Integral formulation of transport phenomena in systems affected by external fields

Abstract

Classical areas of physics such as fluid dynamics continue to present challenges that are by no means less demanding than those encountered at the post-Newtonian frontiers of physics that have been explored since the beginning of the twentieth century, Steven Weinberg was known to have said in a New York Times interview that we know more about about the atomic nucleus than about the behavior of a cubic centimeter of turbulent water.
The traditional approach to studying the motion of fluids starts with the continuum hypothesis and leads to the diffusion equation, the Navier-Stokes equation, and the energy transport equation. It has not lead to an explanation of the origin of turbulence and it is incomplete because there is no prescription for the the pressure tensor for a system not in equilibrium.
In a recent paper, Muriel and Dresden [Physica D 101, 299] proposed to reformulate hydrodynamics using integral equations. In doing so Muriel and Dresden avoided the problem on the definition ofthe pressure tensor encountered in continuum hydrodynamics. They also claimed that doing so resolves the problem of divergent transport coefficients and allows one to study the time evolution of energy, momentum and mass transport. Using projection techniques, they obtained compact expressions for evaluating the time-dependent density, momentum current, and local energy density of a classical many body system.
In this paper, we extend Muriel and Dresden's approach to systems affected by external fields. Our immediate motivation is the analysis of flow around obstacles, but apart from this an extension of the integral formulation to systems affected by external fields can be useful in the study of Rayleigh-Benard flow, gravity driven flow and electrical fluids.