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Pramana

, 93:82 | Cite as

A novelty to the nonlinear rotating Rayleigh–Taylor instability

  • Yusry O El-DibEmail author
  • Galal M Moatimid
  • Amal A Mady
Article
  • 65 Downloads

Abstract

This paper presents a novel approach for studying the nonlinear Rayleigh–Taylor instability (RTI). The system deals with two rotating superposed infinite hydromagnetic Darcian flows through porous media under the influence of a uniform tangential magnetic field. The field allows the presence of currents on the surface of separation. The appropriate linear governing equations are solved and confirmed with the corresponding nonlinear boundary conditions. A nonlinear characteristic of the surface deflection is deducted. Away from the traditional techniques of the stability analysis, the work introduces a new one. The analysis depends mainly on the homotopy perturbation method (HPM). To achieve an analytical approximate periodic solution of the surface deflection, the secular terms are removed. This cancellation resulted in well-known amplitude equations. These equations are utilised to achieve stability criteria of the system. Therefore, the stability configuration is exercised in linear as well as nonlinear approaches. The mathematical procedure adopted here is simple, promising and powerful. The method may be used to treat more complicated nonlinear differential equations that arise in science, physics and engineering applications. A numerical calculation is performed to graph the implication of various parameters on the stability picture. In addition, for more convenience, the surface deflection is depicted.

Keywords

Rayleigh–Taylor instability rotating flow porous media magnetic fluids homotopy perturbation method 

PACS Nos

47.20.Ky 47.20.−k 02.30.Mv 02.30.Jr 12.38.Bx 

References

  1. 1.
    D J Lewis, Proc. R. Soc. London dynamic A 202(1086), 81 (1950)ADSGoogle Scholar
  2. 2.
    S Chandrasekhar, Hydrodynamic and hydromagnetic stability (Oxford University Press, Oxford, 1961)zbMATHGoogle Scholar
  3. 3.
    A R Piriz, J Sanz and L F Ibañez, Phys. Plasmas 4, 1117 (1997)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    A R Piriz, O D Cortăzar and J J Lŏpez Cela, Am. J. Phys. 74(12), 1095 (2006)ADSCrossRefGoogle Scholar
  5. 5.
    A H Nayfeh, J. Appl. Mech. 98, 584 (1976)CrossRefGoogle Scholar
  6. 6.
    A A Mohamed, Y O El-Dib and A A Mady, Chaos Solitons Fractals 14, 1027 (2002)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    Y O El-Dib, J. Magn. Magn. Mater. 260, 1 (2003)ADSCrossRefGoogle Scholar
  8. 8.
    F A I Dullien, Porous media (Academic Press, New York, 1992)zbMATHGoogle Scholar
  9. 9.
    H H Bau, Phys. Fluids 25(10), 1719 (1982)ADSCrossRefGoogle Scholar
  10. 10.
    Y O El-Dib and A Y Ghaly, Chaos Solitons Fractals 18(1), 55 (2003)ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    R C Sharma and P Kumar, Indian J. Pure Appl. Math. 24(9), 563 (1993)Google Scholar
  12. 12.
    G M Moatimid, M H Obied Allah and M A Hassan, Phys. Plasmas 20(10), 102111 (2013)ADSCrossRefGoogle Scholar
  13. 13.
    R E Rosensweig, Ferrohydrodynamics (Cambridge University Press, Cambridge, 1985)Google Scholar
  14. 14.
    R E Zelazo and J R Melcher, J. Fluid Mech. 39(1), 1 (1969)ADSCrossRefGoogle Scholar
  15. 15.
    S K Malik and M Singh, Q. Appl. Math. 47(1), 59 (1989)MathSciNetGoogle Scholar
  16. 16.
    R Kant and S K Singh, Phys. Fluids 28(2), 3534 (1985)ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    A R Elhefnawy, Int. J. Theor. Phys. 31(8), 1505 (1992)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Y O El-Dib, J. Plasma Phys. 51(1), 1 (1994)ADSCrossRefGoogle Scholar
  19. 19.
    A R F Elhefnawy, M A Mahmoud, M A A Mahmoud and G M Khedr, Can. Appl. Math. Q. 12(3), 323 (2004)MathSciNetGoogle Scholar
  20. 20.
    P K Bhatia and R P Mathur, Z. Naturforsch. 60a, 484 (2005)Google Scholar
  21. 21.
    S Chandrasekhar, Astrophys. J. 119, 7 (1954),  https://doi.org/10.1086/145790 ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    M M Scase, K A Baldwin and R J A Hill, Phys. Rev. Fluids 2(2), 024801 (2017)ADSCrossRefGoogle Scholar
  23. 23.
    P K Sharma, R P Prajapti and R K Chhajlani, Acta Phys. Pol. A 118(4), 576 (2010)CrossRefGoogle Scholar
  24. 24.
    R P Prajapati, Phys. Plasmas 23, 022106 (2016)ADSCrossRefGoogle Scholar
  25. 25.
    J J Tao, X T He and W H Ye, Phys. Rev. E 87, 013001 (2013)ADSCrossRefGoogle Scholar
  26. 26.
    Y O El-Dib and A A Mady, J. Comput. Appl. Mech. 49(2), 50 (2018)Google Scholar
  27. 27.
    J H He, Comput. Method. Appl. Mech. Eng. 178, 257 (1999)ADSCrossRefGoogle Scholar
  28. 28.
    J H He, Appl. Math. Comput. 151(1), 287 (2004)MathSciNetGoogle Scholar
  29. 29.
    J H He, Int. J. Nonlinear Sci. Numer. Simul. 6(2), 207 (2005)MathSciNetGoogle Scholar
  30. 30.
    Z Ayati and J Biazar, J. Egypt. Math. Soc. 23(2), 424 (2015)CrossRefGoogle Scholar
  31. 31.
    Y O El-Dib, Nonlinear Sci. Lett. A 8(4), 352 (2017)Google Scholar
  32. 32.
    Y O El-Dib, Pramana – J. Phys. 92: 7 (2019)ADSCrossRefGoogle Scholar
  33. 33.
    Y O El-Dib and G M Moatimid, Nonlinear Sci. Lett. A 9(3), 220 (2018)Google Scholar
  34. 34.
    Y Murakami, Phys. Lett. A 131, 368 (1988)ADSMathSciNetCrossRefGoogle Scholar
  35. 35.
    P T Hemamalini and S P Anjali Devi, Fluid Dyn. Mater. Process. 10(4), 491 (2014)Google Scholar
  36. 36.
    G M Moatimid and D R Mostapha, AIP Adv. 9(5), 055302 (2019)ADSCrossRefGoogle Scholar
  37. 37.
    J R Melcher, Field coupled surface waves (MIT Press, Cambridge, 1963)Google Scholar
  38. 38.
    H H Woodson and J R Melcher, Electromechanical dynamics (John Wiley and Sons, New York, 1968)Google Scholar
  39. 39.
    G K Batchelor, An introduction to fluid dynamics (Cambridge University Press, Cambridge, 1967)zbMATHGoogle Scholar
  40. 40.
    Y O El-Dib, J. Plasma Phys. 65(1), 1 (2001)ADSCrossRefGoogle Scholar
  41. 41.
    Y O El-Dib, J. Colloid Interface Sci. 259, 309 (2003)ADSCrossRefGoogle Scholar
  42. 42.
    G M Moatimid, Y O El-Dib and M H Zekry, Pramana – J. Phys. 92: 22 (2019)ADSCrossRefGoogle Scholar
  43. 43.
    K A Baldwin, M M Scase and R J A Hill, Sci. Rep. 5, 11706 (2015)ADSCrossRefGoogle Scholar
  44. 44.
    B Dolai and R P Prajapati, Phys. Plasmas 25, 083708 (2018)ADSCrossRefGoogle Scholar
  45. 45.
    C Nash and S Sen, Topology and geometry for physicists (Academic Press, London, 1983)zbMATHGoogle Scholar
  46. 46.
    Y O El-Dib and G M Moatimid, Z. Naturforsch. 57a, 159 (2002)Google Scholar
  47. 47.
    G M Moatimid and Y O El-Dib, Physica A 333, 41 (2004)ADSCrossRefGoogle Scholar

Copyright information

© Indian Academy of Sciences 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of EducationAin Shams UniversityRoxyEgypt

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