Abstract
The combined effects of vertical heterogeneity of permeability and local thermal non-equilibrium (LTNE) on the onset of ferromagnetic convection in a ferrofluid saturated Darcy porous medium in the presence of a uniform vertical magnetic field are investigated. A two-field model for temperature representing the solid and fluid phases separately is used. The eigenvalue problem is solved numerically using the Galerkin method for different forms of permeability heterogeneity function Γ(z) and their effect on the stability characteristics of the system has been analyzed in detail. It is observed that the general quadratic variation of Γ(z) with depth has more destabilizing effect on the system when compared to the homogeneous porous medium case. Besides, the influence of LTNE and magnetic parameters on the criterion for the onset of ferromagnetic convection is also assessed.
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Abbreviations
- \({a=\sqrt{\ell ^{2}+m^{2}}}\) :
-
Overall horizontal wave number
- \({\vec {B}}\) :
-
Magnetic induction
- c :
-
Specific heat
- c a :
-
Acceleration coefficient
- d :
-
Thickness of the porous layer
- D = d/dz :
-
Differential operator
- \({\vec {g}}\) :
-
Acceleration due to gravity
- h t :
-
heat transfer coefficient
- \({\vec {H}}\) :
-
Magnetic field intensity
- H 0 :
-
Imposed uniform vertical magnetic field
- \({H_{\rm t} =hd^{2}/\varepsilon k_{\rm tf}}\) :
-
Scaled inter-phase heat transfer coefficient
- \({\hat{{k}}}\) :
-
Unit vector in z-direction
- K 0 :
-
The mean value of K(z)
- K(z):
-
Permeability of the porous medium
- k f :
-
Thermal conductivity of the fluid
- k s :
-
Thermal conductivity of the solid
- \({K_{\rm p} =- (\partial M/\partial T_{\rm f} )_{H_0 },_{T_{\rm a}}}\) :
-
Pyromagnetic co-efficient
- ℓ, m:
-
Wave numbers in the x and y directions
- \({\vec {M}}\) :
-
Magnetization
- M0 = M(H0, Ta):
-
Constant mean value of magnetization
- \({M_1 =\mu _0 K^{2}\beta /(1+\chi ){\kern 1pt} \alpha _{\rm t}\rho _0 g}\) :
-
Magnetic number
- M3 = (1 + M0 /H0)/(1 + χ):
-
Non-linearity of magnetization parameter
- p :
-
Pressure
- \({\vec {q}=(u,v,w)}\) :
-
Velocity vector
- \({R= \rho _0 \alpha _{\rm t} g\beta k d^{2}/\varepsilon \mu _{\rm f}\kappa _{\rm f}}\) :
-
Darcy–Rayleigh number
- t :
-
Time
- T :
-
Temperature
- T L :
-
Temperature of the lower boundary
- T u :
-
Temperature of the upper boundary
- \({T_{\rm a} =\left( {T_{\rm l} +T_{\rm u}}\right)/2}\) :
-
Reference temperature
- W :
-
Amplitude of vertical component of perturbed velocity
- (x, y, z):
-
Cartesian co-ordinates
- α t :
-
Thermal expansion coefficient
- β = ΔT/d:
-
Temperature gradient
- \({\chi =(\partial M/\partial H)_{H_0 }, {T_0}}\) :
-
Magnetic susceptibility
- \({\nabla ^{2}=\partial ^{2}/\partial x^{2}+\partial ^{2}/\partial y^{2}+\partial ^{2}/\partial z^{2}}\) :
-
Laplacian operator
- \({\nabla _{\rm h}^2 =\partial ^{2}/\partial x^{2}+\partial ^{2}/\partial y^{2}}\) :
-
Horizontal Laplacian operator
- \({\varepsilon }\) :
-
Porosity of the porous medium
- Γ(z):
-
Non-dimensional permeability heterogeneity function
- \({\kappa _{\rm f} =k_{\rm tf} /(\rho _0 c)_{\rm f}}\) :
-
Thermal diffusivity of the fluid
- μ f :
-
Dynamic viscosity
- μ 0 :
-
Free space magnetic permeability of vacuum
- \({\varphi }\) :
-
Magnetic potential
- \({\Phi }\) :
-
Amplitude of perturbed magnetic potential
- \({\gamma =\varepsilon k_{\rm tf} /\left( {1-\varepsilon }\right) k_{\rm ts}}\) :
-
Porosity modified conductivity ratio
- ρ f :
-
Fluid density
- ρ 0 :
-
Reference density at T a
- \({\Theta}\) :
-
Amplitude of temperature
- b:
-
Basic state
- f:
-
Fluid
- s:
-
Solid
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Acknowledgments
One of the authors (ISS) wishes to thank the University of Hong Kong for inviting him as a visiting Professor and also the Bangalore University for sanctioning sabbatical leave. The work was financially supported by the Research Grants Council of the Hong Kong Special Administrative Region, China, through Project Nos. HKU 715609E and HKU 715510E. We thank the reviewers for useful suggestions.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Shivakumara, I.S., Ng, CO. & Ravisha, M. Ferromagnetic Convection in a Heterogeneous Darcy Porous Medium Using a Local Thermal Non-equilibrium (LTNE) Model. Transp Porous Med 90, 529–544 (2011). https://doi.org/10.1007/s11242-011-9798-6
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DOI: https://doi.org/10.1007/s11242-011-9798-6