Mixed convection in a nanofluid-filled sloshing porous cavity including inner heated rose


In this paper, study of the heat transfer by mixed convection in sloshing enclosures heated from inside by an inner rose and filled with a porous medium using nanofluids is carried out. The Hazen–Dupain–Darcy model is applied to represent the porous medium, while the nanofluids are investigated using the one-phase model. The governing equations are converted to the dimensionless form and then solved numerically using the finite volume method. The controlling parameters in this simulation are numbers of the rose petals k, lengths of the rose petals a, the Richardson number Ri, the sloshing cavity amplitude A, the resonance frequency ω, the nanoparticle volume fraction ϕ and the Darcy number Da. It is found that rate of the heat transfer in case of the even number of the rose petals is greater than that of the odd number case. Also, the increase in either the sloshing cavity amplitude A or lengths of the rose petals a enhances the local Nusselt number, while as the resonance frequency ω increases, the average Nusselt number around the rose is reduced.

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
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19


\(A\) :

Wavy amplitude

b, c :

Ergun’s constants

\(C_{\text{p}}\) :

Specific heat

\({\text{Da}}\) :

Darcy parameter

\(d_{\text{P}}\) :

Average particle size

\(F\) :

Forchheimer coefficient

\({\text{Gr}}\) :

Grashof number

\(g\) :

Gravitational acceleration \(\left( {{\text{m s}}^{-2} } \right)\)

\(K\) :

Permeability \(( {\text{m}}^{2} )\)

\(k\) :

Thermal conductivity \(({\text{W}}\,{\text{m}}^{ - 1} {\text{K}}^{ - 1} )\)

\({\text{Nu}}\) :

Nusselt number

\(P\) :

Pressure \(({\text{N m}}^{-2})\)

\({ \Pr }\) :

Prandtl number

\({\text{Re}}\) :

Reynolds number

\({\text{Ri}}\) :

Richardson parameter

\(T\) :

Temperature \(({\text{K}})\)

\(t\) :

Time (s)

\(\varvec{u}\) :

Velocity vector \(({\text{m s}}^{-1})\)

\(u, v\) :

Dimension velocity components \(({\text{m s}}^{-1})\)

\(U, V\) :

Dimensionless velocity components

\(x, y\) :

Cartesian coordinates (m)

\(X, Y\) :

Dimensionless coordinates

\(\alpha\) :

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

\(\beta\) :

Thermal expansion coefficient (K−1)

\(\varepsilon\) :


\(\phi\) :

Solid volume fraction

\(\mu\) :


\(\nu\) :

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

\(\rho\) :

Density \(({\text{kg m}}^{-3})\)

\(\sigma\) :

Capacity ratio

\(\tau\) :

Dimensionless time

\(\omega\) :

Resonance frequency






Porous medium


Dispersed nanoparticles


Solid porous matrix


Porous medium with nanofluid


  1. 1.

    Rashad AM, Ismael MA, Chamkha AJ, Mansour MA. MHD mixed convection of localized heat source/sink in a nanofluid-filled lid-driven square cavity with partial slip. J Taiwan Inst Chem Eng. 2016;68:173–86. https://doi.org/10.1016/j.jtice.2016.08.033.

    CAS  Article  Google Scholar 

  2. 2.

    Rahmati AR, Rayat Roknabadi A, Abbaszadeh M. Numerical simulation of mixed convection heat transfer of nanofluid in a double lid-driven cavity using lattice Boltzmann method. Alex Eng J. 2016;55(4):3101–14. https://doi.org/10.1016/j.aej.2016.08.017.

    Article  Google Scholar 

  3. 3.

    Garoosi F, Rashidi MM. Conjugate-mixed convection heat transfer in a two-sided lid-driven cavity filled with nanofluid using Manninen’s two phase model. Int J Mech Sci. 2017;131:1026–48. https://doi.org/10.1016/j.ijmecsci.2017.08.030.

    Article  Google Scholar 

  4. 4.

    Kareem AK, Mohammed HA, Hussein AK, Gao S. Numerical investigation of mixed convection heat transfer of nanofluids in a lid-driven trapezoidal cavity. Int Commun Heat Mass Transfer. 2016;77:195–205. https://doi.org/10.1016/j.icheatmasstransfer.2016.08.010.

    CAS  Article  Google Scholar 

  5. 5.

    Kareem AK, Gao S. Computational study of unsteady mixed convection heat transfer of nanofluids in a 3D closed lid-driven cavity. Int Commun Heat Mass Transf. 2017;82:125–38. https://doi.org/10.1016/j.icheatmasstransfer.2017.02.009.

    CAS  Article  Google Scholar 

  6. 6.

    Elshehabey HM, Ahmed SE. MHD mixed convection in a lid-driven cavity filled by a nanofluid with sinusoidal temperature distribution on the both vertical walls using Buongiorno’s nanofluid model. Int J Heat Mass Transf. 2015;88:181–202. https://doi.org/10.1016/j.ijheatmasstransfer.2015.04.039.

    CAS  Article  Google Scholar 

  7. 7.

    Mansour MA, Mohamed RA, Abd-Elaziz MM, Ahmed SE. Numerical simulation of mixed convection flows in a square lid-driven cavity partially heated from below using nanofluid. Int Commun Heat Mass Transfer. 2010;37(10):1504–12.

    CAS  Article  Google Scholar 

  8. 8.

    Mansour M, Ahmed SE. Mixed convection in double lid-driven enclosures filled with Al2O3–water nanofluid. J Thermophys Heat Transf. 2013;27(4):707–18. https://doi.org/10.2514/1.t4102.

    CAS  Article  Google Scholar 

  9. 9.

    Chamkha A, Rashad A, Mansour M, Armaghani T, Ghalambaz M. Effects of heat sink and source and entropy generation on MHD mixed convection of a Cu-water nanofluid in a lid-driven square porous enclosure with partial slip. Phys Fluids. 2017;29(5):052001. https://doi.org/10.1063/1.4981911.

    CAS  Article  Google Scholar 

  10. 10.

    Devendiran DK, Amirtham VA. A review on preparation, characterization, properties and applications of nanofluids. Renew Sustain Energy Rev. 2016;60:21–40. https://doi.org/10.1016/j.rser.2016.01.055.

    CAS  Article  Google Scholar 

  11. 11.

    Ganvir RB, Walke PV, Kriplani VM. Heat transfer characteristics in nanofluid—a review. Renew Sustain Energy Rev. 2017;75:451–60. https://doi.org/10.1016/j.rser.2016.11.010.

    Article  Google Scholar 

  12. 12.

    Mahian O, Kianifar A, Kalogirou SA, Pop I, Wongwises S. A review of the applications of nanofluids in solar energy. Int J Heat Mass Transf. 2013;57(2):582–94. https://doi.org/10.1016/j.ijheatmasstransfer.2012.10.037.

    CAS  Article  Google Scholar 

  13. 13.

    Khanafer K, Vafai K. A review on the applications of nanofluids in solar energy field. Renew Energy. 2018;123:398–406. https://doi.org/10.1016/j.renene.2018.01.097.

    CAS  Article  Google Scholar 

  14. 14.

    Muhammad MJ, Muhammad IA, Sidik NAC, Yazid MNAWM, Mamat R, Najafi G. The use of nanofluids for enhancing the thermal performance of stationary solar collectors: a review. Renew Sustain Energy Rev. 2016;63:226–36. https://doi.org/10.1016/j.rser.2016.05.063.

    Article  Google Scholar 

  15. 15.

    Mahian Omid, Kolsi Lioua, Amani Mohammad, Estellé Patrice, Ahmadi Goodarz, Kleinstreuer Clement, Marshall Jeffrey S, Siavashi Majid, Taylor Robert A, Niazmand Hamid, Wongwises Somchai, Hayat Tasawar, Kolanjiyil Arun, Kasaeian Alibakhsh, Pop Ioan. Recent advances in modeling and simulation of nanofluid flows-part I: fundamentals and theory. Phys Rep. 2019;790:1–48. https://doi.org/10.1016/j.physrep.2018.11.004.

    CAS  Article  Google Scholar 

  16. 16.

    Mahian Omid, Kolsi Lioua, Amani Mohammad, Estellé Patrice, Ahmadi Goodarz, Kleinstreuer Clement, Marshall Jeffrey S, Taylor Robert A, Abu-Nada Eiyad, Rashidi Saman, Niazmand Hamid, Wongwises Somchai, Hayat Tasawar, Kasaeian Alibakhsh, Pop Ioan. Recent advances in modeling and simulation of nanofluid flows—part II: applications. Phys Rep. 2019;791:1–59. https://doi.org/10.1016/j.physrep.2018.11.003.

    CAS  Article  Google Scholar 

  17. 17.

    Rashidi S, Mahian O, Languri EM. Applications of nanofluids in condensing and evaporating systems. J Therm Anal Calorim. 2018;131:2027–39.

    CAS  Article  Google Scholar 

  18. 18.

    Rashidi S, Eskandarian M, Mahian O, Poncet S. Combination of nanofluid and inserts for heat transfer enhancement. J Therm Anal Calorim. 2019;135:437–60.

    CAS  Article  Google Scholar 

  19. 19.

    Rashidi S, Esfahani JA, Maskaniyan M. Applications of magnetohydrodynamics in biological systems-a review on the numerical studies. J Magn Magn Mater. 2017;439:358–72. https://doi.org/10.1016/j.jmmm.2017.05.014.

    CAS  Article  Google Scholar 

  20. 20.

    Nguyen MT, Aly AM, Lee S-W. Natural convection in a non-Darcy porous cavity filled with Cu–water nanofluid using the characteristic-based split procedure in finite-element method. Numer Heat Transf Part A Appl Int J Comput Methodol. 2015;67(2):224–47. https://doi.org/10.1080/10407782.2014.923225.

    CAS  Article  Google Scholar 

  21. 21.

    Chattopadhyay A, Pandit SK, Sen Sarma S, Pop I. Mixed convection in a double lid-driven sinusoidally heated porous cavity. Int J Heat Mass Transf. 2016;93:361–78. https://doi.org/10.1016/j.ijheatmasstransfer.2015.10.010.

    Article  Google Scholar 

  22. 22.

    Begum AS, Nithyadevi N, Öztop HF, Al-Salem K. Numerical simulation of MHD mixed convection in a nanofluid filled non-darcy porous enclosure. Int J Mech Sci. 2017;130:154–66. https://doi.org/10.1016/j.ijmecsci.2017.06.008.

    Article  Google Scholar 

  23. 23.

    Hatami M, Zhou J, Geng J, Song D, Jing D. Optimization of a lid-driven T-shaped porous cavity to improve the nanofluids mixed convection heat transfer. J Mol Liq. 2017;231:620–31. https://doi.org/10.1016/j.molliq.2017.02.048.

    CAS  Article  Google Scholar 

  24. 24.

    Aly AM. Double-diffusive natural convection in a non-Darcy porous cavity filled with nanofluid under the effects of chemical reaction. J Porous Media. 2017;20(2):111–26. https://doi.org/10.1615/jpormedia.v20.i2.20.

    Article  Google Scholar 

  25. 25.

    Aly AM, Raizah ZAS, Ahmed SE. Mixed convection in a cavity saturated with wavy layer porous medium: entropygeneration. J Thermophys Heat Transfer. 2018;32(3):764–80. https://doi.org/10.2514/1.t5369.

    CAS  Article  Google Scholar 

  26. 26.

    Aly AM, Raizah ZAS. Mixed convection in an inclined nanofluid filled-cavity saturated with a partially layered porous medium. J Therm Sci Eng Appl. 2019;11(4):041002. https://doi.org/10.1115/1.4042352.

    CAS  Article  Google Scholar 

  27. 27.

    Aly AM, Ahmed SE, Raizah Z. Double-diffusive natural convection in a square porous cavity with sinusoidal distributions side walls filled with a nanofluid. J Porous Media. 2018. https://doi.org/10.1615/jpormedia.v21.i2.10.

    Article  Google Scholar 

  28. 28.

    Raizah Z, Aly AM, Ahmed SE. Natural convection flow of a power-law non-Newtonian nanofluid in inclined open shallow cavities filled with porous media. Int J Mech Sci. 2018;140:376–93. https://doi.org/10.1016/j.ijmecsci.2018.03.017.

    Article  Google Scholar 

  29. 29.

    Aly AM, Raizah Z, Ahmed SE. Natural convection in an enclosure saturatedwith multilayer porous medium and nanofluid over circular cylinders: entropy generation. J Porous Media. 2018;21(10):1. https://doi.org/10.1615/jpormedia.2018021357.

    Article  Google Scholar 

  30. 30.

    Rashidi S, Kashefi MH, Kim KC, Samimi-Abianeh O. Potentials of porous materials for energy management in heat exchangers—a comprehensive review. Appl Energy. 2019;243:206–32. https://doi.org/10.1016/j.apenergy.2019.03.200.

    Article  Google Scholar 

  31. 31.

    Shamsabadi H, Rashidi S, Esfahani JA. Entropy generation analysis for nanofluid flow inside a duct equipped with porous baffles. J Therm Anal Calorim. 2019;135:1009–19.

    CAS  Article  Google Scholar 

  32. 32.

    Sheikholeslami M, Gorji-Bandpy M, Pop I, Soleimani S. Numerical study of natural convection between a circular enclosure and a sinusoidal cylinder using control volume based finite element method. Int J Therm Sci. 2013;72:147–58. https://doi.org/10.1016/j.ijthermalsci.2013.05.004.

    Article  Google Scholar 

  33. 33.

    Sheikholeslami M, Gorji-Bandpy M, Pop I, Soleimani S, Seyyedi SM. Natural convection of nanofluids in an enclosure between a circular and a sinusoidal cylinder in the presence of magnetic field. Int Commun Heat Mass Transf. 2012;39(9):1435–43. https://doi.org/10.1016/j.icheatmasstransfer.2012.07.026.

    CAS  Article  Google Scholar 

  34. 34.

    Hoghoughi G, Izadi M, Oztop HF, Abu-Hamdeh N. Effect of geometrical parameters on natural convection in a porous undulant-wall enclosure saturated by a nanofluid using Buongiorno’s model. J Mol Liq. 2018;255:148–59. https://doi.org/10.1016/j.molliq.2018.01.145.

    CAS  Article  Google Scholar 

  35. 35.

    Sheikholeslami M, Haq R-U, Shafee A, Li Z. Heat transfer behavior of nanoparticle enhanced PCM solidification through an enclosure with V shaped fins. Int J Heat Mass Transf. 2019;130:1322–42. https://doi.org/10.1016/j.ijheatmasstransfer.2018.11.020.

    CAS  Article  Google Scholar 

  36. 36.

    Sheikholeslami M. Numerical approach for MHD Al2O3-water nanofluid transportation inside a permeable medium using innovative computer method. Comput Methods Appl Mech Eng. 2019;344:306–18. https://doi.org/10.1016/j.cma.2018.09.042.

    Article  Google Scholar 

  37. 37.

    Aly AM, Nguyen MT, Lee S-W. Numerical analysis of liquid sloshing using the incompressible smoothed particle hydrodynamics method. Adv Mech Eng. 2015;7(2):765741. https://doi.org/10.1155/2014/765741.

    Article  Google Scholar 

  38. 38.

    Nguyen MT, Aly AM, Lee S-W. A numerical study on unsteady natural/mixed convection in a cavity with fixed and moving rigid bodies using the ISPH method. Int J Numer Meth Heat Fluid Flow. 2018;28(3):684–703. https://doi.org/10.1108/hff-02-2017-0058.

    Article  Google Scholar 

  39. 39.

    Aly AM, Raizah Z. Incompressible smoothed particle hydrodynamics (ISPH) method for natural convection in a nanofluid-filled cavity including rotating solid structures. Int J Mech Sci. 2018;146:125–40. https://doi.org/10.1016/j.ijmecsci.2018.07.044.

    Article  Google Scholar 

  40. 40.

    Aly AM, Chamkha AJ, Lee S-W, Al-Mudhaf AF. On mixed convection in an inclined lid-driven cavity with sinusoidal heated walls using the ISPH method. Comput Therm Sci Int J. 2016;8(4):337–54. https://doi.org/10.1615/computthermalscien.2016016527.

    Article  Google Scholar 

  41. 41.

    Maxwell J. A treatise on electricity and magnetism. 2nd ed. Cambridge: Oxford University Press; 1904.

    Google Scholar 

  42. 42.

    Brinkman HC. The viscosity of concentrated suspensions and solutions. J Chem Phys. 1952;20(4):571. https://doi.org/10.1063/1.1700493.

    CAS  Article  Google Scholar 

  43. 43.

    Kim BS, Lee DS, Ha MY, Yoon HS. A numerical study of natural convection in a square enclosure with a circular cylinder at different vertical locations. Int J Heat Mass Transf. 2008;51(8):1888–906. https://doi.org/10.1016/j.ijheatmasstransfer.2007.06.033.

    CAS  Article  Google Scholar 

Download references


The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through General Research Project under grant number (G.R.P./51/41).

Author information



Corresponding author

Correspondence to Abdelraheem M. Aly.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Ahmed, S.E., Aly, A.M. Mixed convection in a nanofluid-filled sloshing porous cavity including inner heated rose. J Therm Anal Calorim 143, 275–291 (2021). https://doi.org/10.1007/s10973-019-09216-2

Download citation


  • Mixed convection
  • Nanofluid
  • Porous medium
  • Sloshing enclosure
  • Rose shape