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


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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