Advertisement

Heat transfer analysis in a Maxwell fluid over an oscillating vertical plate using fractional Caputo-Fabrizio derivatives

  • Ilyas KhanEmail author
  • Nehad Ali Shah
  • Yasir Mahsud
  • Dumitru Vieru
Regular Article

Abstract.

This article is focused on heat transfer analysis in the unsteady flow of a generalized Maxwell fluid over an oscillating vertical flat plate with constant temperature. The well-known equation of the Maxwell fluid with classical derivatives, describing the unidirectional and one-dimensional flow, has been generalized to a non-integer-order derivative, known as fractional derivative, with free convection term of buoyancy. A new definition of the fractional derivative introduced by Caputo and Fabrizio has been used in the mathematical formulation of the problem. Exact solution of the dimensionless problem has been obtained by using the Laplace transform. These solutions are expressed with complementary error and modified Bessel functions. Similar solutions for classical Maxwell and Newtonian fluids and generalized Newtonian fluid performing the same motion are obtained as limiting cases of our general results. Graphical illustrations show that the velocity profiles corresponding to a generalized Maxwell fluid are similar to those for an ordinary Maxwell fluid when the fraction order approaches 1. A comparison amongst four different types of fluids is also shown graphically.

References

  1. 1.
    M. Sheikholeslami, T. Hayat, A. Alsaedi, Int. J. Heat Mass Transfer 96, 513 (2016)CrossRefGoogle Scholar
  2. 2.
    Mohsen Sheikholeslami, Mohammad Mehdi Rashidi, J. Taiwan Inst. Chem. Eng. 56, 6 (2015)CrossRefGoogle Scholar
  3. 3.
    Mohsen Sheikholeslami, Kuppalapalle Vajravelu, Mohammad Mehdi Rashidi, Int. J. Heat Mass Transfer 92, 339 (2016)CrossRefGoogle Scholar
  4. 4.
    M. Sheikholeslami, M. Gorji-Bandpy, D.D. Ganji, P. Rana, Soheil Soleimani, Comput. Fluids 94, 147 (2014)MathSciNetCrossRefGoogle Scholar
  5. 5.
    M. Sheikholeslami, R. Ellahi, Int. J. Heat Mass Transfer 89, 799 (2015)CrossRefGoogle Scholar
  6. 6.
    Mohsen Sheikholeslami, J. Mol. Liq. 229, 137 (2017)CrossRefGoogle Scholar
  7. 7.
    M. Sheikholeslami, Phys. Lett. A 381, 494 (2017)ADSCrossRefGoogle Scholar
  8. 8.
    M. Sheikholeslami, K. Vajravelu, Appl. Math. Comput. 298, 272 (2017)MathSciNetGoogle Scholar
  9. 9.
    Mohsen Sheikholeslami, J. Mol. Liq. 225, 903 (2017)CrossRefGoogle Scholar
  10. 10.
    M. Sheikholeslami, P. Rana, Soheil Soleimani, Heat Transfer Res. 48, 121 (2017)CrossRefGoogle Scholar
  11. 11.
    M. Takashima, Phys. Lett. A 33, 371 (1970)ADSCrossRefGoogle Scholar
  12. 12.
    J.C. Maxwell, Philos. Trans. R. Soc. London A 157, 26 (1866)Google Scholar
  13. 13.
    C.H.R. Friedrich, Rheol. Acta 30, 151 (1991)CrossRefGoogle Scholar
  14. 14.
    F. Olsson, J. Yström, J. Non-Newtonian Fluid Mech. 48, 125 (1993)CrossRefGoogle Scholar
  15. 15.
    J.J. Choi, Z. Rusak, J.A. Tichy, J. Non-Newtonian Fluid Mech. 85, 165 (1999)CrossRefGoogle Scholar
  16. 16.
    C. Fetecau, C. Fetecau, Int. J. Non-Linear Mech. 38, 423 (2003)ADSCrossRefGoogle Scholar
  17. 17.
    C. Fetecau, C. Fetecau, Int. J. Non-Linear Mech. 38, 603 (2003)ADSCrossRefGoogle Scholar
  18. 18.
    P.M. Jordan, A. Puri, G. Boros, Int. J. Non-Linear Mech. 39, 1371 (2004)CrossRefGoogle Scholar
  19. 19.
    J. Zierep, C. Fetecau, Int. J. Eng. Sci. 45, 617 (2007)CrossRefGoogle Scholar
  20. 20.
    C. Fetecau, M. Jamil, C. Fetecau, I. Siddique, Int. J. Non-Linear Mech. 44, 1085 (2009)ADSCrossRefGoogle Scholar
  21. 21.
    F. Salah, Z.A. Aziz, D.L.C. Ching, Results Phys. 1, 9 (2011)ADSCrossRefGoogle Scholar
  22. 22.
    M. Jamil, C. Fetecau, N.A. Khan, A. Mahmood, Int. J. Chem. Reactor Eng. 9, 20 (2011)Google Scholar
  23. 23.
    D. Vieru, A. Rauf, Can. J. Phys. 89, 1061 (2011)ADSCrossRefGoogle Scholar
  24. 24.
    D. Vieru, A.A. Zafar, Appl. Math. Inf. Sci. 7, 209 (2013)MathSciNetCrossRefGoogle Scholar
  25. 25.
    I. Khan, F. Ali, U.S. Haq, S. Shafie, Z. Naturforsch. A (2013) DOI:10.5560/ZNA.2013-0040
  26. 26.
    F. Ali, S.A.A. Jan, I. Khan, M. Gohar, N.A. Sheikh, Eur. Phys. J. Plus 131, 310 (2016)CrossRefGoogle Scholar
  27. 27.
    M.A. Imran, I. Khan, M. Ahmad, N.A. Shah, M. Nazar, J. Mol. Liq. 229, 67 (2016)CrossRefGoogle Scholar
  28. 28.
    I. Khan, N.A. Shah, D. Vieru, Eur. Phys. J. Plus 131, 181 (2016)CrossRefGoogle Scholar
  29. 29.
    D. Vieru, C. Fetecau, Fetecau Corina, Therm. Sci. 19, S85 (2015)CrossRefGoogle Scholar
  30. 30.
    C.H.R. Friedrich, Rheol. Acta 30, 151 (1991)CrossRefGoogle Scholar
  31. 31.
    R. Gorenflo, F. Mainardi, D. Moretti, P. Paradisi, Nonlinear Dyn. 29, 129 (2002)CrossRefGoogle Scholar
  32. 32.
    W.C. Tan, F. Xian, L. Wei, China Sci. Bull. 47, 1226 (2002)CrossRefGoogle Scholar
  33. 33.
    H. Qi, H. Jin, Nonlinear Anal. Real World Appl. 10, 2700 (2009)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Q. Haitao, X. Mingyu, Mech. Res. Commun. 34, 210 (2007)CrossRefGoogle Scholar
  35. 35.
    M. Jamil, C. Fetecau, C. Fetecau, Acta Mech. Sin. 28, 274 (2012)ADSMathSciNetCrossRefGoogle Scholar
  36. 36.
    M. Jamil, K.A. Abro, N.A. Khan, Nonlinear Eng. 4, 191 (2015)Google Scholar
  37. 37.
    L. Zheng, F. Zhao, X. Zhang, Nonlinear Anal. Real World Appl. 11, 3744 (2010)MathSciNetCrossRefGoogle Scholar
  38. 38.
    H.T. Qi, J.G. Liu, Eur. Phys. J. ST 193, 71 (2011)CrossRefGoogle Scholar
  39. 39.
    D. Tripathi, Comput. Math. Appl. 62, 1116 (2011)MathSciNetCrossRefGoogle Scholar
  40. 40.
    I. Podlubny, Fractional differential equations (Academic Press, New York, 1999)Google Scholar
  41. 41.
    R. Garra, F. Polito, Commun. Nonlinear Sci. Numer. Simulat. 17, 5073 (2012)ADSCrossRefGoogle Scholar
  42. 42.
    N.A. Shah, I. Khan, Eur. Phys. J. C 76, 362 (2016)ADSCrossRefGoogle Scholar
  43. 43.
    F. Ali, M. Saqib, I. Khan, N.A. Sheikh, Eur. Phys. J. Plus 131, 377 (2016)CrossRefGoogle Scholar

Copyright information

© Società Italiana di Fisica and Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Basic Engineering Sciences DepartmentCollege of Engineering Majmaah UniversityMajmaahSaudi Arabia
  2. 2.Abdus Salam School of Mathematical SciencesGC UniversityLahorePakistan
  3. 3.Department of Theoretical MechanicsTechnical University of IasiIaşiRomania

Personalised recommendations