Skip to main content
Log in

A modern approach of Caputo–Fabrizio time-fractional derivative to MHD free convection flow of generalized second-grade fluid in a porous medium

  • Original Article
  • Published:
Neural Computing and Applications Aims and scope Submit manuscript

Abstract

The present analysis represents the concept of the Caputo–Fabrizio derivatives of fractional order to MHD flow of a second-grade fluid together with radiative heat transfer. The fluid flow is subjected to an infinite oscillating vertical plate embedded in a saturated porous media. The fluid starts motion due to an oscillating boundary and temperature difference between the plate and the fluid. The problem is modeled in terms of partial differential equations, which consist of momentum equation and heat equation. The Laplace transform method is used to obtain the closed-form solutions for velocity and temperature profiles. In order to understand the physics of the problem under consideration, numerical results are obtained using Mathcad software and brought into light through graphical representations. The influence of various physical parameters is studied and displayed in various figures. The corresponding skin friction coefficient and Nusselt number are provided in tables. A graphical comparison is provided showing a strong agreement with the published results in the open literature.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  1. Sabatier J, Agrawal, OP, Machado JT (2007) Advances in fractional calculus, vol 4, no 9. Springer, Dordrecht

  2. Oldham K, Spanier J (1974) The fractional calculus theory and applications of differentiation and integration to arbitrary order, vol 111. Elsevier, New York

    MATH  Google Scholar 

  3. Miller KS, Ross B (1993) An introduction to the fractional calculus and fractional differential equations. John Wiley & Sons, New York, USA

  4. Samko SG, Kilbas AA, Marichev OI (1993) Fractional integrals and derivatives. Theory and applications. Gordon and Breach, Yverdon

    MATH  Google Scholar 

  5. Podlubny I (1998) Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, vol 198. Academic Press, London

    MATH  Google Scholar 

  6. Tan WC, Xu MY (2002) The impulsive motion of flat plate in a generalized second grade fluid. Mech Res Commun 29(1):3–9

    Article  MathSciNet  MATH  Google Scholar 

  7. Caputo M, Fabrizio M (2015) A new definition of fractional derivative without singular kernel. Progr Fract Differ Appl 1(2):1–13

    Google Scholar 

  8. Vieru D, Fetecau C, Fetecau C (2015) Time-fractional free convection flow near a vertical plate with newtonian heating and mass diffusion. Therm Sci 19(1):S85–S98

    Article  Google Scholar 

  9. Alkahtani BST, Atangana A (2016) Analysis of non-homogeneous heat model with new trend of derivative with fractional order. Chaos, Solitons Fractals 89:566–571

    Article  MathSciNet  MATH  Google Scholar 

  10. Atangana A (2016) On the new fractional derivative and application to nonlinear Fisher’s reaction–diffusion equation. Appl Math Comput 273:948–956

    MathSciNet  Google Scholar 

  11. Alkahtani BST, Atangana A (2016) Controlling the wave movement on the surface of shallow water with the Caputo-Fabrizio derivative with fractional order. Chaos, Solitons Fractals 89:539–546

    Article  MathSciNet  MATH  Google Scholar 

  12. Kataria HR, Patel HR (2016) Soret and heat generation effects on MHD Casson fluid flow past an oscillating vertical plate embedded through porous medium. Alexandria Eng J 55:2125–2137

    Article  Google Scholar 

  13. Ibrahim W, Haq RU (2016) Magnetohydrodynamic (MHD) stagnation point flow of nanofluid past a stretching sheet with convective boundary condition. J Braz Soc Mech Sci Eng 38(4):1155–1164

    Article  Google Scholar 

  14. Shercliff JA (1965) Textbook of magnetohydrodynamics. Pergamon Press, London

    Google Scholar 

  15. Hartmann J, Lazarus F (1937) Hg dynamics. Levin & Munksgaard, Copenhagen

    Google Scholar 

  16. Haq RU, Rajotia D, Noor NFM (2016) Thermophysical effects of water driven copper nanoparticles on MHD axisymmetric permeable shrinking sheet: dual-nature study. Eur Phys J E 39(3):1–12

    Google Scholar 

  17. Abdulhameed M, Khan I, Khan A, Shafie S (2013) Closed form solutions for unsteady magnetohydrodynamic flow in a porous medium with wall transpiration. J Porous Media 16(9):795–809

    Article  Google Scholar 

  18. Sheikholeslami M, Ellahi R (2015) Three-dimensional mesoscopic simulations of magnetic field effect on natural convection of nanofluid. Int J Heat Mass Transf 89:799–808

    Article  Google Scholar 

  19. Vafai K, Kim SJ (1990) Fluid mechanics of the interface region between a porous medium and a fluid layer—an exact solution. Int J Heat Fluid Flow 11(3):254–256

    Article  Google Scholar 

  20. Tan W, Masuoka T (2005) Stokes’ first problem for a second grade fluid in a porous half-space with heated boundary. Int J Non-Linear Mech 40(4):515–522

    Article  MATH  Google Scholar 

  21. Aldoss TK, Al-Nimr MA, Jarrah MA, Al-Sha’er BJ (1995) Magnetohydrodynamic mixed convection from a vertical plate embedded in a porous medium. Numer Heat Transf Part A Appl 28(5):635–645

    Article  Google Scholar 

  22. Cortell R (2007) MHD flow and mass transfer of an electrically conducting fluid of second grade in a porous medium over a stretching sheet with chemically reactive species. Chem Eng Process Process Intensif 46(8):721–728

    Article  Google Scholar 

  23. Rashidi S, Nouri-Borujerdi A, Valipour MS, Ellahi R, Pop I (2015) Stress-jump and continuity interface conditions for a cylinder embedded in a porous medium. Transp Porous Media 107(1):171–186

    Article  Google Scholar 

  24. Ellahi R, Hassan M, Zeeshan A (2015) Aggregation effects on water base Al2O3-nanofluid over permeable wedge in mixed convection. Asia-Pac J Chem Eng 11(2):179–186

    Article  Google Scholar 

  25. Shirvan KM, Ellahi R, Mirzakhanlari S, Mamourian M (2016) Enhancement of heat transfer and heat exchanger effectiveness in a double pipe heat exchanger filled with porous media: numerical simulation and sensitivity analysis of turbulent fluid flow. Appl Therm Eng 109:761–774

    Article  Google Scholar 

  26. Haq RU, Nadeem S, Khan ZH, Noor NFM (2015) Convective heat transfer in MHD slip flow over a stretching surface in the presence of carbon nanotubes. Phys B Condens Matter 457:40–47

    Article  Google Scholar 

  27. Ellahi R, Shivanian E, Abbasbandy S, Hayat T (2015) Analysis of some magnetohydrodynamic flows of third-order fluid saturating porous space. J Porous Media 18(2):89–98

    Article  Google Scholar 

  28. Ellahi R, Zeeshan A, Hassan M (2015) A study of Fe3O4 nanoparticles aggregation in engine oil base nanofluid over the vertical stretching of a permeable sheet in a mixed convection. J Zhejiang Univ Sci A :1–12. doi:10.1631/jzus.A1500119

  29. Makinde OD, Khan WA, Culham JR (2016) MHD variable viscosity reacting flow over a convectively heated plate in a porous medium with thermophoresis and radiative heat transfer. Int J Heat Mass Transf 93:595–604

    Article  Google Scholar 

  30. Sheikholeslami M, Ganji DD, Javed MY, Ellahi R (2015) Effect of thermal radiation on magnetohydrodynamics nanofluid flow and heat transfer by means of two phase model. J Magn Magn Mater 374:36–43

    Article  Google Scholar 

  31. Zhang C, Zheng L, Zhang X, Chen G (2015) MHD flow and radiation heat transfer of nanofluids in porous media with variable surface heat flux and chemical reaction. Appl Math Model 39(1):165–181

    Article  MathSciNet  Google Scholar 

  32. Rashidi MM, Ali M, Freidoonimehr N, Rostami B, Hossain MA (2014) Mixed convective heat transfer for MHD viscoelastic fluid flow over a porous wedge with thermal radiation. Adv Mech Eng 6:735939

    Article  Google Scholar 

  33. Dehghan M, Rahmani Y, Ganji DD, Saedodin S, Valipour MS, Rashidi S (2015) Convection–radiation heat transfer in solar heat exchangers filled with a porous medium: homotopy perturbation method versus numerical analysis. Renew Energy 74:448–455

    Article  Google Scholar 

  34. Haq RU, Nadeem S, Khan ZH, Akbar NS (2015) Thermal radiation and slip effects on MHD stagnation point flow of nanofluid over a stretching sheet. Phys E Low-Dimens Syst Nanostruct 65:17–23

    Article  Google Scholar 

  35. Ulhaq S, Khan I, Ali F, Shafie S (2013) Radiation and magnetohydrodynamics effects on unsteady free convection flow in a porous medium. Math Probl Eng 2013:1–7

  36. Rajesh V, Varma SVK (2010) Radiation effects on MHD flow through a porous medium with variable temperature or variable mass diffusion. J Appl Math Mech 6(1):39–57

    Google Scholar 

  37. Chandrakala P, Bhaskar PN (2009) Thermal radiation effects on MHD flow past a vertical oscillating plate. Int J Appl Mech Eng 14:349–358

    Google Scholar 

  38. Mallikarjuna B, Rashad AM, Chamkha AJ, Raju SH (2016) Chemical reaction effects on MHD convective heat and mass transfer flow past a rotating vertical cone embedded in a variable porosity regime. Afr Mat 27(3–4):645–665

    Article  MathSciNet  MATH  Google Scholar 

  39. Bejan A (2013) Convection heat transfer. Wiley, New York

    Book  MATH  Google Scholar 

  40. Nield DA, Bejan A (2006) Convection in porous media. Springer, New York

    MATH  Google Scholar 

  41. Pop I, Ingham DB (2001) Convective heat transfer: mathematical and computational modelling of viscous fluids and porous media. Elsevier, Oxford

    Google Scholar 

  42. Vafai K (2015) Handbook of porous media. CRC Press, Boca Raton

    MATH  Google Scholar 

  43. Ali F, Khan I, Shafie S (2014) Closed form solutions for unsteady free convection flow of a second grade fluid over an oscillating vertical plate. PLoS ONE 9:e85099

    Article  Google Scholar 

  44. Ali F, Norzieha M, Sharidan S, Khan I, Hayat T (2012) New exact solutions of Stokes’ second problem for an MHD second grade fluid in a porous space. Int J Nonlinear Mech 47:521–525

    Article  Google Scholar 

  45. Atangana A (2016) On the new fractional derivative and application to nonlinear Fisher’s reaction–diffusion equation. Appl Math Comput 273:948–956

    MathSciNet  Google Scholar 

  46. Atangana A, Baleanu D (2016) Caputo–Fabrizio derivative applied to groundwater flow within confined aquifer. J Eng Mech 142:D4016005

  47. Shah NA, Khan I (2016) Heat transfer analysis in a second grade fluid over and oscillating vertical plate using fractional Caputo–Fabrizio derivatives. Eur Phys J C 76(7):1–11

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Farhad Ali.

Ethics declarations

Conflict of interest

The authors declare that there is no conflict of interest regarding the publication of this paper.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sheikh, N.A., Ali, F., Khan, I. et al. A modern approach of Caputo–Fabrizio time-fractional derivative to MHD free convection flow of generalized second-grade fluid in a porous medium. Neural Comput & Applic 30, 1865–1875 (2018). https://doi.org/10.1007/s00521-016-2815-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00521-016-2815-5

Keywords

Navigation