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Ferromagnetic Convection in a Heterogeneous Porous Medium

  • Research Article - Mechanical Engineering
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Abstract

The effect of vertical heterogeneity of permeability on the onset convection in a horizontal layer of magnetized ferrofluid-saturated Darcy porous medium is investigated. Four different forms of vertical heterogeneity permeability function \({\Gamma (z)}\) are considered for discussion. The eigenvalue problem is solved numerically using the Galerkin method for three types of temperature boundary conditions namely, (i) isothermal, (ii) insulated to temperature perturbations, and (iii) lower insulated to temperature perturbations and upper isothermal. The general quadratic variation in the vertical heterogeneity of permeability is to hasten the onset of ferromagnetic convection compared with other forms of \({\Gamma(z)}\). The measure of nonlinearity of magnetization and the magnetic susceptibility is found to influence the onset if the boundaries are either isothermal or lower insulated and upper isothermal. Increasing the magnetic number is to augment the onset of ferromagnetic convection. The system is more stabilizing when the boundaries are isothermal and least stable for insulated ones. Compared to the homogeneous porous medium case, the critical wave number is higher if the permeability of the porous medium is heterogeneous and the boundaries are isothermal.

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Abbreviations

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

Overall horizontal wave number

A :

Ratio of heat capacities

\({\vec{B}}\) :

Magnetic induction (T)

d :

Thickness of the porous layer (m)

D =  d/dz :

Differential operator (m−1)

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

Acceleration due to gravity (m s−2)

H :

Magnitude of \({\vec {H}}\) (Amp m−1)

\({\vec {H}}\) :

Magnetic field (Amp m−1)

H 0 :

Imposed uniform vertical magnetic field (Amp m−1)

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

Unit vector in z-direction

K 0 :

The mean value of K(z) (m−2)

K(z):

Permeability of the porous medium (m−2)

\({K_p =- (\partial M/\partial T_f )_{{H_0 } ,{T_a }} }\) :

Pyromagnetic co-efficient (Amp m−1 K−1)

\({\ell, m }\) :

Wave numbers in the x and y directions (m−1)

\({\vec {M}}\) :

Magnetization (Amp m−1)

\({M_0 =M(H_0 , T_a)}\) :

Constant mean value of magnetization (Amp m−1)

\({M_1 =\mu _0 K_p^2 \beta /(1+\chi )\alpha _t \rho _0 g}\) :

Magnetic number

\({M_{3}=(1+M_{0}/H_{0})/(1+\chi)}\) :

Non-linearity of magnetization parameter

p :

Pressure (N m−2)

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

Velocity vector (m s−1)

\({R_{D}= \rho_{0} \alpha_{t} g\beta K_{0}d^{2}/\varepsilon\mu_{f}\kappa}\) :

Darcy–Rayleigh number

t :

Time (s)

T :

Temperature (K)

T L :

Temperature of the lower boundary (K)

T U :

Temperature of the upper boundary (K)

\({T_{a}=(T_{L}+T_{U})/2}\) :

Average temperature (K)

W :

Amplitude of vertical component of perturbed velocity (m s−2)

(x, y, z):

Cartesian co-ordinates (m)

\({\alpha_{t}}\) :

Thermal expansion coefficient (K−1)

\({\beta = \Delta T/d}\) :

Temperature gradient (K m−1)

\({\chi =(\partial M/\partial H)_{H_0 } ,_{T_a } }\) :

Magnetic susceptibility

\({\delta_{1}, \delta_{2}}\) :

Constants and take the value 0 or 1

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

Laplacian operator (m−2)

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

Horizontal Laplacian operator (m−2)

\({\varepsilon}\) :

Porosity of the porous medium

\({\kappa}\) :

Thermal diffusivity of the fluid (W m−1 K−1)

\({\mu_{f}}\) :

Dynamic viscosity (m2 s−1)

\({\mu_{0}}\) :

Free space magnetic permeability of vacuum (H s−1)

\({\varphi}\) :

Magnetic potential (Amp)

\({\Phi}\) :

Amplitude of perturbed magnetic potential (Amp)

\({\Gamma (z)}\) :

Non-dimensional vertical heterogeneity of permeability function

\({\rho_{f}}\) :

Fluid density (kg m−3)

\({\rho_{0}}\) :

Reference density at T a (kg m−3)

\({\Theta }\) :

Amplitude of perturbed temperature (K)

b :

Basic state

f :

Fluid

s :

Solid

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Authors and Affiliations

  1. Department of Mathematics, Bangalore University, Bangalore, 560001, India

    I. S. Shivakumara

  2. Department of Mechanical Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong, China

    C.-O. Ng

  3. Department of Mathematics, Dr. G. Shankar Government Women’s First Grade College and Post Graduate Study Centre, Ajjarkadu, Udupi, 576101, India

    M. Ravisha

Authors
  1. I. S. Shivakumara
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  2. C.-O. Ng
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  3. M. Ravisha
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Correspondence to I. S. Shivakumara.

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Shivakumara, I.S., Ng, CO. & Ravisha, M. Ferromagnetic Convection in a Heterogeneous Porous Medium. Arab J Sci Eng 39, 7265–7274 (2014). https://doi.org/10.1007/s13369-014-1288-z

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