Kinetic Theory of a Dense Gas of Rough Spheres

After a short discussion on the role of general principles and specific microscopic models in the mechanics of continua, the case of a gas whose molecules are rough spheres is considered in detail. The kinetic equation describing the evolution of the distribution function is shaped after Enskog's equation for a dense gas of rough spheres and hence places no restrictions on the density. The recent modifications on Enskog's approach are also taken into account. The //-theorem and the conservation equations are discussed. In particular it is shown that the stress tensor of such a gas turns out to be, generally speaking, nonsymmetric. Introduction As is well known, in thermodynamics there are essentially two different approaches: one is based on general principles concerning the behaviour of materials imagined as continua and the other taking into account the molecular structure of matter. The philosophy of the first approach can be traced back to Anaxagoras, whose Homeomerias are the ancestors of infinitesimal elements of continuum mechanics, the second to Democritus, who coined the word atom. According to Maxwell [1] "The theory that bodies apparently homogeneous and continuous are so in reality is, in its extreme form, a theory incapable of development. To explain the properties of any substance by this theory is impossible. We can only admit the observed properties of such substance as ultimate facts". It is a fact that continuum theory progresses in spite of this drawback. This is due to the fact that there are certain general principles such as conservation of energy, that are satisfied by all the materials, independent of their constitution. This permitted the birth of a branch of knowledge, thermodynamics, whose results are almost as general as those of Euclidean geometry. But, alas, they apply only to equilibrium states, so that it has been remarked, not without reason, that this classical discipline should be called Thermostatics. Modern theories going under the names of thermodynamics of irreversible processes or continuum thermoJ. Non-Equilib. Thermodyn. , Vol. 11, 1986, Nos. 3/4 Copyright © 1986 Walter de Gruyter · Berlin · New York 146 C. Cercignani • mechanics have extended thermodynamics to the treatment of nonequilibrium situations. The extension, however, is not uniquely dictated by the classical structure. Thus one has introduced special principles to restrict the classes of admissible constitutive law such as the principles of frame indifference. It is one of the main merits of I. Müller [2] to have clearly stated that these principles cannot be applied to all the materials in all the physical situations and, in addition, that even the formulation of second principle of thermodynamics, in the form of Clausius-Duhem inequality, has to be modified by introducing, in place ofq/T(T being the absolute temperature and q the heat flux vector) a more general entropy flux, given by a costitutive relation. The source of examples for Müller's criticism of these principles was the kinetic theory of gases, i.e. the theory of the thermomechanical behavior of gases based on the molecular structure of matter. This theory, founded by Clausius, Maxwell and Boltzmann, had its strenuous supporter in the Austrian scientist, who was for one year student in Berlin. When he wrote his "Lectures on Kinetic Theory", hard days were facing the theory; the theory of "energetics", the official doctrine of thermodynamics at the turn of last century, and his leader Ostwald were launching strong attacks against the theories based on the molecular structure of the matter. This explains the rather cautious opening of Boltzmann's Lectures [3]: "Clausius has made a sharp distinction between the general theory of heat, based essentially on the two fundamentals principles of thermodynamics, and the special theory of heat, which starts out by making a definite assumption that heat is molecular motion and then attemps to construct a more precise description of the nature of this motion. The general theory of heat also requires hypotheses which go beyond the bare facts of nature. Nevertheless, it is obviously less dependent on special assumptions than the special theory and it is desirable and necessary to be able to separate its exposition from that of the special theory, and to show that it is independent of the subjective assumptions of the latter. Clausius has already done this very clearly, and indeed has based the division of his book into two parts on just this principle, and it would be useless to repeat his procedure. Recently, the mutual relations of these two branches of the theory of heat have changed somewhat. Through the exploitation of some very interesting analogies and differences, which the properties of energy exhibit in various phenomena of physics, there has arisen the so-called Energetics, which is unfavorable to the view that heat is molecular motion. This view is in fact unnecessary for the general theory of heat and, as is well known, was not held by Robert Mayer. The further development of Energetics is certainly very important for science; however, up to now its concepts are still rather unclear, and its theorems not very precisely expressed, so that it cannot replace the older theory of heat in dealing with new special cases where the results are not already known. (.) Experience teaches that one will be led to new discoveries almost exclusively by means of special mechanical models. (.) J. Non-Equilib. Thermodyij. , Vol. 11, 1986, Nos. 3/4