Thermophoresis of Axially Symmetric Bodies

Thermophoresis of axially symmetric bodies is investigated to first order in the Knudsen-mimber, Kn. The study is made in the limit where the typical length of the immersed body is small compared to the mean free path. It is shown that in this case, in contrast to what is the case for spherical bodies, the arising thermal force on the body is not in general antiparallel to the temperature gradient. It is also shown that the gas excerts a torque on the body, which in magnitude and direction depends on the body geometry. Equations of motion describing the body movement are derived. Asymptotic solutions are studied. INTRODUCTION Thermophoresis as a phenomenon has been known for a long time, and several authors have approached the problem. For example, Einstein calculated the final velocity of a spherical particle in a heat conducting gas using elementary kinetic theory. A recent review of the phenomenon is given in an article by Sone, see [1]. Another review is given by Talbot, see [2]. The first systematic attempt to describe the thermophoresis phenomenon using kinetic theory is found in an article by Waldmann from 1959, see [4]. That work was made under the assumption that the mean free path of the gas is much larger than the dimension of the body. Further results are found in a variety of articles, but these mostly apply to larger bodies, using asymptotic methods, and often to spheres, cf [3]. In this work, thermophoresis of axially symmetric bodies is considered. The typical macroscopical length L over which the temperature varies is assumed to be much larger than A, the mean free path of the gas. To this end we define the Knudsen number according to Kn = X/L. Thus Kn 0 / dS (1) Here /W describes the incident molecular stream and f^ describes the reflected stream, m is the mass of a gas molecule, and c is the velocity. The surface element is taken to be impermeable, and thus the stream of gas molecules incident to the surface element equals the stream of outgoing molecules. Therefore CP585, Rarefied Gas Dynamics: 22 International Symposium, edited by T. J. Bartel and M. A. Gallis © 2001 American Institute of Physics 0-7354-0025-3/01/$18.00 867 , ö