Thermal effects of magnetohydrodynamic micropolar fluid embedded in porous medium with Fourier sine transform technique

  • Kashif Ali AbroEmail author
  • Ilyas Khan
  • J. F. Gómez-Aguilar
Technical Paper


This paper investigates the thermal effects of magnetohydrodynamic micropolar fluid with hidden phenomenon of heat and mass transfer via Caputo–Fabrizio fractional derivative. Analytical solutions are obtained for velocity field, mass concentration, microrotation and temperature distribution by implementing Fourier Sine and Laplace transform. The general solutions have been expressed in terms of simple elementary functions involving the convolution theorem for the Laplace transform. The graphical illustration is depicted in order to explore the influence of rheological parameters, i.e., Grashof, Prandtl, Schmidt numbers, transverse magnetic field, microrotation parameter, porosity and few other parameters on micropolar fluid flow.


Fractional calculus Caputo–Fabrizio fractional derivative Fourier Sine and Laplace transforms Micropolar fluid 

List of symbols

\(C_\infty \)

Species concentration away from plate


Nonzero constant


Species concentration near plate


Acceleration due to gravity

\(\beta _T\)

Volumetric coefficient of thermal expansion

\(\beta _c\)

Volumetric coefficient of mass expansion


Magnetic field

\(\gamma _0\)

Spin gradient viscosity


Thermal conductivity


Heat capacity at a constant pressure


Heat transfer coefficient

\(\eta \)

Spin gradient viscosity parameter


Schmidt number


Thermal Grashof number




Temperature distribution

\(\xi \)

Fourier Sine transform parameter

\(T_\infty \)

Ambient temperature


Unit step function

\(\rho \)

Density of fluid

\(\alpha \)

Vortex viscosity

\(\mu \)

Dynamics viscosity of fluid

\(\Phi \)


\(\epsilon \)

Letting parameter




Radiative heat flux


Mass diffusivity

\(\delta \)

Fractional parameter


Prandtl number

\(\beta \)

Microrotation parameter


Modified Grashof number


Mass concentration


Velocity field


Laplace transform parameter



The author Kashif Ali Abro is highly thankful and grateful to Mehran University of Engineering and Technology, Jamshoro, Pakistan, for generous support and facilities of this research work. José Francisco Gómez Aguilar acknowledges the support provided by CONACyT: Cátedras CONACyT para jóvenes investigadores 2014 and SNI-CONACyT.

Author Contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

Compliance with ethical standards

Conflict of interest

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


  1. 1.
    Eringen AC (1966) Theory of micropolar fluids. J Math Mech 16:1–18MathSciNetGoogle Scholar
  2. 2.
    Eringen AC (1964) Simple microfluids. Int J Eng Sci 2:205–217MathSciNetCrossRefGoogle Scholar
  3. 3.
    Mahmoud MAA, Waheed SE (2012) MHD flow and heat transfer of a micropolar fluid over a stretching surface with heat generation (absorption) and slip velocity. J Egypt Math Soc 20:20–27MathSciNetCrossRefGoogle Scholar
  4. 4.
    Qasim M, Khan I, Shafie S (2013) Heat transfer in a micropolar fluid over a stretching sheet with Newtonian heating. PLoS ONE 8(4):1–16CrossRefGoogle Scholar
  5. 5.
    Siddheshwar PG, Pranesh S (1998) Effect of a non-uniform basic temperature gradient on Rayleigh–Benard convection in a micropolar fluid. Int J Eng Sci 36:1183–1196CrossRefGoogle Scholar
  6. 6.
    Ishak A, Nazar R, Pop I (2008) Heat transfer over a stretching surface with variable heat flux in micropolar fluids. Phys Lett A 372:559–561CrossRefGoogle Scholar
  7. 7.
    Ishak A, Nazar R, Pop I (2008) Magnetohydrodynamic (MHD) flow of a micropolar fluid towards a stagnation point on a vertical surface. Comput Math Appl 56:3188–3194MathSciNetCrossRefGoogle Scholar
  8. 8.
    Nehad AS, Khan I (2016) Heat transfer analysis in a second grade fluid over and oscillating vertical plate using fractional Caputo–Fabrizio derivatives. Eur Phys J Plus 76:1–13CrossRefGoogle Scholar
  9. 9.
    Zafar AA, Fetecau C (2016) Flow over an infinite plate of a viscous fluid with non-integer order derivative without singular kernel. Alex Eng J 1:1–15Google Scholar
  10. 10.
    Abro KA, Hussain M, Baig MM (2016) Impacts of magnetic field on fractionalized viscoelastic fluid. J Appl Environ Biol Sci 6:84–93Google Scholar
  11. 11.
    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 9:1–12Google Scholar
  12. 12.
    Ali F, Khan I, Ul Haq S, Shafie S (2013) Influence of thermal radiation on unsteady free convection MHD flow of Brinkman type fluid in a porous medium with Newtonian heating. Math Probl Eng 1:1–15MathSciNetzbMATHGoogle Scholar
  13. 13.
    Ali F, Khan I, Shafie S, Musthapa N (2013) Heat and mass transfer with free convection MHD flow past a vertical plate embedded in a porous medium. Math Probl Eng 1:1–12MathSciNetzbMATHGoogle Scholar
  14. 14.
    Ali F, Khan I, Mustapha N, Shafie S (2012) Unsteady magnetohydrodynamic oscillatory flow of viscoelastic fluids in a porous channel with heat and mass transfer. J Phys Soc Jpn 81(6):1–11CrossRefGoogle Scholar
  15. 15.
    Ali F, Khan I, Shafie S (2012) A note on new exact solutions for some unsteady flows of Brinkman-type fluids over a plane wall. Z Naturforsch A 67(6–7):377–380Google Scholar
  16. 16.
    Hussanan A, Ismail Z, Khan I, Hussein AG, Shafie S (2014) Unsteady boundary layer MHD free convection flow in a porous medium with constant mass diffusion and Newtonian heating. Eur Phys J Plus 129:1–16CrossRefGoogle Scholar
  17. 17.
    Abid H, Zakaria MN, Khan I, Sharidan S (2013) Radiation effect on unsteady MHD free convection flow in a porous medium with Newtonian heating. Int J Appl Math Stat 42:474–480Google Scholar
  18. 18.
    Abro KA, Shaikh AA, Junejo IA, Chandio M (2015) Analytical solutions under no slip effects for accelerated flows of Maxwell fluids. Sindh Univ Res J (Sci Ser) 47:613–618Google Scholar
  19. 19.
    Atangana A (2018) Non validity of index law in fractional calculus: a fractional differential operator with Markovian and non-Markovian properties. Phys A Stat Mech Appl 505:688–706MathSciNetCrossRefGoogle Scholar
  20. 20.
    Atangana A, Gómez-Aguilar JF (2018) Decolonisation of fractional calculus rules: breaking commutativity and associativity to capture more natural phenomena. Eur Phys J Plus 133:1–22CrossRefGoogle Scholar
  21. 21.
    Saad KM, Gómez-Aguilar JF (2018) Analysis of reaction–diffusion system via a new fractional derivative with non-singular kernel. Phys A Stat Mech Appl 509:703–716MathSciNetCrossRefGoogle Scholar
  22. 22.
    Atangana A (2018) Blind in a commutative world: simple illustrations with functions and chaotic attractors. Chaos Solitons Fractals 114:347–363MathSciNetCrossRefGoogle Scholar
  23. 23.
    Morales-Delgado VF, Gómez-Aguilar JF, Saad KM, Escobar-Jiménez RF (2019) Application of the Caputo–Fabrizio and Atangana–Baleanu fractional derivatives to mathematical model of cancer chemotherapy effect. Math Methods Appl Sci 1:1–28Google Scholar
  24. 24.
    Saad KM (2018) Comparing the Caputo, Caputo–Fabrizio and Atangana–Baleanu derivative with fractional order: fractional cubic isothermal auto-catalytic chemical system. Eur Phys J Plus 133(3):1–12CrossRefGoogle Scholar
  25. 25.
    Kumar A, Kumar S, Yan SP (2017) Residual power series method for fractional diffusion equations. Fundam Inform 151(1–4):213–230MathSciNetCrossRefGoogle Scholar
  26. 26.
    Saad KM, Baleanu D, Atangana A (2018) New fractional derivatives applied to the Korteweg–de Vries and Korteweg–de Vries–Burger’s equations. Comput Appl Math 1:1–14MathSciNetzbMATHGoogle Scholar
  27. 27.
    Kumar A, Kumar S (2016) Residual power series method for fractional Burger types equations. Nonlinear Eng 5(4):235–244MathSciNetCrossRefGoogle Scholar
  28. 28.
    Saad KM, Atangana A, Baleanu D (2018) New fractional derivatives with non-singular kernel applied to the Burgers equation. Chaos Interdiscip J Nonlinear Sci 28(6):1–10MathSciNetCrossRefGoogle Scholar
  29. 29.
    Zhang Y, Kumar A, Kumar S, Baleanu D, Yang XJ (2016) Residual power series method for time-fractional Schrödinger equations. J Nonlinear Sci Appl 9:5821–5829MathSciNetCrossRefGoogle Scholar
  30. 30.
    Guo ZH, Acan O, Kumar S (2016) Sumudu transform series expansion method for solving the local fractional Laplace equation in fractal thermal problems. Therm Sci 20:739–742CrossRefGoogle Scholar
  31. 31.
    Shakir A, Gul T, Islam S (2014) Analysis of MHD and thermally conducting unsteady thin film flow in a porous medium. J Appl Environ Biol Sci 5:109–121Google Scholar
  32. 32.
    Hussanan A, Salleh MZ, Khan I, Tahar RM (2018) Heat and mass transfer in a micropolar fluid with Newtonian heating: an exact analysis. Neural Comput Appl 29(6):59–67CrossRefGoogle Scholar
  33. 33.
    Caputo M, Fabrizio M (2015) A new definition of fractional derivative without singular kernel. Prog Fract Differ Appl 1(2):73–85Google Scholar
  34. 34.
    Zafar AA, Fetecau C (2016) Flow over an infinite plate of a viscous fluid with non-integer order derivative without singular kernel. Alex Eng J 55(3):2789–2796CrossRefGoogle Scholar
  35. 35.
    Narahari M, Nayan MY (2011) Free convection flow past an impulsively started infinite vertical plate with Newtonian heating in the presence of thermal radiation and mass diffusion. Turk J Eng Environ Sci 35:187–198Google Scholar

Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2019

Authors and Affiliations

  1. 1.Department of Basic Sciences and Related StudiesMehran University of Engineering and TechnologyJamshoroPakistan
  2. 2.Faculty of Mathematics and StatisticsTon Duc Thang UniversityHo Chi Minh CityVietnam
  3. 3.CONACyT-Tecnológico Nacional de México/CENIDETCuernavacaMexico
  4. 4.Universidad Virtual CNCIMonterreyMexico

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