Analysis of the effect of generalized fractional Fourier’s and Fick’s laws on convective flows of non-Newtonian fluid subject to Newtonian heating

  • Imran SiddiqueEmail author
  • Syeda Mahwish Bukhari
Regular Article


The aim of this report is to study an unsteady mixed convection flow of an incompressible differential type fluid occurrence of chemical reaction that is first order, heat source and radiative heat source with fractional mass diffusion and thermal transports over an infinite vertical plate. The fractional derivative Caputo–Fabrizio which is defined recently with non-singular kernel is used in constitutive laws for the mass and thermal flux, respectively. Semi analytical solutions of the dimensionless concentration, temperature, and velocity fields in addition the rates of heat and mass transfer from the plate to the fluid are established by virtue of the Laplace inversion numerical algorithms Stehfest’s and Tzou’s. Some solutions for ordinary case and obvious results from articles are retrieved as limiting cases. Finally, an impact of flow and fractionalize parameters \(\alpha \) and \(\beta \) on concentration, temperature and velocity profiles is tabularly and graphically underlined and discussed . We present a valuation between second grade (fractional and ordinary) and viscous (fractional and ordinary) fluids is also interpreted. It is identified that the ordinary fluid has high velocity as comparable to fractional fluids.



The second author is greatly obliged and gratifying to University of Management and Technology, Lahore Campus, Pakistan for the generous support expediting to research work.


  1. 1.
    Y. Jaluria, Natural convection: heat and mass transfer 5 (1980)Google Scholar
  2. 2.
    P.S. Ghoshdastidar, Heat Transfer (Oxford University Press, Oxford, 2004)Google Scholar
  3. 3.
    M.A. Imran, I. Khan, M. Ahmad, N.A. Shah, M. Nazar, Heat and mass transport of differential type fluid with non-integer order time-fractional Caputo derivatives. J. Mol. Liq. 229, 67–75 (2017)CrossRefGoogle Scholar
  4. 4.
    C. Fetecau, D. Vieru, C. Fetecau, I. Pop, Slip effects on the unsteady radiative MHD free convection flow over a moving plate with mass diffusion and heat source. Eur. Phys. J. Plus 130(1), 6 (2017)CrossRefGoogle Scholar
  5. 5.
    S. Eskinazi, Fluid Mechanics and Thermodynamics of Our Environment (Academic Press, New York, 1975)Google Scholar
  6. 6.
    V. Kulish, J. Luis Lage, Application of fractional calculus to fluid mechanics. J. Fluids Engg. 124(3), 4 (2002)Google Scholar
  7. 7.
    P. Ganesan, P. Loganathan, Radiation and mass transfer effects on flow of an incompressible viscous fluid past a moving cylinder. Int. J. Heat Mass Transf. 45(21), 4281–4288 (2002)CrossRefGoogle Scholar
  8. 8.
    A.J. Chamkha, H.S. Tahkr, V.M. Soundalgekar, Radiation effects on free convection flow past a semi infinite vertical plate with mass transfer. Chem. Eng. J. 84, 335–342 (2001)CrossRefGoogle Scholar
  9. 9.
    A.A. Zafar, C. Fetecau, Flow over an infinite plate of a viscous fluid with non-integer order derivative without singular kernel. Alex. Eng. J. 55(3), 2789–2796 (2016)CrossRefGoogle Scholar
  10. 10.
    N.A. Shah, I. Khan, Heat transfer analysis in a second grade fluid over and oscillating vertical plate using fractional Caputo–Fabrizio derivatives. Eur. Phys. J. Plus 76, 362 (2016)ADSCrossRefGoogle Scholar
  11. 11.
    I. Khan, N.A. Shah, Y. Mahsud, D. Vieru, Heat transfer analysis in a Maxwell fluid over an oscillating vertical plate using fractional Caputo-Fabrizio derivatives. Eur. Phys. J. Plus 132(4), 194 (2017)CrossRefGoogle Scholar
  12. 12.
    N. Shah, N. Ahmed, T. Elnaqeeb, M.M. Rashidi, Magnetohydrodynamic free convection flows with thermal memory over a moving vertical plate in porous medium. J. Appl. Comput. Mech. 5, 150–161 (2018)Google Scholar
  13. 13.
    F. Mainardi, Application of fractional calculus in mechanics, transform method and special functions (Bulgarian Academy of Sciences, Bulgaria, 1998)zbMATHGoogle Scholar
  14. 14.
    B.S.T. Alkahtani, A. Atangana, Controlling the wave movement on the surface of shallow water with the Caputo–Fabrizio derivative with fractional order. Chaos Solitons Fract. 89, 539–546 (2016)ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel. Progr. Fract. Differ. Appl. 1(2), 1–13 (2015)Google Scholar
  16. 16.
    K.L. Kuhlman, Review of inverse Laplace transform algorithms for Laplace-space numerical approaches. Numer. Algor. 63(2), 339–355 (2013)MathSciNetCrossRefGoogle Scholar
  17. 17.
    E. Magyari, A. Pantokratoras, Note on the effect of thermal radiation in the linearized rosseland approximation on the heat transfer characteristics of various boundary layer flows. Int. Commun. Heat Mass Transf. 38(5), 554–556 (2011)CrossRefGoogle Scholar
  18. 18.
    J. Hristove, Frontiers in Fractional Calculus, 1st edition, edited by Sanchin Bhalekar, Bentham Science Publishers, 10, 235–295 (2017)Google Scholar
  19. 19.
    Y. Povstenko, Fractional thermoelasticity, in Encyclopedia of Thermal Stresses, vol. 4, ed. by R.B. Hetnarski (Springer, New York, 2014), pp. 1778–1787CrossRefGoogle Scholar
  20. 20.
    M. Narahari, B.K. Dutta, Effect of thermal radiation and mass diffusion on free convection flow near a vertical plate with Newtonian heating. Chem. Eng. Commun. 199, 628–643 (2012)CrossRefGoogle Scholar
  21. 21.
    M.A. Imran, M.B. Riaz, N.A. Shah, A.A. Zafar, Boundary layer flow of MHD generalized Maxwellfluid over an exponentially accelerated infinite vertical surface with slip and Newtonian heating at the boundary. Res. Phys. 8, 1061–1067 (2018)Google Scholar
  22. 22.
    M.A. Imran, Shakila Sarwar, M. Abdullah, I. Khan, An analysis of the semi analytic solutions of a viscous fluid with old and new definitions of fractional derivatives. Chin. J. Phys. 56, 1853–1871 (2018)CrossRefGoogle Scholar
  23. 23.
    H. Sheng, Y. Li, Y.Q. Chen, Application of numerical inverse Laplace transform algorithms in fractional calculus. J. Franklin Inst. 348, 317–330 (2011)MathSciNetCrossRefGoogle Scholar
  24. 24.
    D.K. Tong, X.M. Zhang, X.H. Zhang, Unsteady helicalflows of a generalized Oldroyd-B fluid. J. Non-Newtonian Fluid Mech. 156, 75–83 (2009)CrossRefGoogle Scholar
  25. 25.
    Y. Jiang, H. Qi, H. Xu, X. Jiang, Transient electro osmotic slipflow of fractional Oldroyd-B fluids. Micro Nano Fluid. 21, 1–10 (2017)Google Scholar
  26. 26.
    H. Stehfest, Algorithm 368: Numerical inversion of Laplace transforms. Commun. ACM 13, 47–49 (1970)CrossRefGoogle Scholar
  27. 27.
    D.Y. Tzou, Macro to Microscale Heat transfer: The Lagging Behavior (Taylor & Francis, Washington, 1970)Google Scholar
  28. 28.
    G. Honig, U. Hirdes, A method for the numerical inversion of Laplace transforms. J. Comp. Appl. Math. 10(1), 113–132 (1984)MathSciNetCrossRefGoogle Scholar
  29. 29.
    D. Vieru, C. Fetecau, Corina Fetecau, Time fractional free convection flow near a vertical plate with newtonian heating and mass diffusion. Therm. Sci. 19(1), 85–98 (2015)CrossRefGoogle Scholar

Copyright information

© Società Italiana di Fisica (SIF) and Springer-Verlag GmbH Germany, part of Springer Nature 2020

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Management and TechnologyLahorePakistan

Personalised recommendations