A Proposal for Fundamental Equations of Dynamics of Gases under High Stress (A Proposal for Statistical Mechanical Treatment of Systems Not in Thermal Equilibrium Associated with Transport Phenomena)

Abstract

In order to derive the fundamental equations of gas dynamics which are valid for the flow at high shearing stress and low gas pressure, an attempt was made to eliminate the assumption that the state concerned departs only slightly from thermal equilibrium, as we frequently find phenomena which cannot be treated under this assumption. First, the state of gas in an intermediate volume element was microscopically represented in terms of the quantities of transport. Considering their averages over an intermediate time interval, an attempt was made to find the statistical character of the system. In other words, under the assumption that the macroscopic observation is possible or that the macroscopic law can be established, it was shown that the Boltzmann-Planck method in a new sense is applicable to the present systems. The new distribution function thus derived, f=exp(α+Σβξcξ+ΣΣγξηcξcη+···) (where c denotes the velocity of a particle), should satisfy the Boltzmann-Maxwell equation. This can be shown by means of Kirkwood's verification accepted in some extended meaning. In solving the B-M equation (hereafter, the B-M equation is the abbreviation of the Boltzmann-Maxwell equation) from this new standpoint, the stress components were taken together with mass density and velocity of mass motion as the independent variables in the zeroth approximation, although usually temperature is taken in place of stress. This method of treatment is analogous to that of Enskog and Chapman, although some important modifications are found in the latter half of the treatment. The fundamental equations of gas dynamics have been derived. Instead of viscosity and thermal conductivity, a new conception arises in the equations. That conception is conductivity of stress.