Molecular theory of barycentric velocity: monatomic fluids

The notion of barycentric velocity appears in irreversible thermodynamics and fluid mechanics, in which it is a field variable obeying the hydrodynamic equations or, more specifically, the momentum balance equation, which is coupled to the rest of hydrodynamic equations. Therefore, its behavior is not known until the hydrodynamic equations are solved for the flow problem of interest. Unlike diffusion fluxes, heat fluxes, or stresses, it does not have its own constitutive relation similar to Fick's law, Fourier's law, and Newton's law of viscosity. In this work, the constitutive equation is derived for it. In parallel to the phenomenological notion of barycentric velocity, the notion of mean fluid velocity appears in statistical mechanics of irreversible dynamic processes according to the theory of Irving and Kirkwood [J. Chem. Phys. 18, 817 (1950)], and plays the same role of the phenomenological counterpart. In this work, we investigate the statistical mechanical meanings of the mean fluid velocity of a fluid in flow beyond its formal connection with the barycentric velocity. We show that it consists of two components; the center-of-gravity velocity of the packet of fluid molecules, which may be identified with the barycentric velocity in the phenomenological theory, and the diffusive contribution of its collective modes relative to the center of gravity. If the fluid is uniform in space or if the packet of fluid mass is rigid, the diffusive component vanishes. The statistical mechanical (molecular theory) formula for the mean fluid velocity provides the constitutive relation for it in terms of density and temperature gradients present in the fluid in flow. The constitutive relation obtained for the mean fluid velocity can be an important component in the theory of transport processes in liquids. Its significance to fluid mechanics is briefly discussed.