Influence of a uniform transverse magnetic field on the thermo-hydrodynamic stability in water-based nanofluids with metallic nanoparticles using the generalized Buongiorno’s mathematical model

  • Abderrahim Wakif
  • Zoubair Boulahia
  • S. R. Mishra
  • Mohammad Mehdi Rashidi
  • Rachid Sehaqui
Regular Article


The onset of nanofluid convection in the presence of an externally applied magnetic field is investigated numerically based on the non-homogeneous Buongiorno’s mathematical model. In this study, we use the latest experimental correlations and powerful analytical models for expressing the thermo-physical properties of some electrically conducting nanofluids, such as copper-water, sliver-water and gold-water nanofluids, in which the Brownian motion and thermophoresis effects on slip flow in nanofluids are taken into account in this model (i.e., two-phase transport model). In this paper, we assume that the nanofluid has Newtonian behavior, confined horizontally between two infinite impermeable boundaries and heated from below, in such a way that the nanoparticles tend to concentrate near the upper wall. Considering the basic state of the nanofluidic system, the linear stability theory has been successfully applied to obtain the principal stability equations, which are solved numerically for an imposed volumetric fraction of nanoparticles and no-slip impermeable conditions at the isothermal walls bounding the nanofluid layer. The linear boundary-value problem obtained in this investigation is converted into a pure initial-value problem, so that we can solve it numerically by the fourth-fifth-order Runge-Kutta-Fehlberg method. The generalized Buongiorno’s mathematical model proposed in this study allows performing a highly accurate computational analysis. In addition, the obtained results show that the stability of the studied nanofluidic system depends on several parameters, namely, the magnetic Chandrasekhar number Q , the reference value for the volumetric fraction of nanoparticles \( \phi_0\) and the size of nanoparticles \( d_p\) . In this analysis, the thermo-hydrodynamic stability of the studied nanofluid is controlled through the critical thermal Rayleigh number \( R_{ac}\) , which characterizes the onset of convection cells, whose size is \( L_c=2\pi/a_c\) . Furthermore, the effects of various pertinent parameters on the critical stability parameters \( R_{ac}\) and \( a_c\) are discussed in more detail through graphical and tabular illustrations, for three types of nanofluids including copper-water, sliver-water, and gold-water.


  1. 1.
    S.U.S. Choi, Enhancing thermal conductivity of fluids with nanoparticles, in Proceedings of the 1995 ASME International Mechanical Engineering Congress and Exposition, Vol. 231 (ASME, 1995) pp. 99--105Google Scholar
  2. 2.
    J. Buongiorno, J. Heat Transf. 128, 240 (2006)CrossRefGoogle Scholar
  3. 3.
    P.G. Siddheshwar, C. Kanchana, Y. Kakimoto, A. Nakayama, J. Heat Transf. 139, 012402 (2017)CrossRefGoogle Scholar
  4. 4.
    A. Wakif, Z. Boulahia, R. Sehaqui, Int. J. Adv. Comput. Sci. Appl. 7, 299 (2016)Google Scholar
  5. 5.
    A. Wakif, Z. Boulahia, R. Sehaqui, J. Nanofluids 6, 136 (2017)CrossRefGoogle Scholar
  6. 6.
    B.S. Bhadauria, S. Agarwal, A. Kumar, Transp. Porous Media 90, 605 (2011)MathSciNetCrossRefGoogle Scholar
  7. 7.
    B.S. Bhadauria, P. Kiran, M. Belhaq, MATEC Web Conf. 16, 09003 (2014)CrossRefGoogle Scholar
  8. 8.
    D. Yadav, C. Kim, J. Lee, H.H. Cho, Comput. Fluids 121, 26 (2015)MathSciNetCrossRefGoogle Scholar
  9. 9.
    T. Hayat, T. Muhammad, S.A. Shehzad, A. Alsaedi, Int. J. Therm. Sci. 111, 274 (2017)CrossRefGoogle Scholar
  10. 10.
    A. Wakif, Z. Boulahia, M. Zaydan et al., Int. J. Innov. Appl. Stud. 14, 1048 (2016)Google Scholar
  11. 11.
    A. Wakif, Z. Boulahia, R. Sehaqui, Results Phys. 7, 2134 (2017)CrossRefGoogle Scholar
  12. 12.
    S. Chandrasekhar, Hydrodynamic and hydromagnetic stability (Oxford University Press, Oxford, 1961)Google Scholar
  13. 13.
    D.A. Nield, A.V. Kuznetsov, Eur. J. Mech. 29, 217 (2010)CrossRefGoogle Scholar
  14. 14.
    D. Yadav, J. Lee, Eur. Phys. J. Plus 130, 162 (2015)CrossRefGoogle Scholar
  15. 15.
    D. Yadav, J. Wang, R. Bhargava et al., Appl. Therm. Eng. 103, 1441 (2016)CrossRefGoogle Scholar
  16. 16.
    G.S. McNab, A. Meisen, J. Colloid Interface Sci. 44, 339 (1973)ADSCrossRefGoogle Scholar
  17. 17.
    M. Corcione, Energy Convers. Manag. 52, 789 (2011)CrossRefGoogle Scholar
  18. 18.
    T. Armaghani, A. Kasaeipoor, N. Alavi, M.M. Rashidi, J. Mol. Liq. 223, 243 (2016)CrossRefGoogle Scholar
  19. 19.
    M. Sheikholeslami, D.D. Ganji, M. Gorji-Bandpy, S. Soleimani, J. Taiwan Inst. Chem. Eng. 45, 795 (2014)CrossRefGoogle Scholar
  20. 20.
    K. Mehmood, S. Hussain, M. Sagheer, Int. J. Heat Mass Transfer 109, 397 (2017)CrossRefGoogle Scholar
  21. 21.
    J.C.M. Garnett, Philos. Trans. R. Soc. London Ser. A 203, 385 (1904)ADSCrossRefGoogle Scholar
  22. 22.
    A.H. Sihvola, I.V. Lindell, Effective permeability of mixtures (Helsinki University of Technology, 1989)Google Scholar
  23. 23.
    D. Yadav, R. Bhargava, G.S. Agrawal, J. Eng. Math. 80, 147 (2013)CrossRefGoogle Scholar
  24. 24.
    A. Wakif, Z. Boulahia, R. Sehaqui, Results Phys. (2018)
  25. 25.
    J.F. Schenck, Med. Phys. 23, 815 (1996)CrossRefGoogle Scholar
  26. 26.
    P.G. Siddheshwar, N. Meenakshi, Int. J. Appl. Comput. Math. 3, 271 (2017)MathSciNetCrossRefGoogle Scholar
  27. 27.
    T. Hayat, B. Ahmed, F.M. Abbasi, A. Alsaedi, J. Mol. Liq. 234, 324 (2017)CrossRefGoogle Scholar

Copyright information

© Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Abderrahim Wakif
    • 1
  • Zoubair Boulahia
    • 1
  • S. R. Mishra
    • 2
  • Mohammad Mehdi Rashidi
    • 3
  • Rachid Sehaqui
    • 1
  1. 1.Hassan II UniversityFaculty of Sciences Aïn Chock, Laboratory of MechanicsMâarif, CasablancaMorocco
  2. 2.Department of MathematicsSiksha ‘O’Anusandhan UniversityBhubaneswarIndia
  3. 3.University of BirminghamSchool of Civil EngineeringBirminghamUK

Personalised recommendations