Advertisement

A comparative study of Atangana-Baleanu and Caputo-Fabrizio fractional derivatives to the convective flow of a generalized Casson fluid

  • Nadeem Ahmad Sheikh
  • Farhad AliEmail author
  • Muhammad Saqib
  • Ilyas Khan
  • Syed Aftab Alam Jan
Regular Article

Abstract.

Based on exponential kernel, Caputo and Fabrizio suggested a new definition for fractional order derivatives in 2015. Recently, in 2016, Atangana and Baleanu proposed another version of fractional derivatives, which uses the generalized Mittag-Leffler function as the non-singular and non-local kernel. Moreover, the Atangana-Balaenu (AB) version has all properties of fractional derivatives. Therefore, this articles aims to use the AB fractional derivative idea for the first time to study the free convection flow of a generalized Casson fluid due to the combined gradients of temperature and concentration. Hence, heat and mass transfer are considered together. For the sake of comparison, this problem is also solved via the Caputo-Fabrizio (CF) derivatives technique. Exact solutions in both cases (AB and CF derivatives) are obtained via the Laplace transform and compared graphically as well as in tabular form. In the case of AB approach, the influence of pertinent parameters on velocity field is displayed in plots and discussed. It is found that for unit time, the velocities obtained via AB and CF derivatives are identical. Velocities for time less than 1 show little variation and, for time higher than 1, this variation increases.

References

  1. 1.
    M. Caputo, M. Fabrizio, Progr. Fract. Differ. Appl. 1, 73 (2015)Google Scholar
  2. 2.
    A. Atangana, Appl. Math. Comput. 273, 948 (2016)MathSciNetGoogle Scholar
  3. 3.
    A. Atangana, J.J. Nieto, Adv. Mech. Eng. 7, 1687814015613758 (2015)Google Scholar
  4. 4.
    A. Atangana, R.T. Alqahtani, Adv. Differ. Equ. 2016, 156 (2016)CrossRefGoogle Scholar
  5. 5.
    N.A. Shah, I. Khan, Eur. Phys. J. C 76, 362 (2016)ADSCrossRefGoogle Scholar
  6. 6.
    F. Ali, M. Saqib, I. Khan, N.A. Sheikh, Eur. Phys. J. Plus 131, 377 (2016)CrossRefGoogle Scholar
  7. 7.
    A. Atanganaa, I. Kocab, J. Nonlinear Sci. Appl. 9, 2467 (2016)Google Scholar
  8. 8.
    B.S.T. Alkahtani, A. Atangana, Chaos, Solitons Fractals 89, 539 (2016)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    B.S.T. Alkahtani, A. Atangana, Chaos, Solitons Fractals 89, 566 (2016)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    J. Hristov, Therm. Sci. 20, 757 (2016)CrossRefGoogle Scholar
  11. 11.
    J. Hristov, Therm. Sci. (2016) DOI:10.2298/TSCI160229115H
  12. 12.
    A., Atangana, D. Baleanu, arXiv:1602.03408 (2016)
  13. 13.
    A., Atangana, D. Baleanu, J. Eng. Mech. (2016) DOI:10.1061/(ASCE)EM.1943-7889.0001091
  14. 14.
    A. Atangana, I. Koca, Chaos, Solitons Fractals 89, 447 (2016)ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    O.J.J. Algahtani, Chaos, Solitons Fractals 89, 552 (2016)ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    B.S.T. Alkahtani, Chaos, Solitons Fractals 89, 547 (2016)ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    R. Chhabra, J. Richardson, R Chhabra, Non-Newtonian Flow and Applied Rheology (Butterworth-Heinemann/Elsevier, 2008)Google Scholar
  18. 18.
    R. Bagley, J. Rheol. 27, 201 (1983)ADSCrossRefGoogle Scholar
  19. 19.
    J. Dunn, R. Fosdick, Arch. Ration. Mech. Anal. 56, 191 (1974)CrossRefGoogle Scholar
  20. 20.
    L. Pakzad, F. Ein-Mozaffari, S. Upreti, A. Lohi, Can. J. Chem. Eng. 91, 90 (2011)CrossRefGoogle Scholar
  21. 21.
    I. Khan, F. Ali, S. Shafie, M. Qasim, Bull. Malaysian Math. Sci. Soc. 37, 437 (2014)MathSciNetGoogle Scholar
  22. 22.
    M. Qasim, I. Khan, S. Shafie, Plos One 8, e59393 (2013)ADSCrossRefGoogle Scholar
  23. 23.
    C. Fetecau, C. Fetecau, Int. J. Eng. Sci. 43, 781 (2005)MathSciNetCrossRefGoogle Scholar
  24. 24.
    C. Fetecau, C. Fetecau, Int. J. Eng. Sci. 44, 788 (2006)CrossRefGoogle Scholar
  25. 25.
    M. Khan, S. Ali, H. Qi, Math. Comput. Model. 49, 1519 (2009)CrossRefGoogle Scholar
  26. 26.
    W. Casson, A flow equation for pigment-oil suspensions of printing of the printing ink type (Pergamon Press, 1959)Google Scholar
  27. 27.
    S. Pramanik, Ain Shams Eng. J. 5, 205 (2014)CrossRefGoogle Scholar
  28. 28.
    F. Ali, N.A. Sheikh, I. Khan, M. Saqib, J. Magn. & Magn. Mater 423, 327 (2017)ADSCrossRefGoogle Scholar
  29. 29.
    K. Bhattacharyya, M.S. Uddin, G.C. Layek, Alex. Eng. J. 55, 1703 (2016)CrossRefGoogle Scholar
  30. 30.
    I. Khan, N.A. Shah, D. Vieru, Eur. Phys. J. Plus 131, 181 (2016)CrossRefGoogle Scholar
  31. 31.
    S. Shaw, G. Mahanta, P. Sibanda, Alex. Eng. J. 55, 1295 (2016)CrossRefGoogle Scholar
  32. 32.
    Z. Abbas, M. Sheikh, S.S. Motsa, Energy 95, 12 (2016)CrossRefGoogle Scholar
  33. 33.
    C.S.K. Raju, N. Sandeep, V. Sugunamma, M.J. Babu, J.R. Reddy, Eng. Sci. Technol. 19, 45 (2016)Google Scholar
  34. 34.
    Agathoklis D. Passos, Vasileios-Alexandros Chatzieleftheriou, Aikaterini A. Mouza, Spiros V. Paras, Chem. Eng. Sci. 148, 229 (2016)CrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  • Nadeem Ahmad Sheikh
    • 1
  • Farhad Ali
    • 1
    Email author
  • Muhammad Saqib
    • 1
  • Ilyas Khan
    • 2
  • Syed Aftab Alam Jan
    • 1
  1. 1.Department of MathematicsCity University of Science and Information TechnologyPeshawarPakistan
  2. 2.Basic Engineering Sciences DepartmentCollege of Engineering Majmaah UniversityMajmaahSaudi Arabia

Personalised recommendations