Advertisement

The motion of a thin liquid layer on the outer surface of a rotating cylinder

  • A. M. MoradEmail author
  • M. Yu. Zhukov
Regular Article

Abstract.

We derive the shallow water equations describing the motion of a thin liquid film on the outer surface of a rotating cylinder. These equations are an analogue of the modified Boussinesq equations describing shallow water flows with constant vorticity. The standard multi-scale methods are employed to construct asymptotic equations in the long-wave approximation. These asymptotic equations are analyzed using the hodograph method. It is found that for the particular case of a dispersionless irrotational flow, the equations describing flows on the outer surface of a cylinder reduce to elliptic equations. Numerical evaluation of the exact solutions obtained shows that the asymptotic equations possess a rich variety of solutions representing various wave patterns.

Keywords

Soliton Free Boundary Shallow Water Equation Boussinesq Equation Asymptotic Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    G. Seiden, P.J. Thomas, Rev. Mod. Phys. 83, 1323 (2011)ADSCrossRefGoogle Scholar
  2. 2.
    J. Ashmore, A.E. Hosoi, H.A. Stone, J. Fluid Mech. 479, 65 (2003)ADSCrossRefzbMATHGoogle Scholar
  3. 3.
    E.S. Benilov, M.S. Benilov, N. Kopteva, J. Fluid Mech. 597, 91 (2008)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    E.S. Benilov, M.S. Benilov, S.G.B. O’Brien, J. Eng. Math. 63, 197 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    G.A. Leslie, S.K. Wilson, B.R. Duffy, J. Fluid Mech. 716, 51 (2013)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    S.T. Thoroddsen, L. Mahadevan, Exp. Fluids 23, 1 (1997)CrossRefGoogle Scholar
  7. 7.
    B.R. Duffy, S.K. Wilson, J. Fluid Mech. 394, 29 (1999)ADSCrossRefzbMATHGoogle Scholar
  8. 8.
    C.J. Noakes, J.R. King, D.S. Riley, Q. J. Mech. Appl. Math. 59, 163 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    L. Preziosi, D.D. Joseph, J. Fluid Mech. 187, 99 (1988)ADSCrossRefzbMATHGoogle Scholar
  10. 10.
    U. Thiele, J. Fluid Mech. 671, 121 (2011)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    M. Villegas-Díaz, H. Power, D.S. Riley, J. Fluid Mech. 541, 317 (2005)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    S.K. Wilson, R. Hunt, B.R. Duffy, Q. J. Mech. Appl. Math. 55, 357 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    G.A. Leslie, S.K. Wilson, B.R. Duffy, Phys. Fluids 23, 062101 (2011)ADSCrossRefGoogle Scholar
  14. 14.
    G.A. Leslie, S.K. Wilson, B.R. Duffy, Q. J. Mech. Appl. Math. 65, 483 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    L.W. Schwartz, D.E. Weidner, J. Eng. Math. 29, 91 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    P.L. Evans, L.W. Schwartz, R.V. Roy, Phys. Fluids 16, 2742 (2004)ADSCrossRefMathSciNetGoogle Scholar
  17. 17.
    A. Acrivos, B. Jin, J. Eng. Math. 50, 99 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    V.V. Pukhnachev, J. Appl. Mech. Tech. Phys. 18, 344 (1977)ADSCrossRefGoogle Scholar
  19. 19.
    M.A. Kelmanson, J. Fluid Mech. 633, 327 (2009)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    M. Chugunova, R.M. Taranet, Int. J. Differ. Equ. 2012, 570283 (2012)Google Scholar
  21. 21.
    E.A. Karabut, J. Appl. Mech. Tech. Phys. 48, 55 (2007)ADSCrossRefGoogle Scholar
  22. 22.
    E.J. Hinch, M.A. Kelmanson, P.D. Metcalfe, Proc. R. Soc. London A 460, 2975 (2004)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    D. Badali, M. Chugunova, D.E. Pelinovsky, S. Pollack, arXiv:1101.3033v1 (2011)
  24. 24.
    D. Takagi, H.E. Huppert, J. Fluid Mech. 647, 221 (2010)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    H.K. Moffatt, J. Mec. 16, 651 (1977)ADSGoogle Scholar
  26. 26.
    K. Seshan (Editor), Handbook of Thin-film deposition techniques, 2nd edition (CRC Press, 2002)Google Scholar
  27. 27.
    G.B. Whithem, Linear and nonlinear wave (John Willey & Sons Inc., New-York, London, Sydney, Toronto, 1974)Google Scholar
  28. 28.
    D. Korteweg, G. de Vries, Philos. Mag. 395, 422 (1985)Google Scholar
  29. 29.
    L. Rayleigh, Proc. R. Soc. A 84, 247 (1910)ADSCrossRefzbMATHGoogle Scholar
  30. 30.
    A.C. Newell, Solitons in mathematics and physics (SIAM, 1985)Google Scholar
  31. 31.
    L.V. Ovsyannikov, N.I. Makarenko, V.I. Nalimov, V.Yu. Liapidevskii, P.I. Plotnikov, I.V. Sturova, V.I. Bukreev, V.A. Vladimirov, Nonlinear Problems in the Theory of Surface and Internal Waves (Nauka, Novosibirsk, 1985) (in Russian)Google Scholar
  32. 32.
    B.A. Trubnikov, S.K. Zhdanov, S.M. Zverev, Hydrodynamics of Unstable Media: General Theory and Applied Problems, 1st edition (CRC Press LLC, 1996)Google Scholar
  33. 33.
    R.K. Dodd, J.C. Eilbeck, J.D. Gibbon, H.C. Morris, Solitons and Nonlinear Wave Equations (Academic Press, 1984)Google Scholar
  34. 34.
    M.Yu. Zhukov, A.M. Morad, arXiv:1303.2327 (2013)
  35. 35.
    V.A. Vladimirov, M.Yu. Zhukov, V.I. Yudovich, P.V. Denissenko, Matem. Mod. 13, 27 (2001)zbMATHGoogle Scholar
  36. 36.
    M.A. Goldshtik, V.N. Shtern, N.I. Yavorsky, Viscous flows with paradoxical features (Novosibirsk, Nauka, 1989)Google Scholar
  37. 37.
    E.V. Shiryaeva, V.A. Vladimirov, M.Yu. Zhukov, Phys. Rev. E 80, 041603 (2009)ADSCrossRefGoogle Scholar
  38. 38.
    A.M. Abourabia, K.M. Hassan, A.M. Morad, Chaos Solitons Fractals 42, 1170 (2009)ADSCrossRefzbMATHGoogle Scholar
  39. 39.
    P.A. Clarkson, A.S. Fokas, M.J. Ablowitz, SIAM J. Appl. Math. 49, 1188 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  40. 40.
    S. Senashov, A. Yakhno, SIGMA 8, 071 (2012)MathSciNetGoogle Scholar
  41. 41.
    B.L. Rozhdestvenskii, N.N. Yanenko, Systems of quasilinear equations and their application to gas dynamics (American Mathematical Society, Providence, 1983)Google Scholar
  42. 42.
    M. Abramowitz, I. Stegun (Editors), Handbook of mathematical functions (National Bureau of Standards, 1972)Google Scholar
  43. 43.
    E.V. Shiryaeva, M.Yu. Zhukov, arXiv:1410.2832 (2014)

Copyright information

© Società Italiana di Fisica and Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceMenoufiya UniversityMenoufiyaEgypt
  2. 2.Southern Federal UniversityRostov-on-DonRussia

Personalised recommendations