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Interfacial fluid instabilities and Kapitsa pendula

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Abstract.

The onset and development of instabilities is one of the central problems in fluid mechanics. Here we develop a connection between instabilities of free fluid interfaces and inverted pendula. When acted upon solely by the gravitational force, the inverted pendulum is unstable. This position can be stabilized by the Kapitsa phenomenon, in which high-frequency low-amplitude vertical vibrations of the base creates a fictitious force which opposes the gravitational force. By transforming the dynamical equations governing a fluid interface into an appropriate pendulum-type equation, we demonstrate how stability can be induced in fluid systems by properly tuned vibrations. We construct a “dictionary”-type relationship between various pendula and the classical Rayleigh-Taylor, Kelvin-Helmholtz, Rayleigh-Plateau and the self-gravitational instabilities. This makes several results in control theory and dynamical systems directly applicable to the study of tunable fluid instabilities, where the critical wavelength depends on the external forces or the instability is suppressed entirely. We suggest some applications and instances of the effect ranging in scale from microns to the radius of a galaxy.

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References

  1. M.C. Cross, P.C. Hohenberg, Rev. Mod. Phys. 65, 851 (1993)

    Article  ADS  Google Scholar 

  2. S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability (Dover Publications, Inc., Mineola, NY, 1981).

  3. P.G. Drazin, Introduction to Hydrodynamic Stability (Cambridge University Press, Cambridge, UK, 2002)

  4. H.J. Kull, Phys. Rep. 206, 197 (1991)

    Article  ADS  Google Scholar 

  5. K. Baldwin, M. Scase, R. Hill, Nat. Sci. Rep. 5, 11706 (2015)

    Article  ADS  Google Scholar 

  6. A. Poehlmann, R. Richter, I. Rehberg, J. Fluid Mech. 732, 1 (2013)

    Article  Google Scholar 

  7. X. Chen, F. Eliot, J. Fluid Mech. 560, 395 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  8. Mathew J. Russo, Paul H. Steen, Phys. Fluids A 1, 1926 (1989)

    Article  ADS  Google Scholar 

  9. M. Marr-Lyon, D. Thiessen, P. Marston, J. Fluid Mech. 351, 345 (1997)

    Article  ADS  Google Scholar 

  10. M. Marr-Lyon, D. Thiessen, P. Marston, Phys. Rev. Lett. 86, 2293 (2001)

    Article  ADS  Google Scholar 

  11. N. Bertin, R. Wunenburger, E. Brasselet, J.-P. Delville, Phys. Rev. Lett. 105, 164501 (2010)

    Article  ADS  Google Scholar 

  12. D.V. Lyubimov, A.A. Cherepanov, Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza 6, 3 (1991)

    ADS  Google Scholar 

  13. G. Gandikota, D. Chatain, T. Lyubimova, D. Beysens, Phys. Rev. E 89, 063003 (2014)

    Article  ADS  Google Scholar 

  14. V. Shevtsova, Y.A. Gaponenko, V. Yasnou, A. Mialdun, A. Nepomnyashchy, J. Fluid Mech. 795, 409 (2016)

    Article  ADS  Google Scholar 

  15. V.I. Arnold, Mathematical Methods of Classical Mechanics (Springer, New York, NY, 2010)

  16. A. Stephenson, Mem. Proc. Manchester Lit. Philos. Soc. 52, 1 (1908)

    Google Scholar 

  17. A. Stephenson, Philos. Mag. 17, 765 (1909)

    Article  Google Scholar 

  18. P.L. Kapitsa, Dynamical stability of a pendulum when its point of suspension vibrates, Collected Papers of P.L. Kapitsa, Vol. II (Pergamon Press, 1965) pp. 714--725

  19. P.L. Kapitsa, Pendulum with a vibrating suspension, Collected Papers of P.L. Kapitsa, Vol. II (Pergamon Press, 1965) pp. 726--737

  20. V.N. Chelomei, Dokl. Akad. Nauk SSSR 110, 345 (1983)

    Google Scholar 

  21. M. Levi, SIAM Rev. 30, 639 (1988)

    Article  MathSciNet  Google Scholar 

  22. M. Levi, H. Broer, Arch. Ration. Mech. Anal. 131, 225 (1995)

    Article  Google Scholar 

  23. M. Levi, Int. J. Bifurcat. Chaos 15, 2747 (2005)

    Article  Google Scholar 

  24. W. Paul, Rev. Mod. Phys. 62, 531 (1990)

    Article  ADS  Google Scholar 

  25. J. Holyst, W. Wojciechowski, Physica A 324, 388 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  26. T.B. Benjamin, F. Ursell, Proc. R. Soc. Lond. A 225, 505 (1954)

    Article  ADS  Google Scholar 

  27. F. Troyon, R. Gruber, Phys. Fluids 14, 2069 (1971)

    Article  ADS  Google Scholar 

  28. G.H. Wolf, Z. Phys. 227, 291 (1969)

    Article  ADS  Google Scholar 

  29. G.H. Wolf, Phys. Rev. Lett. 24, 444 (1970)

    Article  ADS  Google Scholar 

  30. Inga Koszalka, Vibrating pendulum and stratified fluids, in Geophysical Fluid Dynamics Proceedings Volumes (WHOI, 2005)

  31. A. Weathers, B. Folie, B. Liu, S. Childress, J. Zhang, J. Fluid Mech. 650, 415 (2010)

    Article  ADS  Google Scholar 

  32. L.D. Landau, E.M. Lifshitz, Mechanics, 3rd edition (Pergamon Press, Oxford, 1986)

  33. A. Seyranian, A. Mailybaev, Multiparameter Stability Theory with Mechanical Applications (World Scientific, NJ, 2004)

  34. A. Mailybaev, A. Seyranian, J. Sound Vibrat. 323, 1016 (2009)

    Article  ADS  Google Scholar 

  35. H. Broer, I. Hoveijn, M. van Noort, C. Simó, G. Vegter, J. Dyn. Differ. Equ. 16, 897 (2004)

    Article  Google Scholar 

  36. H.W. Broer, I. Hoveijn, M. van Noort, G. Vegter, J. Differ. Equ. 157, 120 (1999)

    Article  ADS  Google Scholar 

  37. J. Wesson, Phys. Fluids 13, 762 (1970)

    Article  ADS  Google Scholar 

  38. K. Kumar, L. Tuckerman, J. Fluid Mech. 279, 49 (1994)

    Article  ADS  MathSciNet  Google Scholar 

  39. D. Horsley, L. Forbes, J. Eng. Math. 79, 13 (2013)

    Article  Google Scholar 

  40. P. Chen, Z. Luo, S. Güven, S. Tasoglu, D.V. Ganesan, A. Weng, U. Demirci, Adv. Mater. 26, 5936 (2014)

    Article  Google Scholar 

  41. R. Krechetnikov, J.E. Marsden, Physica D 214, 25 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  42. D.J. Acheson, T. Mullin, Nat. Corresp. 366, 215 (1993)

    Google Scholar 

  43. D.J. Acheson, Proc. R. Soc. Lond. A 443, 239 (1993)

    Article  ADS  Google Scholar 

  44. T. Mullin, A. Champneys, B. Fraser, J. Galan, D. Acheson, Proc. R. Soc. Lond. A 459, 539 (2003)

    Article  ADS  Google Scholar 

  45. B. Fraser, A. Champneys, Proc. R. Soc. Lond. A 458, 1353 (2002)

    Article  ADS  Google Scholar 

  46. M.-R. Alam, Y. Liu, D.K.P. Yue, J. Fluid Mech. 624, 191 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  47. H. Jeffreys, Proc. R. Soc. Lond. A 107, 189 (1925)

    Article  ADS  Google Scholar 

  48. A. Jenkins, Phys. Rep. 525, 167 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  49. M. Ruijgrok, A. Tondl, F. Verhulst, ZAMM 73, 255 (1993)

    Article  ADS  Google Scholar 

  50. A. Bloch, P. Hagerty, A.G. Rojo, M.I. Weinstein, J. Stat. Phys. 115, 1073 (2004)

    Article  ADS  Google Scholar 

  51. G.K. Batchelor, An Introduction to Fluid Mechanics, 3rd edition (Cambridge University Press, Cambridge, 1976)

  52. D. Merkt, A. Pototsky, M. Bestehorn, U. Thiele, Phys. Fluids 17, 064104 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  53. W.Y. Jiang, S.P. Lin, Phys. Fluids 17, 054105 (2005)

    Article  ADS  Google Scholar 

  54. J. Atencia, D. Beebe, Nature 437, 648 (2005)

    Article  ADS  Google Scholar 

  55. P. Trinh, H. Kim, N. Hammoud, P. Howell, S.J. Chapman, H. Stone, Phys. Fluids 26, 051704 (2014)

    Article  ADS  Google Scholar 

  56. E.R. Harrison, Phys. Rev. D 1, 2726 (1970)

    Article  ADS  Google Scholar 

  57. Ya.B. Zel’dovich, Mon. Not. R. Astron. Soc. 160, 1P (1972)

    Article  ADS  Google Scholar 

  58. G.F. Smoot et al., Astrophys. J. 396, L1 (1992)

    Article  ADS  Google Scholar 

  59. Carl H. Gibson, Primordial viscosity, diffusivity, Reynolds number and turbulence in the beginnings of gravitational structure formation, PhD Thesis, UCSD, San Diego, CA, 1996

  60. K. Subramanian, The origin, evolution and signatures of primordial magnetic fields, arXiv:1504.02311v1 (2015)

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Authors and Affiliations

  1. School of Engineering, Brown University, 02912, Providence, RI, USA

    Madison S. Krieger

  2. Program for Evolutionary Dynamics, Harvard University, 02138, Cambridge, MA, USA

    Madison S. Krieger

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  1. Madison S. Krieger
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Correspondence to Madison S. Krieger.

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Krieger, M.S. Interfacial fluid instabilities and Kapitsa pendula. Eur. Phys. J. E 40, 67 (2017). https://doi.org/10.1140/epje/i2017-11556-x

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