A force acting on an oblate spheroid with discontinuous surface temperature in a slightly rarefied gas

Abstract

An oblate spheroid, the respective hemispheroids of which are kept at different uniform temperatures, placed in a rarefied gas at rest is considered. The explicit formula for the force acting on the spheroid (radiometric force) is obtained for small Knudsen numbers. This is a model of a vane of the Crookes radiometer. The analysis is performed for a general axisymmetric distribution of the surface temperature of the spheroid, allowing abrupt changes. Although the generalized slip flow theory, established by Sone (Rarefied Gas Dynamics, vol. 1, 1969, pp. 243–253), is available for general rarefied gas flows at small Knudsen numbers, it cannot be applied to the present problem because of the abrupt temperature changes. However, if it is combined with the symmetry relations for the linearized Boltzmann equation developed recently by Takata (J. Stat. Phys., vol. 136, 2009, pp. 751–784), one can bypass the difficulty. To be more specific, the force acting on the spheroid in the present problem can be generated from the solution of the adjoint problem to which the generalized slip flow theory can be applied, i.e. the problem in which the same spheroid with a uniform surface temperature is placed in a uniform flow of a rarefied gas. The analysis of the present paper follows this strategy.