Advertisement

Thermodynamics of magnetohydrodynamic Brinkman fluid in porous medium

Applications to thermal science
  • Ambreen Siyal
  • Kashif Ali Abro
  • Muhammad Anwar Solangi
Article
  • 14 Downloads

Abstract

This research article investigates that how heat flow changes versus temperature or time on the rheology of magnetohydrodynamic Brinkman fluid embedded in porous medium for the oscillations of heated plate. A fractional approach namely Caputo–Fabrizio fractional operator is applied for developing the governing partial differential equations of Brinkman fluid flow. The fractional governing partial differential equations have been modeled for temperature distribution, mass concentration and velocity field along with imposed initial and boundary conditions. The solutions are obtained by integral transforms and presented in special and elementary functions. In the limiting sense, the analytical solutions are particularized in the presence and absence of heat and mass transfer, magnetic field and porous medium. The parametric graphs have been depicted for the influence of different embedded rheological parameters on fluid flow. The results show few interesting differences and similarities by comparative analysis for fractional and ordinary Brinkman fluid flow, such as physically higher Prandtl (Pr) number that leads to decay thermal diffusivity which results in the reduction in thermal field; this means that better quality of production can be achieved through proper choice of Prandtl (Pr) and Schmidt (Sc) numbers.

Keywords

Caputo–Fabrizio fractional operator Brinkman fluid Porous medium Analytic solutions Parametric study 

Notes

Acknowledgements

The authors appreciate the constructive remarks and suggestions of the anonymous referees that helped to improve the quality of paper. The authors are highly thankful and grateful to Mehran University of Engineering and Technology, Jamshoro, Pakistan, for generous support and facilities of this research work.

References

  1. 1.
    Koca I, Atangana A. Solutions of Cattaneo-Hristov model of elastic heat diffusion with Caputo–Fabrizio and Atangana–Baleanu fractional derivatives. Therm Sci. 2016;21:2299–305.  https://doi.org/10.2298/TSCI160102102M.CrossRefGoogle Scholar
  2. 2.
    Khan U, Khan SI, Ahmed N, Bano S, Mohyudin ST. Heat transfer analysis for squeezing flow of a Casson fluid between parallel plates. Ain Shams Eng J. 2016;7:497–504.CrossRefGoogle Scholar
  3. 3.
    Khan I, Gul A, Shafie S. Effects of magnetic field on molybdenum disulfide nanofluids in mixed convection flow inside a channel filled with a saturated porous medium. J Porous Media. 2017;20:435–48.CrossRefGoogle Scholar
  4. 4.
    Sheikh NA, Ali F, Khan I, Saqib M, Khan A. MHD flow of micropolar fluid over an oscillating vertical plate embedded in porous media with constant temperature and concentration. Math Probl Eng. 2017; Article ID 9402964.  https://doi.org/10.1155/2017/9402964.CrossRefGoogle Scholar
  5. 5.
    Shakeel A, Ahmad S, Khan H, Vieru D. Solutions with wright functions for time fractional convection flow near a heated vertical plate. Adv Differ Equ. 2016;2016:51.  https://doi.org/10.1186/s13662-016-0775-9.CrossRefGoogle Scholar
  6. 6.
    Zakaria MN, Hussanan A, Khan I, Shafie S. The effects of radiation on free convection flow with ramped wall temperature in Brinkman-type fluid. J Teknol. 2013;62:33–9.Google Scholar
  7. 7.
    Kashif AA, Shaikh HS, Norzieha M, Khan I, Asifa T. A mathematical study of magnetohydrodynamic Casson fluid via special functions with heat and mass transfer embedded in porous plate. Mal J Fund Appl Sci. 2018;14:20–38.Google Scholar
  8. 8.
    Ali F, Aftab SAJ, Khan I, Gohar M, Sheikh NA. Solutions with special functions for time fractional free convection flow of Brinkman-type fluid. Eur Phys J Plus. 2016;131:310.CrossRefGoogle Scholar
  9. 9.
    Kashif AA, Rashidi MM, Khan I, Irfan AA, Asifa T. Analysis of stokes’ second problem for nanofluids using modern fractional derivatives. J Nanofuids. 2018;7:738–47.CrossRefGoogle Scholar
  10. 10.
    Shirazi M, Shateri A, Bayareh M. Numerical investigation of mixed convection heat transfer of a nanofluid in a circular enclosure with a rotating inner cylinder. J Therm Anal Calorim. 2018;133:1061–73.  https://doi.org/10.1007/s10973-018-7186-y.CrossRefGoogle Scholar
  11. 11.
    Kashif AA, Hussain M, Baig MM. An analytic study of molybdenum disulfide nanofluids using the modern approach of Atangana–Baleanu fractional derivatives. Eur Phys J Plus. 2017;132:439.  https://doi.org/10.1140/epjp/i2017-11689-y.CrossRefGoogle Scholar
  12. 12.
    Atangana A, Baleanu D. New fractional derivatives with nonlocal and nonsingular kernel: theory and application to heat transfer model. Therm Sci. 2016;20:763–9.CrossRefGoogle Scholar
  13. 13.
    Abro AK, Khan I. Effects of CNTs on magnetohydrodynamic flow of methanol based nanofluids via Atangana–Baleanu and Caputo–Fabrizio fractional derivatives. Therm Sci. 2018.  https://doi.org/10.2298/tsci180116165a.CrossRefGoogle Scholar
  14. 14.
    Kashif AA, Mukarrum H, Mirza MB. A mathematical analysis of magnetohydrodynamic generalized Burger fluid for permeable oscillating plate. Punjab Univ J Math. 2018;50:97–111.Google Scholar
  15. 15.
    Magomedov RA, Meilanov RR, Meilanov RP, Akhmedov EN, Beybalaev VD, Aliverdiev AA. Generalization of thermodynamics in of fractional-order derivatives and calculation of heat-transfer properties of noble gases. J Therm Anal Calorim. 2018;133:1189–94.  https://doi.org/10.1007/s10973-018-7024-2.CrossRefGoogle Scholar
  16. 16.
    Kashif AA, Irfan AA, Almani SM, Khan I. On the thermal analysis of magnetohydrodynamic Jeffery fluid via modern non integer order derivative. J King Saud Univ Sci. 2018.  https://doi.org/10.1016/j.jksus.2018.07.012.CrossRefGoogle Scholar
  17. 17.
    Zhuo L, Liu L, Dehghan S, Yang QC, Xue D. A review and evaluation of numerical tools for fractional calculus and fractional order controls. Int J Control. 2016;90:1165–81.Google Scholar
  18. 18.
    Gomez-Aguilar JF, Morales-Delgado VF, Taneco-Hernandez MA, Baleanu D, Escobar Jimenez RF, Al Qurashi MM. Analytical solutions of the electrical RLC circuit via Liouville–Caputo operators with local and non-local kernels. Entropy. 2016;18:402.  https://doi.org/10.3390/e18080402.CrossRefGoogle Scholar
  19. 19.
    Kashif AA, Anwar AM, Muhammad AU. A comparative mathematical analysis of RL and RC electrical circuits via Atangana–Baleanu and Caputo–Fabrizio fractional derivatives. Eur Phys J Plus. 2018;133:113.  https://doi.org/10.1140/epjp/i2018-11953-8.CrossRefGoogle Scholar
  20. 20.
    Sopasakis P, Sarimveis H, Macheras P, Dokoumetzidis A. Fractional calculus in pharmacokinetics. J Pharmacokin Pharmacodyn. 2018;45:107–14.CrossRefGoogle Scholar
  21. 21.
    Jahanbakhshi A, Nadooshan AA, Bayareh M. Magnetic field effects on natural convection flow of a non-Newtonian fluid in an L-shaped enclosure. J Therm Anal Calorim. 2018;133:1407–16.  https://doi.org/10.1007/s10973-018-7219-6.CrossRefGoogle Scholar
  22. 22.
    Kashif AA, Khan I, Gómez-Aguilar JF. A mathematical analysis of a circular pipe in rate type fluid via Hankel transform. Eur Phys J Plus. 2018;133:397.  https://doi.org/10.1140/epjp/i2018-12186-7.CrossRefGoogle Scholar
  23. 23.
    Kashif AA, Hussain M, Baig MM. Influences of magnetic field in viscoelastic fluid. Int J Nonlinear Anal Appl. 2018;9:99–109.  https://doi.org/10.22075/ijnaa.2017.1451.1367.CrossRefGoogle Scholar
  24. 24.
    Abro AK, Muhammad AS, Muzaffar HL. Influence of slippage in heat and mass transfer for fractionalized MHD flows in porous medium. Int J Adv Appl Math Mech. 2017;4:5–14.Google Scholar
  25. 25.
    Bahiraei M, Hangi M. Natural convection of magnetic nanofluid in a cavity under non-uniform magnetic field: a novel application. J Supercond Nov Magn. 2014;27:587–94.  https://doi.org/10.1007/s10948-013-2317-y.CrossRefGoogle Scholar
  26. 26.
    Bahiraei M, Hangi M. Flow and heat transfer characteristics of magnetic nanofluids: a review. J Magn Magn Mater. 2015;374:125–38.CrossRefGoogle Scholar
  27. 27.
    Bahiraei M, Hangi M. Automatic cooling by means of thermomagnetic phenomenon of magnetic nanofluid in a toroidal loop. Appl Therm Eng. 2016;107:700–8.  https://doi.org/10.1016/j.applthermaleng.2016.07.021.CrossRefGoogle Scholar
  28. 28.
    Bahiraei M, Hangi M. Investigating the effect of line dipole magnetic field on hydrothermal characteristics of a temperature-sensitive magnetic nanofluid using two-phase simulation. Nanoscale Res Lett. 2016;11:443.CrossRefGoogle Scholar
  29. 29.
    Mugheri DM, Kashif AA, Solangi MA. Application of modern approach of Caputo–Fabrizio fractional derivative to MHD second grade fluid through oscillating porous plate with heat and mass transfer. Int J Adv Appl Sci. 2018;5:97–105.CrossRefGoogle Scholar
  30. 30.
    Bahiraei M, Hangi M, Monavari A. Assessment of hydrothermal characteristics of Mn–Zn ferrite nanofluid as a functional material under quadrupole magnetic field. Powder Technol. 2017;305:174–82.CrossRefGoogle Scholar
  31. 31.
    Kashif AA, Hussain M, Baig MM. Analytical solution of MHD generalized Burger’s fluid embedded with porosity. Int J Adv Appl Sci. 2017;4:80–9.Google Scholar
  32. 32.
    Kashif AA, Hussain M, Baig MM. Slippage of fractionalized Oldroyd-B fluid with magnetic field in porous medium. Progr Fract Differ Appl Int J. 2017;3:69–80.CrossRefGoogle Scholar
  33. 33.
    Hristov J. Steady-state heat conduction in a medium with spatial non-singular fading memory: derivation of Caputo–Fabrizio space-fractional derivative with Jeffrey’s kernel and analytical solutions. Therm Sci. 2017;21:827–39.CrossRefGoogle Scholar
  34. 34.
    Ali F, Saqib M, Khan I, Sheikh NA. Application of Caputo–Fabrizio derivatives to MHD free convection flow of generalized Walters’-B fluid model. Eur Phys J Plus. 2016;131:377.CrossRefGoogle Scholar
  35. 35.
    Atangana A, Baleanu D. Caputo–Fabrizio derivative applied to groundwater flow within confined aquifer. J Eng Mech. 2016.  https://doi.org/10.1061/(asce)em.1943-7889.0001091.CrossRefGoogle Scholar
  36. 36.
    Kashif AA, Khan I. Analysis of heat and mass transfer in MHD flow of generalized Casson fluid in a porous space via non-integer order derivative without singular kernel. Chin J Phys. 2017;55:1583–95.CrossRefGoogle Scholar
  37. 37.
    Shah NA, Khan I. Heat transfer analysis in a second grade fluid over and oscillating vertical plate using fractional Caputo–Fabrizio derivatives. Eur Phys J C. 2016;76:1–11.CrossRefGoogle Scholar
  38. 38.
    Nadeem AS, Ali F, Saqib M, Khan I, Aftab SAJ, Alshomrani AS, Alghamdi MS. Comparison and analysis of the Atangana–Baleanu and Caputo–Fabrizio fractional derivatives for generalized Casson fluid model with heat generation and chemical reaction. Results Phys. 2017;7:789–800.CrossRefGoogle Scholar
  39. 39.
    Owolabi KM, Atangana A. Numerical approximation of nonlinear fractional parabolic differential equations with Caputo–Fabrizio derivative in Riemann–Liouville sense. Chaos Solitons Fractals. 2017;99:171–9.CrossRefGoogle Scholar
  40. 40.
    Al-Salti N, Karimov E, Sadarangani K. On a differential equation with Caputo–Fabrizio fractional derivative of order and application to mass-spring-damper system. Progr Fract Differ Appl. 2015;2:257–63.  https://doi.org/10.18576/pfda/020403.CrossRefGoogle Scholar
  41. 41.
    Saqib M, Ali F, Khan I, Sheikh NA, Aftab SAJ, Samiulhaq. Exact solutions for free convection flow of generalized Jeffrey fluid: a Caputo–Fabrizio fractional model. Alex Eng J. 2017.  https://doi.org/10.1016/j.aej.2017.03.017.CrossRefGoogle Scholar
  42. 42.
    Imran MA, Khan I, Ahmad M, Shah NA, Nazar M. Heat and mass transport of differential type fluid with non-integer order time-fractional Caputo derivatives. J Mol Liq. 2017;229:67–75.CrossRefGoogle Scholar
  43. 43.
    Sheikh NA, Ali F, Khan I, Saqib M. 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 Appl. 2016.  https://doi.org/10.1007/s00521-016-2815-5.CrossRefGoogle Scholar
  44. 44.
    Owolabi KM, Atangana A. Analysis and application of new fractional Adams–Bashforth scheme with Caputo–Fabrizio derivative. Chaos Solitons Fractals. 2017;105:111–9.CrossRefGoogle Scholar
  45. 45.
    Kashif AA, Solangi MA. Heat transfer in magnetohydrodynamic second grade fluid with porous impacts using Caputo–Fabrizio fractional derivatives. Punjab Univ J Math. 2017;49:113–25.Google Scholar
  46. 46.
    Ali F, Sheikh NA, Khan I, Saqib M. Magnetic field effect on blood flow of Casson fluid in axisymmetric cylindrical tube: a fractional model. J Magn Magn Mater. 2017;423:327–36.CrossRefGoogle Scholar
  47. 47.
    Atangana A, Koca I. Model of thin viscous fluid sheet flow within the scope of fractional calculus: fractional derivative with and no singular kernel. Fundam Inform. 2017;151:145–59.  https://doi.org/10.3233/FI-2017-1484.CrossRefGoogle Scholar
  48. 48.
    Kashif AA, Chandio AD, Irfan AA, Khan I. Dual thermal analysis of magnetohydrodynamic flow of nanofluids via modern approaches of Caputo–Fabrizio and Atangana–Baleanu fractional derivatives embedded in porous medium. J Therm Anal Calorim. 2018.  https://doi.org/10.1007/s10973-018-7302-z.CrossRefGoogle Scholar
  49. 49.
    Ali F, Khan I, Samiulhaq, Sharidan S. A note on new exact solutions for some unsteady flows of Brinkman-Type fluids over a plane wall. Z Naturforsch. 2012;67:377–80.  https://doi.org/10.5560/zna.2012-0039.CrossRefGoogle Scholar
  50. 50.
    Khan A, Kashif AA, Asifa T, Khan I. Atangana–Baleanu and Caputo–Fabrizio analysis of fractional derivatives for heat and mass transfer of second grade fluids over a vertical plate: a comparative study. Entropy. 2017;19:1–12.CrossRefGoogle Scholar
  51. 51.
    Khan I, Kashif AA. Thermal analysis in Stokes’ second problem of nanofluid: applications in thermal engineering. Case Stud Therm Eng. 2018;12:271–5.  https://doi.org/10.1016/j.csite.2018.04.005.CrossRefGoogle Scholar
  52. 52.
    Al-Mdallal Q, Kashif A A, Khan I. Analytical solutions of fractional Walter’s-B fluid with applications. Complexity. 2018; Article ID 8918541.Google Scholar
  53. 53.
    Caputo M, Fabrizio M. A new definition of fractional derivative without singular kernel. Progr Fract Differ Appl. 2015;1:1–13.Google Scholar
  54. 54.
    Kashif AA, Shaikh AA, Dehraj S. Exact solutions on the oscillating plate of Maxwell fluids. Mehran Univ Res J Eng Technol. 2016;35:157–62.Google Scholar
  55. 55.
    Kashif AA, Sumera D, Baig MM. Effects of transverse magnetic field on oscillating plate of second grade fluid. Sindh Univ Res J Sci Ser. 2016;48:605–10.Google Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2018

Authors and Affiliations

  • Ambreen Siyal
    • 1
  • Kashif Ali Abro
    • 1
  • Muhammad Anwar Solangi
    • 1
  1. 1.Department of Basic Sciences and Related StudiesMehran University of Engineering and TechnologyJamshoroPakistan

Personalised recommendations