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Penetrative Electro-Thermal-Convection in a Dielectric Fluid-Saturated Porous Layer Via Internal Heating: Effect of Boundary Conditions

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Abstract

The effect of boundary conditions on the onset of penetrative electro-thermal-convection (ETC) in a dielectric fluid-saturated porous medium via internal heating is investigated. The lower and upper boundaries are considered to be either rigid or stress-free. The lower boundary is isothermal, while a general Robin-type of thermal boundary condition is invoked at the upper boundary. The eigenvalue problem is formulated and numerical solutions via Galerkin-type of weighted residual technique for (1) lower and upper-rigid (R–R), (2) lower-rigid and upper-free (R–F), (3) lower and upper-free (F–F) boundary combinations. It is found that the values of gravity thermal Rayleigh or electric thermal Rayleigh number influencing the onset of ETC is higher for R–R boundaries and lower for F–F boundaries. As strength of internal heating increases, it is observed that, there is a hasten in the onset of ETC, while an increase in the Biot number, ratio of viscosities and decrease in the Darcy number is to delay the onset of ETC. Several standard results are recovered as special cases from the current study.

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Abbreviations

\( a \) :

Overall horizontal wave number

\( Bi \) :

Biot number

d :

Thickness of the layer

\( Da \) :

Darcy number

\( D \) :

Differential operator

\( \vec{E} \) :

Electric force field

\( \vec{g} \) :

Acceleration due to gravity

\( k_{1} \) :

Permeability of the porous medium

\( M \) :

Ratio of heat capacities

\( N_{s} \) :

Internal heat source strength

\( p \) :

Pressure

\( \vec{q} \) :

Velocity vector

\( Q \) :

Constant internal heat source strength

\( R_{e} \) :

Electric thermal Rayleigh number

\( R_{t} \) :

Thermal Rayleigh number

\( t \) :

Time

\( T \) :

Temperature

\( W \) :

Amplitude of vertical component of velocity

\( \alpha \) :

Thermal expansion coefficient

\( \nabla^{2} \) :

Laplacian operator

\( \nabla_{h}^{2} \) :

Horizontal Laplacian operator

\( \varepsilon \) :

Dielectric constant

\( \eta \) :

Dielectric constant expansion coefficient

\( \kappa \) :

Thermal diffusivity

\( \varLambda \) :

Ratio of viscosity

\( \mu \) :

Coefficient of dynamic viscosity

\( \mu \) :

Coefficient of effective viscosity

\( \nu \) :

Kinematic viscosity

\( \phi \) :

Electric potential

\( \varphi_{p} \) :

Porosity of the porous medium

\( \varPhi \) :

Amplitude of electric field potential

\( \rho \) :

Density of the fluid

\( \rho_{e} \) :

Free charge

\( \varTheta \) :

Amplitude of temperature field

\( x,y,z \) :

Cartesian co-ordinates

\( 0 \) :

Reference value

\( f \) :

Fluid

\( b \) :

Basic state

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Acknowledgements

The authors wish to thank the referees for their useful comments. Also, gratefully acknowledge the financial support received in the form of a “Research Fund for Talented Teacher Scheme” from Vision Group of Science & Technology, Government of Karnataka, Bengaluru, India.

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Authors and Affiliations

  1. Department of Mathematics, Dr. Ambedkar Institute of Technology, Bengaluru, 560056, India

    R. Ashwini & C. E. Nanjundappa

  2. Department of Mathematics, Bangalore University, Bengaluru, 560001, India

    I. S. Shivakumara

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  1. R. Ashwini
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  2. C. E. Nanjundappa
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  3. I. S. Shivakumara
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Correspondence to C. E. Nanjundappa.

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Ashwini, R., Nanjundappa, C.E. & Shivakumara, I.S. Penetrative Electro-Thermal-Convection in a Dielectric Fluid-Saturated Porous Layer Via Internal Heating: Effect of Boundary Conditions. Int. J. Appl. Comput. Math 5, 37 (2019). https://doi.org/10.1007/s40819-019-0619-x

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