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Finite element simulations for natural convective flow of nanofluid in a rectangular cavity having corrugated heated rods


In this study, numerical simulations of natural convection in a partially heated rectangular cavity containing water-based copper oxide nanofluid (CuO–water) have been carried out. The flow field and heat transfer inside the cavity are influenced by two corrugated heated rods. The governing partial differential equations are transformed to dimensionless coupled nonlinear partial differential equations using some suitable variables. For the thermophysical properties of nanofluid, Koo and Kleinstreuer–Li model is implemented in the governing equation. Numerical solutions of the resulting system of equations are obtained utilizing finite element method. The simulations for flow field and thermal distribution are portrayed in terms of line graphs, streamlines and isotherms. The results are executed for various Rayleigh numbers \((10^4\le {\mathrm{Ra}} \le 10^6)\), nanoparticle volume fractions \((0.0\le \phi \le 0.2)\), amplitudes of the corrugated heated rods \((0.05\le A_{\mathrm{m}} \le 0.2)\) and wavelength numbers \((0 \le n \le 20)\). Results depict that the thermal distribution and flow field are getting stronger because of increasing Ra and n. The impact of nanoparticle volume fraction is found to be useful in intensifying the heat transfer rate because of dominant convection. It is worth mentioning that with an increase in \(A_{\mathrm{m}}\) the thermal distribution in the entire cavity is control by convection.

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\({\overline{x}}, {\overline{y}}\) :

Dimensional Cartesian coordinates \(\left[ \text {L}\right]\)

\(\overline{u}\), \({\overline{v}}\) :

Velocity components \(\left[ \text {L T}^{-1}\right]\)

g :

Gravitational acceleration \(\left[ \text {L T}^{-2}\right]\)

UV :

Dimensionless velocity components

k :

Thermal conductivity \(\left[ \text {ML}\,\text {K}^{-1}\text {T}^{-3}\right]\)

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

Specific heat capacity \(\left[ \text {ML}^{2}\,\text {K}^{-1}\text {T}^{-2}\right]\)


Rayleigh number


Dimensionless number

\(\omega\) :

Frequency of the wavy rods \(\left[ \text {T}^{-1}\right]\)

\(n_1\) :

Dimensionless wavelength number

XY :

Dimensionless variable

\({\overline{P}}\) :

Pressure \(\left[ \text {M}\,\text {L}^ {-1}\,\,\text {T}^ {-2}\right]\)

\({\overline{T}}\) :

Temperature \(\left[ \text{ K }\right]\)

T :

Dimensionless temperature

l :

Length of cavity \(\left[ \text {L}\right]\)

\(\Pr\) :

Prandtl number

P :

Dimensionless pressure

\({\mathrm{Nu}}_{\mathrm{m}}\) :

Mean Nusselt number

\(\lambda\) :

Amplitude of inner rods \(\left[ \text {L}\right]\)

\(A_{\mathrm{m}}\) :

Amplitude ratio

\(\mu\) :

Dynamic viscosity \(\left[ \text {M}\,\text {L}^ {-1}\text {T}^{-1}\right]\)

\(\beta\) :

Thermal expansion coefficient \(\left[ \text {K}^{-1}\right]\)

\(\phi\) :

Nanoparticle volume fraction

\(\nu\) :

Kinematic viscosity \(\left[ \text {L}^ {-2}\,\text {T}^{-1}\right]\)

\(\rho\) :

Density \(\left[ \text {M}\,\text {L}^ {-3}\right]\)

\(\gamma\) :

Penalty parameter

f :


C :


nf :


h :



  1. 1.

    Sheikholeslami M, Gorji-Bandpy M, Pop I, Soleimani Soheil. 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.

  2. 2.

    Wu F, Wang G. Numerical simulation of natural convection in an inclined porous cavity under time-periodic boundary conditions with a partially active thermal side wall. RSC Adv. 2017;7(28):17519–30.

  3. 3.

    Aparna K, Seetharamu KN. Investigations on the effect of non-uniform temperature on fluid flow and heat transfer in a trapezoidal cavity filled with porous media. Int J Heat Mass Transf. 2017;108:63–78.

  4. 4.

    Maxwell JC. A treatise on electricity and magnetism, vol. 1. Oxford: Clarendon Press; 1873.

  5. 5.

    Masuda H, Ebata A, Teramae K. Alteration of thermal conductivity and viscosity of liquid by dispersing ultra-fine particles (dispersion of γ-Al\(_2\)O\(_3\), SiO\(_2\) and TiO\(_2\) ultra-fine particles). Netsu Bussei (Japan) 4227233; 1993.

  6. 6.

    Choi SUS. Enhancing conductivity of fluids with nanoparticles, ASME fluid eng. Division. 1995;231:99–105.

  7. 7.

    Barrett TR, Robinson S, Flinders K, Sergis A, Hardalupas Y. Investigating the use of nanofluids to improve high heat flux cooling systems. Fusion Eng Des. 2013;88(9–10):2594–7.

  8. 8.

    Leong KY, Saidur R, Kazi SN, Mamun AH. Performance investigation of an automotive car radiator operated with nanofluid-based coolants (nanofluid as a coolant in a radiator). Appl Therm Eng. 2010;30(17–18):2685–92.

  9. 9.

    Hussein AM, Bakar RA, Kadirgama K. Study of forced convection nanofluid heat transfer in the automotive cooling system. Case Stud Therm Eng. 2014;2:50–61.

  10. 10.

    Turgut A, Elbasan E. Nanofluids for electronics cooling. In: 2014 IEEE 20th international symposium for design and technology in electronic packaging (SIITME). IEEE; 2014. p. 35–37.

  11. 11.

    Hsieh SS, Leu HY, Liu HH. Spray cooling characteristics of nanofluids for electronic power devices. Nanoscale Res Lett. 2015;10(1):139.

  12. 12.

    Ijam A, Saidur R. Nanofluid as a coolant for electronic devices (cooling of electronic devices). Appl Therm Eng. 2012;32:76–82.

  13. 13.

    Bi S, Guo K, Liu Z, Wu J. Performance of a domestic refrigerator using TiO\(_2\)-R600a nano-refrigerant as working fluid. Energy Convers Manag. 2011;52(1):733–7.

  14. 14.

    Saidur R, Kazi SN, Hossain MS, Rahman MM, Mohammed HA. A review on the performance of nanoparticles suspended with refrigerants and lubricating oils in refrigeration systems. Renew Sustain Energy Rev. 2011;15(1):310–23.

  15. 15.

    Mahbubul IM, Saidur R, Amalina MA. Thermal conductivity, viscosity and density of R141b refrigerant based nanofluid. Proc Eng. 2013;56:310–5.

  16. 16.

    Saidur R, Meng TC, Said Z, Hasanuzzaman M, Kamyar A. Evaluation of the effect of nanofluid-based absorbers on direct solar collector. Int J Heat Mass Transf. 2012;55(21–22):5899–907.

  17. 17.

    Kasaeian A, Eshghi AT, Sameti M. A review on the applications of nanofluids in solar energy systems. Renew Sustain Energy Rev. 2015;43:584–98.

  18. 18.

    Kumar V, Tiwari A Kumar, Ghosh S K. Application of nanofluids in plate heat exchanger: a review. Energy Convers Manag. 2015;105:1017–36.

  19. 19.

    Said Z, Saidur R, Sabiha MA, Hepbasli A, Rahim NA. Energy and exergy efficiency of a flat plate solar collector using ph treated Al\(_2\)O\(_3\) nanofluid. J of Clean Prod. 2016;112:3915–26.

  20. 20.

    Sheikholeslami M, Shehzad SA, Li Z, Shafee A. Numerical modeling for alumina nanofluid magnetohydrodynamic convective heat transfer in a permeable medium using Darcy law. Int J Heat Mass Transf. 2018;127:614–22.

  21. 21.

    Saleem S, Shafee A, Nawaz M, Dara RN, Tlili I, Bonyah E. Heat transfer in a permeable cavity filled with a ferrofluid under electric force and radiation effects. AIP Adv. 2019;9(9):095107.

  22. 22.

    Subhani M, Nadeem S. Numerical analysis of micropolar hybrid nanofluid. Appl Nanosci. 2019;9:447.

  23. 23.

    Lu D, Ramzan M, Ullah N, Chung JD, Farooq U. A numerical treatment of radiative nanofluid 3D flow containing gyrotactic microorganism with anisotropic slip, binary chemical reaction and activation energy. Sci Rep. 2017;7(1):17008.

  24. 24.

    Hayat T, Nadeem S, Khan AU. Numerical analysis of Ag–Cuo/water rotating hybrid nanofluid with heat generation/absorption. Can J Phys. 2018;97:644–50.

  25. 25.

    Irshad N, Saleem A, Nadeem S, Shahzadi I. Endoscopic analysis of wave propagation with Ag-nanoparticles in curved tube having permeable walls. Curr Nanosci. 2018;14(5):384–402.

  26. 26.

    Nadeem S, Ahmed Z, Saleem S. Carbon nanotubes effects in magneto nanofluid flow over a curved stretching surface with variable viscosity. Microsyst Technol. 2019;25:2881.

  27. 27.

    Ramzan M, Ullah N, Chung JD, Lu D, Farooq U. Buoyancy effects on the radiative magneto micropolar nanofluid flow with double stratification, activation energy and binary chemical reaction. Sci Rep. 2017;7(1):12901.

  28. 28.

    Mebarek-Oudina F. Convective heat transfer of Titania nanofluids of different base fluids in cylindrical annulus with discrete heat source. Heat Transf Asian Res. 2018;48:135–47.

  29. 29.

    Nayak MK, Bhatti M Mubashir, Makinde OD, Akbar NS. Transient magneto-squeezing flow of Nacl-CNP nanofluid over a sensor surface inspired by temperature dependent viscosity. In: Defect and diffusion forum, vol 387. Trans Tech Publications; 2018. p. 600–614.

  30. 30.

    Ellahi R, Zeeshan A, Hussain F, Abbas T. Study of shiny film coating on multi-fluid flows of a rotating disk suspended with nano-sized silver and gold particles: a comparative analysis. Coatings. 2018;8(12):422.

  31. 31.

    Javed MF, Khan NB, Khan MI, Muhammad R, Rehman MU, Khan SW, Khan TA, Hassan MT. Optimization of SWCNTs and MWCNTs (single and multi-wall carbon nanotubes) in peristaltic transport with thermal radiation in a non-uniform channel. J Mol Liq. 2019;273:383–91.

  32. 32.

    Nawaz M, Saleem S, Rana S. Computational study of chemical reactions during heat and mass transfer in magnetized partially ionized nanofluid. J Braz Soc Mech Sci Eng. 2019;41(8):326.

  33. 33.

    Sadiq MA, Khan AU, Saleem S, Nadeem S. Numerical simulation of oscillatory oblique stagnation point flow of a magneto micropolar nanofluid. RSC Adv. 2019;9(9):4751–64.

  34. 34.

    Saleem S, Nadeem S, Rashidi MM, Raju CSK. An optimal analysis of radiated nanomaterial flow with viscous dissipation and heat source. Microsyst Technol. 2019;25(2):683–9.

  35. 35.

    Ahmed SE, Rashed ZZ. MHD natural convection in a heat generating porous medium-filled wavy enclosures using Buongiorno’s nanofluid model. Case Stud Therm Eng. 2019;14:100430.

  36. 36.

    Vo DD, Saleem S, Alderremy AA, Nguyen TK, Nadeem S, Li Z. Heat transfer enhancement and migration of ferrofluid due to electric force inside a porous media with complex geometry. Phys Scr. 2019;94:115218.

  37. 37.

    Saleem S, Rafiq H, Al-Qahtani A, El-Aziz MA, Malik MY, Animasaun IL. Magneto Jeffrey nanofluid bioconvection over a rotating vertical cone due to gyrotactic microorganism. Math Probl Eng. 2019;2019:3478037.

  38. 38.

    Ahmed Z, Al-Qahtani A, Nadeem S, Saleem S. Computational study of MHD nanofluid flow possessing micro-rotational inertia over a curved surface with variable thermophysical properties. Processes. 2019;7(6):387.

  39. 39.

    Sheikholeslami M, Shamlooei M. Fe\(_3\)O\(_4\)–H\(_2\)O nanofluid natural convection in presence of thermal radiation. Int J Hydrog Energy. 2017;42(9):5708–18.

  40. 40.

    Sheikholeslami M, Vajravelu K. Nanofluid flow and heat transfer in a cavity with variable magnetic field. Appl Math Comput. 2017;298:272–82.

  41. 41.

    Sheikholeslami M, Zeeshan A. Analysis of flow and heat transfer in water based nanofluid due to magnetic field in a porous enclosure with constant heat flux using CVFEM. Comput Methods Appl Mech Eng. 2017;320:68–81.

  42. 42.

    Sheikholeslami M, Arabkoohsar A, Jafaryar M. Impact of a helical-twisting device on the thermal–hydraulic performance of a nanofluid flow through a tube. J Therm Anal Calorim. 2019.

  43. 43.

    Sheikholeslami M. Solidification of NEPCM under the effect of magnetic field in a porous thermal energy storage enclosure using CuO nanoparticles. J Mol Liq. 2018;263:303–15.

  44. 44.

    Sheikholeslami M. Magnetic field influence on CuO–H\(_2\)O nanofluid convective flow in a permeable cavity considering various shapes for nanoparticles. Int J Hydrog Energy. 2017;42(31):19611–21.

  45. 45.

    Sheikholeslami M, Jafaryar M, Ali JA, Hamad SM, Divsalar A, Shafee A, Nguyen-Thoi T, Li Z. Simulation of turbulent flow of nanofluid due to existence of new effective turbulator involving entropy generation. J Mol Liq. 2019;291:111283.

  46. 46.

    Usman M, Hamid M, Zubair T, Haq RU, Wang W. Cu–Al\(_2\)O\(_3\)/water hybrid nanofluid through a permeable surface in the presence of nonlinear radiation and variable thermal conductivity via LSM. Int J Heat Mass Transf. 2018;126:1347–56.

  47. 47.

    Chamkha AJ, Selimefendigil F. MHD free convection and entropy generation in a corrugated cavity filled with a porous medium saturated with nanofluids. Entropy. 2018;20(11):846.

  48. 48.

    Haq RU, Aman S. Water functionalized CuO nanoparticles filled in a partially heated trapezoidal cavity with inner heated obstacle: FEM approach. Int J Heat Mass Transf. 2019;128:401–17.

  49. 49.

    Rahman MM, Pop I, Saghir MZ. Steady free convection flow within a titled nanofluid saturated porous cavity in the presence of a sloping magnetic field energized by an exothermic chemical reaction administered by Arrhenius kinetics. Int J Heat Mass Transf. 2019;129:198–211.

  50. 50.

    Jiang Y, Zhou X. Analysis of flow and heat transfer characteristics of nanofluids surface tension driven convection in a rectangular cavity. Int J Mech Sci. 2019;153–154:154–63.

  51. 51.

    Wang L, Huang C, Yang X, Chai Z, Shi B. Effects of temperature-dependent properties on natural convection of power-law nanofluids in rectangular cavities with sinusoidal temperature distribution. Int J Heat Mass Transf. 2019;128:688–99.

  52. 52.

    Alkanhal TA, Sheikholeslami M, Usman Muhammad, Haq R U, Shafee A, Al-Ahmadi A S, Tlili I. Thermal management of MHD nanofluid within the porous medium enclosed in a wavy shaped cavity with square obstacle in the presence of radiation heat source. Int J Heat Mass Transf. 2019;139:87–94.

  53. 53.

    Pal SK, Bhattacharyya S, Pop I. A numerical study on non-homogeneous model for the conjugate-mixed convection of a Cu–water nanofluid in an enclosure with thick wavy wall. Appl Math Comput. 2019;356:219–34.

  54. 54.

    Haq RU, Soomro FA, Hammouch Z. Heat transfer analysis of CuO–water enclosed in a partially heated rhombus with heated square obstacle. Int J Heat Mass Transf. 2018;118:773–84.

  55. 55.

    Calcagni B, Marsili F, Paroncini M. Natural convective heat transfer in square enclosures heated from below. Appl Therm Eng. 2005;25(16):2522–31.

  56. 56.

    Koo J, Kleinstreuer C. A new thermal conductivity model for nanofluids. J Nanopart Res. 2004;6(6):577–88.

  57. 57.

    Li J. Computational analysis of nanofluid flow in microchannels with applications to micro-heat sinks and bio-MEMS. Raleigh: North Carolina State University; 2008. Ph.D. Thesis.

  58. 58.

    Ahmad S, Rohni AM, Pop I. Blasius and Sakiadis problems in nanofluids. Acta Mech. 2011;218(3–4):195–204.

  59. 59.

    Esfe MH, Arani AAA, Yan WM, Ehteram H, Aghaie A, Afrand M. Natural convection in a trapezoidal enclosure filled with carbon nanotube–EG–water nanofluid. Int J Heat Mass Transf. 2016;92:76–82.

  60. 60.

    Haq Rizwan Ul, Soomro Feroz Ahmed, F Öztop Hakan, Mekkaoui Toufik. Thermal management of water-based carbon nanotubes enclosed in a partially heated triangular cavity with heated cylindrical obstacle. Int J Heat Mass Transf. 2019;131:724–36.

  61. 61.

    Taylor C, Hood P. A numerical solution of the Navier–Stokes equations using the finite element technique. Comput Fluids. 1973;1:73–100.

  62. 62.

    Jiajan W. Solution to incompressible Navier Stokes equations by using finite element method 2010. Ph.D. Thesis.

  63. 63.

    Heinrich JC, Vionnet CA. The penalty method for the Navier–Stokes equations. Arch Comput Methods Eng. 1995;2(2):51–65.

  64. 64.

    Dyne BR, Heinrich JC. Physically correct penalt-like formulations for accurate pressure calculation in finite element algorithms of the Navier–Stokes equations. Int J Numer Methods Eng. 1993;36(22):3883–902.

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Ullah, N., Nadeem, S. & Khan, A.U. Finite element simulations for natural convective flow of nanofluid in a rectangular cavity having corrugated heated rods. J Therm Anal Calorim (2020).

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  • Rectangular cavity
  • Natural convection
  • Nanofluid
  • Corrugated heated rods
  • Finite element method