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The Onset of Darcy–Brinkman Electroconvection in a Dielectric Fluid Saturated Porous Layer

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Abstract

The combined effect of a vertical AC electric field and the boundaries on the onset of Darcy–Brinkman convection in a dielectric fluid saturated porous layer heated either from below or above is investigated using linear stability theory. The isothermal bounding surfaces of the porous layer are considered to be either rigid or free. It is established that the principle of exchange of stability is valid irrespective of the nature of velocity boundary conditions. The eigenvalue problem is solved exactly for free–free (F/F) boundaries and numerically using the Galerkin technique for rigid–rigid (R/R) and lower-rigid and upper-free (F/R) boundaries. It is observed that all the boundaries exhibit qualitatively similar results. The presence of electric field is emphasized on the stability of the system and it is shown that increasing the AC electric Rayleigh number R ea is to facilitate the transfer of heat more effectively and to hasten the onset of Darcy–Brinkman convection. Whereas, increase in the ratio of viscosities Λ and the inverse Darcy number Da −1 is to delay the onset of Darcy–Brinkman electroconvection. Besides, increasing R ea and Da −1 as well as decreasing Λ are to reduce the size of convection cells.

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Abbreviations

\({a=\sqrt{\ell^{2}+m^{2}}}\) :

Overall horizontal wave number

d :

Thickness of the porous layer

D = d/dz :

Differential operator

Da = k/d 2 :

Darcy number

\({\vec{E}}\) :

Electric field

\({\vec{g}}\) :

Acceleration due to gravity

\({\hat{{k}}}\) :

Unit vector in z-direction

k :

Permeability of the porous medium

ℓ, m :

Wave numbers in the x and y directions

M :

Ratio of heat capacities

p :

Pressure

\({Pr=\nu\phi/\kappa}\) :

Prandtl number

\({\vec{q}=(u,v,w)}\) :

Velocity vector

\({R_{\rm ea} =\eta^{2}\varepsilon_0 E_0^2(\Delta T)^{2}d^{2}/\mu\kappa}\) :

AC electric Rayleigh number

R eaD (= R ea Da):

AC electric Darcy–Rayleigh number

R t = α gΔT d 3/νκ :

Thermal Rayleigh number

R tD (= R t Da):

Thermal Darcy–Rayleigh number

t :

Time

T :

Temperature

T 0 :

Temperature of the lower boundary

T 1 :

Temperature of the upper boundary

V :

Electric potential

W :

Amplitude of vertical component of perturbed velocity

(x, y, z):

Cartesian co-ordinates

α :

Thermal expansion coefficient

\({\nabla^{2}=\partial^{2}/\partial x^{2}+\partial^{2}/\partial y^{2}+\partial^{2}/\partial z^{2}}\) :

Laplacian operator

\({\nabla_h^2 =\partial^{2}/\partial x^{2}+\partial^{2}/\partial y^{2}}\) :

Horizontal Laplacian operator

ΔT = T 0 − T 1 :

Temperature difference between the lower upper boundaries

ε :

Dielectric constant

η :

Thermal expansion coefficient of dielectric constant

κ :

Effective thermal diffusivity of the fluid

\({\Lambda =\tilde{\mu}/\mu}\) :

Ratio of viscosities

μ :

Dynamic viscosity of the fluid

\({\tilde{\mu}}\) :

Effective viscosity

ν = μ/ρ 0 :

Kinematic viscosity

\({\phi}\) :

Porosity of the porous medium

Φ:

Amplitude of perturbed electric potential

ρ :

Fluid density

ρ e :

Free charge density

ρ 0 :

Reference density at T 0

Θ:

Amplitude of perturbed temperature

σ :

Electrical conductivity

ω = ω r + i ω i :

Growth rate

b:

Basic state

c:

Critical

f:

Fluid

s:

Solid

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Authors and Affiliations

  1. School of Mechanical Engineering, Yonsei University, Seoul, 120-749, South Korea

    I. S. Shivakumara & Jinho Lee

  2. UGC-Centre for Advanced studies in Fluid Mechanics, Department of Mathematics, Bangalore University, 560 001, Bangalore, India

    I. S. Shivakumara & N. Rudraiah

  3. Department of Mathematics, Mount Carmel College, 560 052, Bangalore, India

    K. Hemalatha

Authors
  1. I. S. Shivakumara
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  2. N. Rudraiah
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  3. Jinho Lee
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  4. K. Hemalatha
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Correspondence to Jinho Lee.

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Shivakumara, I.S., Rudraiah, N., Lee, J. et al. The Onset of Darcy–Brinkman Electroconvection in a Dielectric Fluid Saturated Porous Layer. Transp Porous Med 90, 509–528 (2011). https://doi.org/10.1007/s11242-011-9797-7

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