Dissipation Mechanisms for Fluids and Objects in Relative Motion Described by the Navier-Stokes Equation

This work demonstrates that an additional resistance term should be included in the Navier-Stokes equation when fluids and objects are in relative motion. This is based on an observation that the effect of the microscopic molecular random velocity component parallel to the macroscopic flow direction is neglected. The two components of the random velocity perpendicular to the local mean flow direction are accounted for by the viscous resistance, e.g., by Stokes' law for spherical objects. The relationship between the mean- and the random velocity in the longitudinal direction induces differences in molecular collision velocities and collision frequency rates on the up- and downstream surface areas of the object. This asymmetry therefore induces flow resistance and energy dissipation. The flow resistance resulting from the longitudinal momentum transfer mode is referred to as thermal resistance and is quantified by calculating the net difference in pressure up- and downstream the surface areas of a sphere using a particle velocity distribution that obeys Boltzmann's transport equation. It depends on the relative velocity between the fluid and the object, the number density and the molecular fluctuation statistics of the fluid, and the area of the object and the square root of the absolute temperature. Results show that thermal resistance is dominant compared to viscous resistance considering water and air in slow relative motion to spherical objects larger than nanometer-size at ambient temperature and pressure conditions. Including the thermal resistance term in the conventional expression for the terminal velocity of spherical objects falling through liquids, the Stokes-Einstein relationship and Darcy's law, corroborates its presence, as modified versions of these equations fit observed data much more closely than the conventional expressions. The thermal resistance term can alternatively resolve d'Alembert's paradox as a finite flow resistance is predicted at both low and high relative fluid-object velocities in the limit of vanishing fluid viscosity.