Study of Rayleigh–Bénard Convection of a Newtonian Nanoliquid in a High Porosity Medium Using Local Thermal Non-equilibrium Model

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


In the paper we make linear and non-linear stability analyses of Rayleigh–Bénard convection in a Newtonian, nanoliquid-saturated porous medium using local thermal non-equilibrium model (LTNE). The LTNE assumption results in advanced onset of convection and increase in heat transport when compared to that of local thermal equilibrium assumption. Free–free and rigid–rigid, isothermal boundaries are considered for investigation. The Galerkin method is used to obtain the critical eigen value. The influence of inter-phase heat transfer coefficient, ratio of thermal conductivities, Brinkman number, porous parameter on the onset of convection as well as on heat transport has been presented graphically and discussed in detail. The effect of increasing the value of porosity modified thermal conductivities advances the onset of convection and enhances the amount of heat transport whereas the remaining parameters have an opposing influence on both onset of convection as well as heat transport.


Rayleigh–Bénard convection Nanoliquid Porous medium LTNE Free–free Rigid–rigid Linear Non-linear 

List of Symbols


Amplitudes of convection


Specific heat at constant pressure


Channel depth

\(\mathbf {g}\)

Acceleration due to gravity


Inter-phase heat transfer coefficient


Dimensionless inter-phase heat transfer coefficient


Wavenumber in the x direction






Prandtl number


Filtration velocity or Darcy velocity






Dimensional velocity in the y direction


Cartesian coordinate


Dimensionless coordinates

Greek Symbols

\(\alpha _{nl}\)

Thermal diffusivity of the nanoliquid

\(\alpha _{bl}\)

Thermal diffusivity of the base liquid

\(\gamma \)

Porosity-modified ratio of thermal conductivities

\(\beta \)

Thermal expansion coefficient


Thermal conductivity

\(\Lambda \)

Ratio of viscosities or Brinkman number

\(\mu \)

Viscosity of the nanoliquid

\(\mu ^{'}\)

Viscosity of the nanoliquid in saturated porous medium

\(\phi \)

Porosity(\(0<\phi <1\))

\(\Psi \)

Dimensionless stream function

\(\psi \)

Dimensional stream function

\(\rho \)


\(\sigma ^2\)

Inverse Darcy number or porous parameter

\(\tau \)

Dimensionless time

\(\Theta \)

Dimensionless temperature

Subscripts or Superscripts


Reference value


Basic state




Base liquid


Nano liquid


Local thermal equilibrium


Local thermal non-equilibrium





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© Springer Nature India Private Limited 2019

Authors and Affiliations

  1. 1.Department of MathematicsBangalore UniversityBangaloreIndia
  2. 2.Department of Mathematics, Faculty of Mathematical and Physical SciencesM S Ramaiah University of Applied SciencesPeenya, BangaloreIndia

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