Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-23T16:47:20.608Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  07 May 2014

Kazuo Aoki ,
Shigeru Takata  and
Tatsunori Tomota
Kazuo Aoki
Affiliation:
Department of Mechanical Engineering and Science and Advanced Research Institute of Fluid Science and Engineering, Graduate School of Engineering, Kyoto University, Kyoto 615-8540, Japan
Shigeru Takata*
Affiliation:
Department of Aeronautics and Astronautics and Advanced Research Institute of Fluid Science and Engineering, Graduate School of Engineering, Kyoto University, Kyoto 615-8540, Japan
Tatsunori Tomota
Affiliation:
Department of Mechanical Engineering and Science and Advanced Research Institute of Fluid Science and Engineering, Graduate School of Engineering, Kyoto University, Kyoto 615-8540, Japan
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

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.

JFM classification

Rarefied Gas Flow: Kinetic theory Micro-/Nano-fluid dynamics: Non-continuum effects Rarefied Gas Flow: Rarefied Gas Flow
Type
Papers
Information
Journal of Fluid Mechanics , Volume 748 , 10 June 2014 , pp. 712 - 730
Copyright
© 2014 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

‡

Present address: Toyota Central R&D Labs., Inc., Nagakute, Aichi 480-1192, Japan.

References

Aoki, K., Bardos, C., Dogbe, C. & Golse, F. 2001a A note on the propagation of boundary induced discontinuities in kinetic theory. Math. Models Meth. Appl. Sci. 11, 1581–1595.Google Scholar
Aoki, K., Takata, S., Aikawa, H. & Golse, F. 2001b A rarefied gas flow caused by a discontinuous wall temperature. Phys. Fluids 13, 2645–2661.Google Scholar
Bhatnagar, P. L., Gross, E. P. & Krook, M. 1954 A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems. Phys. Rev. 94, 511–525.CrossRefGoogle Scholar
Chen, S., Xu, K. & Lee, C. 2012 The dynamic mechanism of a moving Crookes radiometer. Phys. Fluids 24, 111701.CrossRefGoogle Scholar
Chernyak, V. & Beresnev, S. 1993 Photophoresis of aerosol particles. J. Aerosol Sci. 24, 857–866.CrossRefGoogle Scholar
Hidy, G. M. & Brock, J. R. 1970 The Dynamics of Aerocolloidal Systems, chap. VI. Pergamon.Google Scholar
Kennard, E. H. 1938 Kinetic Theory of Gases. McGraw-Hill.Google Scholar
Ketsdever, A., Gimelshein, N., Gimelshein, S. & Selden, N. 2012 Radiometric phenomena: from the 19th to the 21st century. Vacuum 86, 1644–1662.CrossRefGoogle Scholar
Ohwada, T., Sone, Y. & Aoki, K. 1989 Numerical analysis of the shear and thermal creep flows of a rarefied gas over a plane wall on the basis of the linearized Boltzmann equation for hard-sphere molecules. Phys. Fluids A 1, 1588–1599.Google Scholar
Payne, L. E. & Pell, W. H. 1960 The Stokes flow problem for a class of axially symmetric bodies. J. Fluid Mech. 7, 529–549.CrossRefGoogle Scholar
Preining, O. 1966 Photophoresis. In Aerosol Science (ed. Davies, C. N.), pp. 111–135. Academic.Google Scholar
Sone, Y. 1966 Thermal creep in rarefied gas. J. Phys. Soc. Japan 21, 1836–1837.CrossRefGoogle Scholar
Sone, Y. 1969 Asymptotic theory of flow of rarefied gas over a smooth boundary I. In Rarefied Gas Dynamics (ed. Trilling, L. & Wachman, H. Y.), vol. 1, pp. 243–253. Academic.Google Scholar
Sone, Y. 1971 Asymptotic theory of flow of rarefied gas over a smooth boundary II. In Rarefied Gas Dynamics (ed. Dini, D.), vol. 2, pp. 737–749. Editrice Tecnico Scientfica.Google Scholar
Sone, Y. 1972 Flow induced by thermal stress in rarefied gas. Phys. Fluids 15, 1418–1423.Google Scholar
Sone, Y. 1991 Asymptotic theory of a steady flow of a rarefied gas past bodies for small Knudsen numbers. In Advances in Kinetic Theory and Continuum Mechanics (ed. Gatignol, R. & Soubbaramayer,  ), pp. 19–31. Springer.Google Scholar
Sone, Y. 2000 Flows induced by temperature fields in a rarefied gas and their ghost effect on the behavior of a gas in the continuum limit. Annu. Rev. Fluid Mech. 32, 779–811.Google Scholar
Sone, Y. 2002 Kinetic Theory and Fluid Dynamics. Birkhäuser, Supplementary Notes and Errata: Kyoto University Research Information Repository (http://hdl.handle.net/2433/66099).CrossRefGoogle Scholar
Sone, Y. 2007 Molecular Gas Dynamics: Theory, Techniques, and Applications. Birkhäuser, Supplementary Notes and Errata: Kyoto University Research Information Repository (http://hdl.handle.net/2433/66098).CrossRefGoogle Scholar
Sone, Y. & Aoki, K. 1977 Forces on a spherical particle in a slightly rarefied gas. In Rarefied Gas Dynamics (ed. Potter, J. L.), pp. 417–433. AIAA.Google Scholar
Sone, Y. & Takata, S. 1992 Discontinuity of the velocity distribution function in a rarefied gas around a convex body and the S layer at the bottom of the Knudsen layer. Transp. Theory Stat. Phys. 21, 501–530.Google Scholar
Sone, Y. & Tanaka, S. 1980 Thermal stress slip flow induced in rarefied gas between non-coaxial circular cylinders. In Theoretical and Applied Mechanics (ed. Rimrott, F. P. J. & Tabarrok, B.), pp. 405–416. North-Holland.Google Scholar
Taguchi, S. & Aoki, K. 2011 Numerical analysis of rarefied gas flow induced around a flat plate with a single heated side. In Rarefied Gas Dynamics (ed. Levin, D. A., Wysong, I. J. & Garcia, A. L.), pp. 790–795. AIP.Google Scholar
Taguchi, S. & Aoki, K. 2012 Rarefied gas flow around a sharp edge induced by a temperature field. J. Fluid Mech. 694, 191–224.Google Scholar
Takata, S. 2009a Symmetry of the linearized Boltzmann equation and its application. J. Stat. Phys. 136, 751–784.Google Scholar
Takata, S. 2009b Symmetry of the linearized Boltzmann equation II. Entropy production and Onsager–Casimir relation. J. Stat. Phys. 136, 945–983.Google Scholar
Takata, S. & Hattori, M. 2012a Asymptotic theory for the time-dependent behavior of a slightly rarefied gas over a smooth solid boundary. J. Stat. Phys. 147, 1182–1215.Google Scholar
Takata, S. & Hattori, M. 2012b On the second-order slip and jump coefficients for the general theory of slip flow. In 28th International Conference on Rarefied Gas Dynamics 2012 (ed. Mareschal, M., Santos, A. & Lafita, A. T.), AIP Conf. Proc., vol. 1501, pp. 59–66. AIP.Google Scholar
Takata, S. & Sone, Y. 1995 Flow induced around a sphere with a non-uniform temperature in a rarefied gas, with application to the drag and thermal force problems of a spherical particle with an arbitrary thermal conductivity. Eur. J. Mech. (B/Fluids) 14, 487–518.Google Scholar
Welander, P. 1954 On the temperature jump in a rarefied gas. Ark. Fys. 7, 507–553.Google Scholar