Kinetic theory and the principle of material frame-indifference in the mechanics of continuous media

Abstract

The property of Boltzmann's kinetic equation in the presence of Euclidean transformations such as non-stationary rotations and translations of the frame of reference is discussed. It is shown that in the transition from an inertial frame to a rotating frame, in Boltzmann's equation additional intertial terms appear, and in the transition from non-inertial to non-inertial, the equation is invariant under the above transitions. The algorithms, and the results of an approximate method for solving the equation, of the Chapman-Enskog method in particular, are also invariant. The additional terms appear, specifically, in the expressions for stresses and the heat fluxes in Barnett's approximation, and in this sense these expressions are frame-dependent. Because of the condition that Knudsen's numbers should be small, this limits the domain of applicability of one of the basic postulates of the axiomatic theory of continuous media, namely the principle of material frame-indifference (or the principle of material objectivity), in accordance with which the constitutive (determining) relations should be invariant under continuous changes of the frame of reference. The existing papers on this subject are critically analysed.