Steady finite-amplitude Rayleigh–Bénard convection of ethylene glycol–copper nanoliquid in a high-porosity medium made of 30% glass fiber-reinforced polycarbonate


In the paper, we make linear and nonlinear stability analyses of Rayleigh–Bénard convection in a Newtonian nanoliquid-saturated high-porosity medium. Single-phase model is used for nanoliquids, and values of thermophysical quantities concerning ethylene glycol–copper nanoliquid-saturated porous medium are calculated using mixture theory or phenomenological relations. The study is carried out for free-free, rigid-rigid and rigid-free isothermal boundaries. Boundary effects on onset of convection are shown to conform to classical predictions. The addition of copper nanoparticles to ethylene glycol is shown to lead to advanced onset of convection in the porous medium and thereby to a substantial increase in heat transport. Theoretical explanation is provided for the enhanced heat transfer situation in the medium. With suitable scaling in quantities, the result concerning heat transfer in ethylene glycol–copper nanoliquid-saturated porous medium is shown to be obtainable from those of ethylene glycol-saturated porous medium without copper nanoparticles. Nanoparticles serve the purpose of cooling and porous matrix retains the heat, thereby meaning that residence time of heat in the system can be regulated by using nanoparticles and porous matrix.

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\(\alpha \) :

Thermal diffusivity of the nanoliquid in saturated porous medium (\({\rm m}^{2}\,{\rm s}^{-1}\))

\(\alpha _{1}\) :

Thermal diffusivity of the base liquid in saturated porous medium

\(\beta \) :

Thermal expansion coefficient of the nanoliquid in saturated porous medium (\({\rm K^{-1}}\))

\(\beta _{1}\) :

Thermal expansion coefficient of the base liquid in saturated porous medium

\(\chi \) :

Nanoparticle volume fraction

\(\Delta T\) :

Temperature difference

\(\Lambda \) :

Brinkman number or ratio of viscosities

\(\mu \) :

Viscosity of the nanoliquid

\(\mu ^{\prime}\) :

Viscosity of the nanoliquid in saturated porous medium (kg (m s)−1)

\(\nu \) :

Wave number (\({\rm m}^{-1}\))

\(\phi \) :


\(\Psi \) :

Non-dimensional stream function

\(\psi \) :

Dimensional stream function

\(\rho \) :

Density of the nanoliquid in saturated porous medium (kg m–4)

\(\sigma ^{2}\) :

Porous parameter

\(\Theta \) :

Non-dimensional temperature


Amplitudes of convection

\(C_\mathrm{p}\) :

Specific heat capacity of the nanoliquid in saturated porous medium at constant pressure (J (kg K)−1)

D :

Ozoe heat transfer diminishment parameter

E :

Ozoe heat transfer enhancement parameter

\(g=(0,0,-g)\) :

Acceleration due to gravity (\({\rm m\,s}^{-2}\))

K :

Permeability of the porous medium

k :

Thermal conductivity of the nanoliquid in saturated porous medium

\(k_{1}\) :

Thermal conductivity of the base liquid in saturated porous medium

\(k_\mathrm{l}\) :

Thermal conductivity of the base liquid

\(k_\mathrm{nl}\) :

Thermal conductivity of the nanoliquid (W (m K)−1)

\(k_\mathrm{np}\) :

Thermal conductivity of the nanoparticle

M :

Ratio of specific heats

\(\mathrm{Nu}\) :

Nusselt number of the nanoliquid in saturated porous medium

\(\mathrm{Nu}_{1}\) :

Nusselt number of the base liquid in saturated porous medium

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

Nusselt number of the nanoliquid

p :


\(q=(u,0,w)\) :

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


Rayleigh number of the nanoliquid in saturated porous medium

T :

Dimensional temperature (K)

\(T_{0}\) :

Reference temperature

uw :

Horizontal and vertical velocity components

xX :

Dimensional and non-dimensional horizontal coordinates

zZ :

Dimensional and non-dimensional vertical coordinates

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

Nusselt number of the base liquid

h :

Distance between the plates (m)


At reference value


Liquid property in saturated porous medium


Basic state




Base liquid







\(\prime\) :

Perturbed quantity


Free-free boundaries


Rigid-free boundaries


Rigid-rigid boundaries


  1. 1.

  2. 2.

    Abu-Nada E. Rayleigh–Bénard convection in nanofluids: effect of temperature dependent properties. Int J Therm Sci. 2011;50(9):1720–30.

    CAS  Article  Google Scholar 

  3. 3.

    Adler P. Porous media: geometry and transports. Boston: Elsevier; 2013.

    Google Scholar 

  4. 4.

    Bergman TL, Incropera FP, Lavine AS, Dewitt DP. Fundamentals of heat and mass transfer. Sixth ed. New York: Wiley; 2011.

    Google Scholar 

  5. 5.

    Bianco V, Manca O, Nardini S, Vafai K. Heat transfer enhancement with nanofluids. New York: CRC Press; 2015.

    Google Scholar 

  6. 6.

    Bourantas GC, Skouras ED, Loukopoulos VC, Burganos VN. Heat transfer and natural convection of nanofluids in porous media. Eur J Mech B Fluids. 2014;43:45–56.

    Article  Google Scholar 

  7. 7.

    Brinkman HC. The viscosity of concentrated suspensions and solutions. J Chem Phys. 1952;20:571.

    CAS  Article  Google Scholar 

  8. 8.

    Buongiorno J. Convective transport in nanofluids. J Heat Transfer. 2006;128:240–50.

    Article  Google Scholar 

  9. 9.

    Chandrasekhar S. Hydrodynamic and hydromagnetic stability. London: Clarendon Press; 1961.

    Google Scholar 

  10. 10.

    Chandrasekhar S, Reid WH. On the expansion of functions which satisfy four boundary conditions. Proc Natl Acad Sci USA. 1957;43:521–7.

    CAS  PubMed  Article  PubMed Central  Google Scholar 

  11. 11.

    Corcione M. Rayleigh–Bénard convection heat transfer in nanoparticle suspensions. Int J Heat Fluid Flow. 2011;32:65–77.

    CAS  Article  Google Scholar 

  12. 12.

    Das SK, Putra N, Thiesen P, Roetzel W. Temperature dependence of thermal conductivity enhancement for nanofluids. ASME J Heat Transfer. 2003;125:567–74.

    CAS  Article  Google Scholar 

  13. 13.

    Dhananjay Y, Agrawal GS, Bhargava R. Rayleigh–Bénard convection in nanofluid. Int J Appl Math Mech. 2011;7:61–76.

    Google Scholar 

  14. 14.

    Dullien FAL. Porous media: fluid transport and pore structure. Second ed. San Diego: Academic Press; 2012.

    Google Scholar 

  15. 15.

    Eastman JA, Choi SUS, Li S, Yu W, Thompson LJ. Anomalously increased effective thermal conductivities of ethylene glycol-based nanofluids containing copper nanoparticles. Appl Phys Lett. 2001;78:718–20.

    CAS  Article  Google Scholar 

  16. 16.

    Hamilton RL, Crosser OK. Thermal conductivity of heterogeneous two-component systems. Ind Eng Chem Fundam. 1962;1:187–91.

    CAS  Article  Google Scholar 

  17. 17.

    Ingham DB, Pop I. Transport phenomena in porous media. Oxford: Elsevier; 1998.

    Google Scholar 

  18. 18.

    Kasaeian A, Azarian RD, Mahian O, Kolsi L, Chamkha AJ, Wongwises S, Pop I. Nanofluid flow and heat transfer in porous media: a review of the latest developments. Int J Heat Mass Transfer. 2017;107:778–91.

    CAS  Article  Google Scholar 

  19. 19.

    Kaviany M. Principles of heat transfer in porous media. New York: Springer; 2012.

    Google Scholar 

  20. 20.

    Khanafer K, Vafai K. Applications of nanofluids in porous medium. J Therm Anal Calorim. 2019;135(2):1479–92.

    CAS  Article  Google Scholar 

  21. 21.

    Khanafer K, Vafai K, Lightstone M. Buoyancy-driven heat transfer enhancement in a two-dimensional enclosure utilizing nanofluids. Int J Heat Mass Transf. 2003;46(19):3639–53.

    CAS  Article  Google Scholar 

  22. 22.

    Kim J, Kang YT, Choi CK. Analysis of convective instability and heat transfer characteristics of nanofluids. Phys Fluids (1994-present). 2004;16:2395–401.

    CAS  Article  Google Scholar 

  23. 23.

    Liu D, Yu L. Single-phase thermal transport of nanofluids in a minichannel. J Heat Transf. 2011;133:031009-1–-11.

    Google Scholar 

  24. 24.

    Minkowycz WJ, Sparrow EM, Abraham JP. Nanoparticle heat transfer and fluid flow, vol. 4. Boca Raton: CRC Press; 2012.

    Google Scholar 

  25. 25.

    Moradi A, Toghraie D, Meghdadi IAH, Hosseinian A. An experimental study on mwcnt-water nanofluids flow and heat transfer in double-pipe heat exchanger using porous media. J Therm Anal Calorim. 2019;132:1–11.

    Google Scholar 

  26. 26.

    Nagata M. Bifurcations at the Eckhaus points in two-dimensional Rayleigh–Bénard convection. Phys Rev E. 1995;52:6141–5.

    CAS  Article  Google Scholar 

  27. 27.

    Nield DA, Bejan A. Convection in porous media. 3rd ed. New York: Springer; 2006.

    Google Scholar 

  28. 28.

    Nield DA, Kuznetsov AV. Thermal instability in a porous medium layer saturated by a nanofluid. Int J Heat Mass Transfer. 2009;52:5796–801.

    CAS  Article  Google Scholar 

  29. 29.

    Nield DA, Kuznetsov AV. The onset of convection in a horizontal nanofluid layer of finite depth. Eur J Mech B Fluids. 2010;29(3):217–23.

    Article  Google Scholar 

  30. 30.

    Platten JK, Legros JC. Convection in liquids. Berlin: Springer; 2012.

    Google Scholar 

  31. 31.

    Savithiri S, Pattamatta A, Das SK. Rayleigh–Bénard convection in water-based alumina nanofluid: a numerical study. Numer Heat Transf Part A Appl. 2017;71:202–14.

    CAS  Article  Google Scholar 

  32. 32.

    Shenoy A, Sheremet MA, Pop I. Convective flow and heat transfer from wavy surfaces: viscous fluids, porous media, and nanofluids. Boca Raton: CRC Press; 2016.

    Google Scholar 

  33. 33.

    Sheremet MA, Cimpean DS, Pop I. Free convection in a partially heated wavy porous cavity filled with a nanofluid under the effects of brownian diffusion and thermophoresis. Appl Therm Eng. 2017;113:413–8.

    CAS  Article  Google Scholar 

  34. 34.

    Sheremet MA, Groşan T, Pop I. Steady-state free convection in right-angle porous trapezoidal cavity filled by a nanofluid: buongiornos mathematical model. Eur J Mech B Fluids. 2015;53:241–50.

    Article  Google Scholar 

  35. 35.

    Sheremet MA, Pop I. Free convection in a porous horizontal cylindrical annulus with a nanofluid using buongiornos model. Comput Fluids. 2015;118:182–90.

    CAS  Article  Google Scholar 

  36. 36.

    Siddheshwar PG, Kanchana C, Kakimoto Y, Nakayama A. Steady finite-amplitude Rayleigh–Bénard convection in nanoliquids using a two-phase model-theoretical answer to the phenomenon of enhanced heat transfer. ASME J Heat Transf. 2017;139:012402-1–8.

    Article  CAS  Google Scholar 

  37. 37.

    Siddheshwar PG, Meenakshi N. Amplitude equation and heat transport for Rayleigh–Bénard convection in Newtonian liquids with nanoparticles. Int J Appl Comput Math. 2015;2:1–22.

    Article  Google Scholar 

  38. 38.

    Siddheshwar PG, Ramachandramurthy V, Uma D. Rayleigh–Bénard and Marangoni magnetoconvection in Newtonian liquid with thermorheological effects. Int J Eng Sci. 2011;49(10):1078–94.

    Article  Google Scholar 

  39. 39.

    Siddheshwar PG, Sekhar GN, Jayalatha G. Effect of time-periodic vertical oscillations of the rayleigh–bénard system on nonlinear convection in viscoelastic liquids. J Non Newton Fluid Mech. 2010;165(19–20):1412–8.

    CAS  Article  Google Scholar 

  40. 40.

    Straughan B. Stability and wave motion in porous media, vol. 165. New York: Springer; 2008.

    Google Scholar 

  41. 41.

    Tzou DY. Thermal instability of nanofluids in natural convection. Int J Heat Mass Transfer. 2008;51:2967–79.

    CAS  Article  Google Scholar 

  42. 42.

    Xuan Y, Roetzel W. Conceptions for heat transfer correlation of nanofluids. Int J Heat Mass Transfer. 2000;43:3701–7.

    CAS  Article  Google Scholar 

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One of the authors, T N Sakshath, is thankful to the Department of Backward Classes Welfare, Government of Karnataka, for the financial support and also to the Bangalore University for supporting his research.

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Correspondence to P. G. Siddheshwar.

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Siddheshwar, P.G., Sakshath, T.N. Steady finite-amplitude Rayleigh–Bénard convection of ethylene glycol–copper nanoliquid in a high-porosity medium made of 30% glass fiber-reinforced polycarbonate. J Therm Anal Calorim 143, 485–502 (2021).

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  • Nanoliquid
  • Rayleigh–Bénard convection
  • Porous medium
  • Linear
  • Nonlinear
  • Stability
  • Single-phase