Skip to main content
SpringerLink
Log in
SpringerLink
Log in

Motion of a Rough Disc in Newtonian Aerodynamics

  • Conference paper
  • First Online:
Optimization in the Natural Sciences (EmC-ONS 2014)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 499))

Included in the following conference series:

Abstract

Dynamics of a rough disc in a rarefied medium is considered. We prove that any finite rectifiable curve can be approximated in the Hausdorff metric by trajectories of centers of rough discs provided that the parameters of the system are carefully chosen. To control the dynamics of the disc, we use the so-called inverse Magnus effect which causes deviation of the trajectory of a spinning body. We study the so-called response laws for scattering billiards e.g. relationship between the velocity of incidence of a particle and that of reflection. We construct a special family of such laws that is weakly dense in the set of symmetric Borel measures. Then we find a shape of cavities that provides selected law of reflections. We write down differential equations that describe motions of rough discs. We demonstrate how a given curve can be approximated by considered trajectories.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Brock, F., Ferone, V., Kawohl, B.: A symmetry problem in the calculus of variations. Calc. Var. 4, 593–599 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bucur, D., Buttazzo, G.: Variational Methods in Shape Optimization Problems, p. 216. Birkhäuser, Boston (2005)

    MATH  Google Scholar 

  3. Buttazzo, G., Ferone, V., Kawohl, B.: Minimum problems over sets of concave functions and related questions. Math. Nachr. 173, 71–89 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  4. Buttazzo, G., Kawohl, B.: On Newton’s problem of minimal resistance. Math. Intell. 15, 7–12 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  5. Comte, M., Lachand-Robert, T.: Newton’s problem of the body of minimal resistance under a single-impact assumption. Calc. Var. Partial Differ. Equ. 12, 173–211 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  6. Lachand-Robert, T., Oudet, E.: Minimizing within convex bodies using a convex hull method. SIAM J. Optim. 16, 368–379 (2006)

    Article  MathSciNet  Google Scholar 

  7. Lachand-Robert, T., Peletier, M.A.: Newton’s problem of the body of minimal resistance in the class of convex developable functions. Math. Nachr. 226, 153–176 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  8. Lachand-Robert, T., Peletier, M.A.: An example of non-convex minimization and an application to Newton’s problem of the body of least resistance. Ann. Inst. H. Poincaré, Anal. Non Lin. 18, 179–198 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  9. Plakhov, A.: Newton’s problem of minimal resistance for bodies containing a half-space. J. Dynam. Control Syst. 10, 247–251 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  10. Plakhov, A.: Optimal roughening of convex bodies. Canad. J. Math. 64, 1058–1074 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  11. Aleksenko, A., Plakhov, A.: Bodies of zero resistance and bodies invisible in one direction. Nonlinearity 22, 1247–1258 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  12. Plakhov A., Exterior Billiards. Systems with Impacts Outside Bounded Domains, p. xiv+284. Springer, New York (2012)

    Google Scholar 

  13. Wolf, E., Habashy, T.: Invisible bodies and uniqueness of the inverse scattering problem. J. Modern Optics 40, 785–792 (1993)

    Article  Google Scholar 

  14. Cercignani, C.: Rarified Gas Dynamics. From Basic Concepts to Actual Calculations. Cambridge University Press, Cambridge (2000)

    Google Scholar 

  15. Kosuge, S., Aoki, K., Takata, S., Hattori, R., Sakai, D.: Steady flows of a highly rarefied gas induced by nonuniform wall temperature. Phys. Fluids 23, 030603 (2011)

    Article  Google Scholar 

  16. Muntz, E.P.: Rarefied gas dynamics. Annu. Rev. Fluid Mech. 21, 387–422 (1989)

    Article  MathSciNet  Google Scholar 

  17. Rudyak, V.Y.: Derivation of equations of motion of a slightly rarefied gas around highly heated bodies from Boltzmann’s equation. J. Appl. Mech. Tech. Phy. 14(5), 646–649 (1973)

    Article  Google Scholar 

  18. Bunimovich, A.I.: Relations between the forces on a body moving in a rarefied gas in a light flux and in a hypersonic Newtonian stream. Fluid Dyn. 8(4), 584–589 (1973)

    Article  Google Scholar 

  19. Bunimovich, A.I., Kuz’menko, V.I.: Aerodynamic and thermal characteristics of three-dimensional star-shaped bodies in a rarefied gas. Fluid Dyn. 18(4), 652–654 (1983)

    Article  Google Scholar 

  20. Ivanov, S.G., Yanshin, A.M.: Forces and moments acting on bodies rotating around a symmetry axis in a free molecular flow. Fluid Dyn. 15, 449–453 (1980)

    Article  MATH  Google Scholar 

  21. Weidman, P.D., Herczynski, A.: On the inverse Magnus effect in free molecular flow. Phys. Fluids 16, L9–L12 (2004)

    Article  MathSciNet  Google Scholar 

  22. Borg, K.I., Söderholm, L.H.: Orbital effects of the Magnus force on a spinning spherical satellite in a rarefied atmosphere. Eur. J. Mech. B/Fluids 27, 623–631 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  23. Borg, K.I., Söderholm, L.H., Essénm, H.: Force on a spinning sphere moving in a rarefied gas. Phys. Fluids 15, 736–741 (2003)

    Article  Google Scholar 

  24. Plakhov, A.: Newton’s problem of the body of minimum mean resistance. Sbornik Math. 195, 1017–1037 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  25. Plakhov, A.: Billiards and two-dimensional problems of optimal resistance. Arch. Ration. Mech. Anal. 194, 349–382 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  26. Plakhov, A.: Billiard scattering on rough sets: Two-dimensional case. SIAM J. Math. Anal. 40, 2155–2178 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  27. Plakhov, A., Tchemisova, T., Gouveia, P.: Spinning rough disk moving in a rarefied medium. Proc. R. Soc. A. 466, 2033–2055 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  28. Bunimovich, L.A.: Mushrooms and other billiards with divided phase space. Chaos 11, 802–808 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  29. Porter, M.A., Lansel, S.: Mushroom Billiards. Not. AMS 53, 334–337 (2006)

    MATH  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was supported by Russian Foundation for Basic Researches under Grants 14-01-00202-a and 15-01-03797-, by Saint-Petersburg State University under Thematic Plans 6.0.112.2010 and 6.38.223.2014, by FEDER funds through COMPETE – Operational Programme Factors of Competitiveness (Programa Operacional Factores de Competitividade) and by Portuguese funds through the Center for Research and Development in Mathematics and Applications (CIDMA) from the “Fundação para a Ciência e a Tecnologia” (FCT), cofinanced by the European Community Fund FEDER/POCTI under FCT research projects (PTDC/MAT/113470/2009 and PEst-C/MAT/UI4106/2011 with COMPETE number FCOMP-01-0124-FEDER-022690). The author is grateful to Prof. Alexandre Plakhov from University of Aveiro for his ideas, remarks and corrections.

Author information

Authors and Affiliations

  1. Department of Mathematics, Center for Research and Development in Mathematics and Applications, University of Aveiro, Aveiro, 3810-193, Portugal

    Sergey Kryzhevich

  2. Faculty of Mathematics and Mechanics, Saint-Petersburg State University, 28, Universitetskiy Pr., 198503, Peterhof, Saint-Petersburg, Russia

    Sergey Kryzhevich

Authors
  1. Sergey Kryzhevich
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sergey Kryzhevich .

Editor information

Editors and Affiliations

  1. Dept. of Mathematics, University of Aveiro, Aveiro, Portugal

    Alexander Plakhov

  2. University of Aveiro, Aveiro, Portugal

    Tatiana Tchemisova

  3. University of Aveiro, Aveiro, Portugal

    Adelaide Freitas

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

Kryzhevich, S. (2015). Motion of a Rough Disc in Newtonian Aerodynamics. In: Plakhov, A., Tchemisova, T., Freitas, A. (eds) Optimization in the Natural Sciences. EmC-ONS 2014. Communications in Computer and Information Science, vol 499. Springer, Cham. https://doi.org/10.1007/978-3-319-20352-2_1

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-20352-2_1

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-20351-5

  • Online ISBN: 978-3-319-20352-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation