Skip to main content
SpringerLink
Log in

Analytical solution of Stokes’ second problem on the behavior of rarefied gas over an oscillating surface

  • Published:
Fluid Dynamics Aims and scope Submit manuscript

Abstract

Stokes’ second problem on the behavior of rarefied gas occupying a half-space is analytically solved. The plane bounding the half-space executes harmonic oscillations. The kinetic equation with the model collision integral in the form of a τ-model is used and the case of diffuse reflection of gas molecules from the wall is considered. The distribution function of gas molecules is constructed and the mass velocity of the gas in the half-space, together with its value directly at the wall, is determined. The drag force acting from the gas on the boundary executing oscillatory motion in its plane is obtained. Moreover, the energy dissipation rate per unit area of the oscillating plate bounding the gas is determined.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. G.G. Stokes, “On the Effect of the Internal Friction of Fluids on the Motion of Pendulums,” Trans. Cambr. Phil. Soc. 9(2), 8 (1851).

    ADS  Google Scholar 

  2. L. Ai and K. Vafai, “An Investigation of Stokes’ Second Problem for Mon-Newtonian Fluids,” Numer. Heat Transfer. Pt. A. Applications 47, 955 (2005).

    Article  ADS  Google Scholar 

  3. M. Khan, Asis Anjum, and C. Fetecau, “On Exact Solutions of Stokes Second Problem for a Burgers’ Fluid. I. The Case γ < λ 2/4,” Zeitschr. Appl. Math. Phys. 61, 697 (2009).

    Article  MathSciNet  Google Scholar 

  4. W.P. Graebel, Engineering Fluid Mechanics, Taylor & Francis, New York (2001).

    Google Scholar 

  5. F. Sharipov and D. Kalempa, “Gas Flow around a Longitudinally Oscillating Plate at Arbitrary Ratio of Collision Frequency to Oscillation Frequency,” in: M.S. Ivanov and A.K. Rebrov (eds.), Rarefied Gas Dynamics: 25th Int. Symp., Novosibirsk, 2007, Novosibirsk (2007), p. 1140

  6. D.M. Karabacak, V. Yakhot, and K.L. Ekinci, “High-Frequency Nanofluids: An Experimental Study Using Nanomechanical Resonators,” Phys. Rev. Lett. 98, 254505 (2007).

    Article  ADS  Google Scholar 

  7. A.N. Cleland and M.L. Roukes, “A Nanometre-Scale Mechanical Electrometer,” Nature 392, 160 (1998).

    Article  ADS  Google Scholar 

  8. E. Steinhell, W. Scherber, M. Seide, and H. Rieger, “Investigation on the Interaction of Gases and Well Defined Solid Surfaces with Respect to Possibilities for Reduction of Aerodynamic Friction and Aerothermal Heating,” in: J.L. Potter (ed.), Rarefied Gas Dynamics, Acad. Press, New York (1977), p. 589.

    Google Scholar 

  9. V.V. Dudko, A.A. Yushkanov, and Yu.I. Yalamov, “Surface Property Effect on Shear Wave Parameters,” Zh. Tekhn. Fiz. 75(4), 135 (2005).

    Google Scholar 

  10. V.V. Dudko, A.A. Yushkanov, and Yu.I. Yalamov, “Shear Wave Generation by a Surface Oscillating in a Gas,” Teplofiz. Vys. Temp. 47, 262 (2009).

    Google Scholar 

  11. V.V. Dudko, “Rarefied Gas Slip along Fixed and Oscillating Surfaces,” Candidate’s Dissertation in Mathematics and Physics, Moscow (2010).

  12. V.A. Akimova, A.V. Latyshev, and A.A. Yushkanov, “Analytical Solution of the Second Stokes Problem on Behaviour of Gas over Oscillation Surface. Pt. 1. Eigenvalues and Eigensolutions,” ArXiv: 1111.3429v1 [math-ph] 15 Nov 2011.

  13. V.A. Akimova, A.V. Latyshev, and A.A. Yushkanov, “Analytical Solution of the Second Stokes Problem on Behaviour of Gas over Oscillation Surface. Pt. 2. Mathematical Apparatus of Solving of Problem,” ArXiv: 1111.5182v1 [math-ph] 22 Nov 2011.

  14. C. Cercignani, Theory and Applications of the Boltzmann Equations, Scott. Acad. Press (1975).

  15. A.V. Latyshev and A.A. Yushkanov, Analytical Methods in Kinetic Theory [in Russian], Moscow (2008).

  16. F.D. Gakhov, Boundary Value Problems [in Russian], Nauka, Moscow (1977).

  17. L.D. Landau and E.M. Lifshitz, Electrodynamics of Continuum Media, Pergamon Press, Oxford (1984).

    Google Scholar 

  18. L.D. Landau and E.M. Lifshitz, Fluid Mechanics, Pergamon Press, Oxford (1987).

    MATH  Google Scholar 

Download references

Authors
  1. V. A. Akimova
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. V. Latyshev
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. A. Yushkanov
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Original Russian Text © V.A. Akimova, A.V. Latyshev, A.A. Yushkanov, 2013, published in Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, 2013, Vol. 48, No. 1, pp. 125–140.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Akimova, V.A., Latyshev, A.V. & Yushkanov, A.A. Analytical solution of Stokes’ second problem on the behavior of rarefied gas over an oscillating surface. Fluid Dyn 48, 109–122 (2013). https://doi.org/10.1134/S0015462813010122

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0015462813010122

Keywords

Search

Navigation