Skip to main content

Numerical simulation of non-uniform heating due to magnetohydrodynamic natural convection in a nanofluid filled rhombic enclosure


Numerical simulation of magnetohydrodynamic natural convection heat transfer in a rhombic enclosure of inclination angle \({\mathrm {45}}^{\mathrm {^{\circ }}}\) containing copper-water nanofluid has been presented in this paper. The top and bottom walls of the enclosure are subjected to non-uniform heating while left wall being subjected to lower temperature and right wall being maintained adiabatic. The finite element strategy (COMSOL Multiphysics) is used to solve the governing equations. The numerical simulations are done for the parametric values: 10\(^{4\, }\le \) Rayleigh number \(\le \) 10\(^{6}\); 0 \(\le \) Hartmann number \(\le \) 100; 0 \(\le \) volume fraction of nanofluid \(\le \) 0.05. The phase deviation angle (top wall) is varied in the range from 0 to \({\uppi }\) with amplitude of non linear heating being maintained constant. The motivation of this research goes with the fact that the associated transport phenomenon conveys the implication of designing an optimal thermal system analogous to the theme of non-uniform heating, with the phase angle being a crucial design parameter. The numerical results depict to the fact, that the rate of heat transfer follows non-monotonic trends and is considerably influenced by interplay of the phase shift angle, Rayleigh number and Hartmann number. The results showed that at Rayleigh number \(\ge 10^{5}\), the heat transfer rate gets inhibited by enhancing the magnetic field intensity. The impact of different types of nano particles is illustrated by comparing the results with the results of three different nanofluids, silver– water, titanium dioxide–water and diamond–water nanofluids.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14


\(\hbox {B}_{0}\) :

Magnetic field strength, N m\(^{-1}\)A\(^{-1}\)

\(\hbox {c}_{\mathrm{p}}\) :

Specific heat, Jkg\(^{-1}\)K\(^{-1}\)


Enclosure length, m


Acceleration due to gravity, ms\(^{-2}\)


Heat transfer coefficient, Wm\(^{-2}\)K\(^{-1}\)


Thermal conductivity Wm\(^{-1}\)K\(^{-1}\)

\(\hbox {Nu}_{\mathrm{l}}\) :

Local Nusselt number

\(\hbox {Nu}_{\mathrm{avg}}\) :

Average Nusselt number


Pressure, Nm\(^{-2}\)


Dimensionless pressure


Prandtl number


Rayleigh number


Hartmann number

\(\hbox {T}_{\mathrm{h}}\) :

Temperature of heated walls, K

\(\hbox {T}_{\mathrm{c}}\) :

Temperature of cold walls of enclosure, K


Velocity components in x and y directions, ms\(^{-1}\)


Dimensionless velocity components in X and Y directions


Cartesian coordinates


Dimensionless coordinates

\(\uprho \) :

Density of fluid, kgm\(^{-3}\)

\(\upalpha \) :

Thermal diffusivity, m\(^{2}\)s\(^{-1}\)

\(\upbeta \) :

Thermal expansion coefficient, K\(^{-1}\)

\(\upnu \) :

Kinematic viscosity, m\(^{2}\)s\(^{-1}\)

\(\uptheta \) :

Dimensionless temperature

\(\upvarphi \) :

Volume fraction

\(\upsigma \) :

Electrical conductivity, Sm\(^{-1}\)

\(\lambda \) :

The phase angle of the top wall of rhombic enclosure

\(\varPhi \) :

Inclination angle of rhombic enclosure

\(\varepsilon \) :

Amplitude parameter of top wall of rhombic enclosure








  1. S. Kakac, W. Aung, R. Viskanta, Natural convection: fundamentals and applications (1985)

  2. A. Baïri, E. Zarco-Pernia, J.M. García de María, Appl. Therm. Eng. 63, 304 (2014)

    Article  Google Scholar 

  3. S.U.S. Choi, J. Eastman, Enhancing thermal conductivity of fluids with nanoparticles (1995)

  4. J. Eastman, S. Choi, S. Li, W. Yu, L. Thompson, Appl. Phys. Lett. 78, 718 (2001)

    ADS  Article  Google Scholar 

  5. Y. Xuan, W. Roetzel, Int. J. Heat Mass Transf. 43, 3701 (2000)

    Article  Google Scholar 

  6. N. Putra, W. Roetzel, S.K. Das, Heat Mass Transf. 39, 775 (2003)

    ADS  Article  Google Scholar 

  7. Y.-S. Lin, P.-Y. Hsiao, C.-C. Chieng, Int. J. Therm. Sci. 62, 56 (2012)

    Article  Google Scholar 

  8. K.S. Hwang, J.-H. Lee, S.P. Jang, Int. J. Heat Mass Transf. 50, 4003 (2007)

    Article  Google Scholar 

  9. H.F. Oztop, E. Abu-Nada, Int. J. Heat Fluid Flow 29, 1326 (2008)

    Article  Google Scholar 

  10. A.K. Santra, S. Sen, N. Chakraborty, Int. J. Therm. Sci. 47, 1113 (2008)

    Article  Google Scholar 

  11. E. Büyük Öğüt, Int. J. Therm. Sci. 48, 2063 (2009)

  12. B. Ghasemi, S. Aminossadati, Int. J. Therm. Sci. 49, 1 (2010)

    Article  Google Scholar 

  13. S. Parvin, A.J. Chamkha, Int. Commun. Heat Mass Transf. 54, 8 (2014)

    Article  Google Scholar 

  14. V. Krishan, in: V. Krishan (Ed.), Astrophysical Plasmas and Fluids (Springer Netherlands, Dordrecht, 1999), p. 117

  15. L. Zhang, M.M. Bhatti, E.E. Michaelides, M. Marin, R. Ellahi, Eur. Phys. J. Spec. Top. (2021)

  16. L. Zhang, M.M. Bhatti, A. Shahid, R. Ellahi, O.A. Bég, S.M. Sait, J. Taiwan Inst. Chem. Eng. 124, 98 (2021)

    Article  Google Scholar 

  17. M.M. Bhatti, L. Phali, C.M. Khalique, Arch. Appl. Mech. 91, 1683 (2021)

    ADS  Article  Google Scholar 

  18. S. Dutta, S. Pati, J. Thermophys. Heat Transf. 1 (2022)

  19. N.A. Shah, S. Wang, T. Elnaqeeb, H. Qi, 24, 45 (2021)

  20. N. Ali Shah, N. Ahmed, T. Elnaqeeb, M.M. Rashidi, J. Appl. Comput. Mech. 5, 150 (2019)

    Google Scholar 

  21. Y. Ma, R. Mohebbi, M.M. Rashidi, Z. Yang, M.A. Sheremet, Int. J. Heat Mass Transf. 130, 123 (2019)

    Article  Google Scholar 

  22. R.U. Haq, F.A. Soomro, T. Mekkaoui, Q.M. Al-Mdallal, Int. J. Heat Mass Transf. 121, 1168 (2018)

    Article  Google Scholar 

  23. B. Ghasemi, S.M. Aminossadati, A. Raisi, Int. J. Therm. Sci. 50, 1748 (2011)

    Article  Google Scholar 

  24. A.H. Mahmoudi, I. Pop, M. Shahi, Int. J. Therm. Sci. 59, 126 (2012)

    Article  Google Scholar 

  25. K.S. Al Kalbani, M.M. Rahman, S. Alam, N. Al-Salti, I.A. Eltayeb, Heat Transf. Eng. 39, 511 (2018)

    ADS  Article  Google Scholar 

  26. M.B. Ben Hamida, K. Charrada, Numer. Heat Transf. Part A: Appl. 67, 902 (2015)

  27. F. Moukalled, H. Diab, S. Acharya, Numer. Heat Transf. Part A: Appl. 24, 89 (1993)

  28. R. Anandalakshmi, T. Basak, Int. J. Heat Mass Transf. 55, 1325 (2012)

    Article  Google Scholar 

  29. R. Anandalakshmi, T. Basak, Eur. J. Mech. B. Fluids 41, 29 (2013)

  30. S. Dutta, N. Goswami, A.K. Biswas, S. Pati, Int. J. Heat Mass Transf. 136, 777 (2019)

    Article  Google Scholar 

  31. S. Dutta, N. Goswami, S. Pati, A.K. Biswas, J. Therm. Anal. Calorim. 144, 1493 (2021)

    Article  Google Scholar 

  32. S. Dutta, S. Pati, L. Baranyi, Case Stud. Thermal Eng. 28, 101507 (2021)

    Article  Google Scholar 

  33. T. Basak, S. Roy, T. Paul, I. Pop, Int. J. Heat Mass Transf. 49, 1430 (2006)

    Article  Google Scholar 

  34. S. Dutta, S. Pati, p. 1 (2021)

  35. M. Torki, N. Etesami, J. Therm. Anal. Calorim. 139, 1565 (2020)

    Article  Google Scholar 

  36. S. Zeinali Heris, M. Pour, O. Mahian, S. Wongwises, Int. J. Heat Mass Transf. 73, 231 (2014)

    Article  Google Scholar 

  37. Y. Hu, Y. He, C. Qi, B. Jiang, H. Inaki Schlaberg, Int. J. Heat Mass Transf. 78, 380 (2014)

    Article  Google Scholar 

  38. E. Khalili, A. Saboonchi, M. Saghafian, Int. J. Therm. Sci. 112, 82 (2017)

    Article  Google Scholar 

  39. X.-Q. Wang, A. Mujumdar, Br. J. Chem. Eng. 25, 613 (2008)

    Article  Google Scholar 

  40. K. Khanafer, K. Vafai, M. Lightstone, Int. J. Heat Mass Transf. 46, 3639 (2003)

    Article  Google Scholar 

  41. O. Abouali, G. Ahmadi, Appl. Therm. Eng. 36, 1 (2012)

    Article  Google Scholar 

  42. T. Elnaqeeb, Eur. Phys. J. Spec. Top. 228, 2695 (2019)

    Article  Google Scholar 

  43. H.C. Brinkman, J. Chem. Phys. 20, 571 (1952)

    ADS  Article  Google Scholar 

  44. J.C. Maxwell, A Treatise on Electricity and Magnetism (Clarendon Press, New York, 1873)

    MATH  Google Scholar 

  45. COMSOL Multiphysics v. 5.2. COMSOL AB, Stockholm, Sweden

  46. R.W.L. Perumal Nithiarasu, Kankanhalli N. Seetharamu, Fundamentals of the Finite Element Method for Heat and Mass Transfer (2016)

  47. M. Thangavelu, N. Nagarajan, R.-J. Yang, Heat Transf. Eng. 1 (2021)

  48. M. Sharifpur, A.B. Solomon, T.L. Ottermann, J.P. Meyer, Int. Commun. Heat Mass Transf. 98, 297 (2018)

    Article  Google Scholar 

  49. M.D. Massoudi, M.B. Ben Hamida, Eur. Phys. J. Plus 135, 1 (2020)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Thanaa Elnaqeeb.

Ethics declarations

Conflict of interest

The authors declare no conflict of interest.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Dutta, S., Elnaqeeb, T. Numerical simulation of non-uniform heating due to magnetohydrodynamic natural convection in a nanofluid filled rhombic enclosure. Eur. Phys. J. Spec. Top. (2022).

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: