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

  • P. G. SiddheshwarEmail author
  • T. N. Sakshath


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.


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

List of symbols

Greek symbols

\(\alpha \)

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

Latin symbols


Amplitudes of convection


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


Ozoe heat transfer diminishment parameter


Ozoe heat transfer enhancement parameter


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


Permeability of the porous medium


Thermal conductivity of the nanoliquid in saturated porous medium


Thermal conductivity of the base liquid in saturated porous medium


Thermal conductivity of the base liquid


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


Thermal conductivity of the nanoparticle


Ratio of specific heats


Nusselt number of the nanoliquid in saturated porous medium


Nusselt number of the base liquid in saturated porous medium


Nusselt number of the nanoliquid




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


Rayleigh number of the nanoliquid in saturated porous medium


Dimensional temperature (K)


Reference temperature


Horizontal and vertical velocity components


Dimensional and non-dimensional horizontal coordinates


Dimensional and non-dimensional vertical coordinates


Nusselt number of the base liquid


Distance between the plates (m)



At reference value


Liquid property in saturated porous medium


Basic state




Base liquid









Perturbed quantity


Free-free boundaries


Rigid-free boundaries


Rigid-rigid boundaries



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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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2020

Authors and Affiliations

  1. 1.Department of MathematicsBangalore UniversityBangaloreIndia
  2. 2.Department of MathematicsM S Ramaiah University of Applied SciencesBangaloreIndia

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