. Such physicochemically low-Reynolds-number flows arise in the latter case in circumstances wherein the Schmidt number v/D (v = kinematic viscosity, D = molecular diffusivity) is large compared with unity, a situation that arises for most liquid-phase diffusion experiments but not for most gases.