Study of Steady, Two-Dimensional, Unicellular Convection in a Water-Copper Nanoliquid-Saturated Porous Enclosure Using Single-Phase Model

  • P. G. SiddheshwarEmail author
  • B. N. Veena
Conference paper
Part of the Trends in Mathematics book series (TM)


In the present paper, we study Brinkman-Bénard convection in nanoliquid-saturated porous enclosure with vertical walls being adiabatic and horizontal walls being isothermal for two velocity boundary combinations, namely, free-free (FF) and rigid-rigid (RR). Brinkman model has been modified in the present study to account for added nanoparticles. Thermophysical properties of nanoliquid in a saturated porous medium as a function of corresponding properties of base liquid, nanoparticle and porous medium are modelled using phenomenological laws and mixture theory. An analytical study has been made of Brinkman-Bénard convection in a porous enclosure using single-phase model. The effect of nanoparticles is to advance onset of convection and enhance heat transfer, whereas porous medium facilitates delayed onset and retainment of heat energy in the system. The present study shows good agreement with those of previous works.


Porous enclosure Free-free Rigid-rigid and single-phase model 



One of the authors (BNV) would like to thank the University Grants Commission, Government of India for awarding her the “National Fellowship for Higher Education” to carry out her research. The authors thank the Bangalore University for encouragement and support.


  1. 1.
    Brinkman, H.C.: The viscosity of concentrated suspensions and solutions. J. Chem. Phys. 20, 571–571 (1952)CrossRefGoogle Scholar
  2. 2.
    Chandrasekhar, S.: Hydrodynamic and Hydromagnetic Stability. Oxford University Press-London (1961)Google Scholar
  3. 3.
    Hamilton, R.L., Crosser, O.K.: Thermal conductivity of heterogeneous two-component systems. Ind. Eng. Chem. Fund. 1, 187–191 (1962)CrossRefGoogle Scholar
  4. 4.
    Masuda, H., Ebata, A., Terama, K., Hishinuma, N.: Alteration of thermal conductivity and viscosity of liquid by dispersing ultra-fine particles dispersion of Al 2 O 3, SiO 2 and TiO 2 ultra-fine particles. Netsu Bussei. 7, 227–233 (1993)CrossRefGoogle Scholar
  5. 5.
    Choi, S.U.S., Eastman, J.A.: Enhancing thermal conductivity of fluids with nanoparticles. D. A. Siginer and H. P. Wang (Eds.) Development and applications of Non-Newtonian Flows, ASME-Publications-Fed. 231, 99–106 (1995)Google Scholar
  6. 6.
    Eastman, J.A., Choi, S.U.S., Li, S., Yu, W., Thompson, L.J.: Anomalously increased effective thermal conductivities of ethylene glycol-based nanofluids containing copper nanoparticles. Appl. Phys. Lett. 78, 718–720 (2001)CrossRefGoogle Scholar
  7. 7.
    Das, S.K., Putra, N., Thiesen, P., Roetzel, W.: Temperature dependence of thermal conductivity enhancement for nanofluids. J. Heat Transfer, ASME. 125, 567–574 (2003)CrossRefGoogle Scholar
  8. 8.
    Khanafer, K., Vafai, K., Lightstone, M.: Buoyancy-driven heat transfer enhancement in a two-dimensional enclosure utilizing nanofluids. Int. J. Heat Mass Transfer. 46, 3639–3653 (2003)CrossRefGoogle Scholar
  9. 9.
    Nield, D.A., Bejan, A.: Convection in Porous Media. Wiley-New York (2006)Google Scholar
  10. 10.
    Buongiorno, J.: Convective transport in nanofluids. J. Heat Transfer, ASME. 128, 240–250 (2006)CrossRefGoogle Scholar
  11. 11.
    Elhajjar, B., Bachir, G., Mojtabi, A., Fakih, C., Charrier-Mojtabi, M.C.: Modeling of Rayleigh-Bénard natural convection heat transfer in nanofluids. C. R. Mécanique. 338, 350–354 (2010)CrossRefGoogle Scholar
  12. 12.
    Siddheshwar, P.G., Meenakshi, N.: Amplitude equation and heat transport for Rayleigh-Bénard convection in Newtonian liquids with nanoparticles. Int. J. Appl. Comput. Math. 2, 1–22 (2015)CrossRefGoogle Scholar
  13. 13.
    Bianco, V., Manca, O., Nardini, S., Vafai, K.: Heat transfer enhancement with nanofluids. CRC Press-New York (2015)Google Scholar
  14. 14.
    Siddheshwar, P.G., Kanchana, C., Kakimoto, Y., Nakayama, A.: Steady finite-amplitude Rayleigh-Bénard convection in nanoliquids using two phase model: Theoretical answer to the phenomenon of enhanced heat transfer. J. Heat Transfer, ASME. 139, 012402–012412 (2017)CrossRefGoogle Scholar
  15. 15.
    Siddheshwar, P.G., Veena, B. N.: Unsteady Rayleigh-Bénard convection of nanoliquids in enclosures. World Academy of Science, Engineering and Technology, International Journal of Mechanical, Aerospace, Industrial, Mechatronic and Manufacturing Engineering. 11, 1051–1060 (2017)Google Scholar
  16. 16.
    Siddheshwar, P.G., Veena, B. N.: A theoretical study of natural convection of water-based nanoliquids in low-porosity enclosures using single-phase model. J. Nanofluids. 7, 1–12 (2018)CrossRefGoogle Scholar
  17. 17.

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsBangalore UniversityBangaloreIndia

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