Ferromagnetic Convection in a Heterogeneous Darcy Porous Medium Using a Local Thermal Non-equilibrium LTNE ModelReport as inadecuate




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Transport in Porous Media

, Volume 90, Issue 2, pp 529–544

First Online: 07 July 2011Received: 02 April 2011Accepted: 21 June 2011

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 Γz and their effect on the stability characteristics of the system has been analyzed in detail. It is observed that the general quadratic variation of Γz 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.

KeywordsHeterogeneous porous medium Local thermal non-equilibrium Ferromagnetic convection List of Symbols\{a=\sqrt{\ell ^{2}+m^{2}}}\Overall horizontal wave number

\{\vec {B}}\Magnetic induction

cSpecific heat

caAcceleration coefficient

dThickness of the porous layer

D = d-dzDifferential operator

\{\vec {g}}\Acceleration due to gravity

htheat transfer coefficient

\{\vec {H}}\Magnetic field intensity

H0Imposed uniform vertical magnetic field

\{H { m t} =hd^{2}-\varepsilon k { m tf}}\Scaled inter-phase heat transfer coefficient

\{\hat{{k}}}\Unit vector in z-direction

K0The mean value of Kz

KzPermeability of the porous medium

kfThermal conductivity of the fluid

ksThermal conductivity of the solid

\{K { m p} =- \partial M-\partial T { m f} {H 0 }, {T { m a}}}\Pyromagnetic co-efficient

ℓ, mWave numbers in the x and y directions

\{\vec {M}}\Magnetization

M0 = MH0, TaConstant mean value of magnetization

\{M 1 =\mu 0 K^{2}\beta -1+\chi {\kern 1pt} \alpha { m t} ho 0 g}\Magnetic number

M3 = 1 + M0 -H0-1 + χNon-linearity of magnetization parameter

pPressure

\{\vec {q}=u,v,w}\Velocity vector

\{R= ho 0 \alpha { m t} g\beta k d^{2}-\varepsilon \mu { m f}\kappa { m f}}\Darcy–Rayleigh number

tTime

TTemperature

TLTemperature of the lower boundary

TuTemperature of the upper boundary

\{T { m a} =\left {T { m l} +T { m u}} ight-2}\Reference temperature

WAmplitude of vertical component of perturbed velocity

x, y, zCartesian co-ordinates

Greek SymbolsαtThermal expansion coefficient

β = ΔT-dTemperature gradient

\{\chi =\partial M-\partial H {H 0 }, {T 0}}\Magnetic susceptibility

\{ abla ^{2}=\partial ^{2}-\partial x^{2}+\partial ^{2}-\partial y^{2}+\partial ^{2}-\partial z^{2}}\Laplacian operator

\{ abla { m h}^2 =\partial ^{2}-\partial x^{2}+\partial ^{2}-\partial y^{2}}\Horizontal Laplacian operator

\{\varepsilon }\Porosity of the porous medium

ΓzNon-dimensional permeability heterogeneity function

\{\kappa { m f} =k { m tf} - ho 0 c { m f}}\Thermal diffusivity of the fluid

μfDynamic viscosity

μ0Free space magnetic permeability of vacuum

\{\varphi }\Magnetic potential

\{\Phi }\Amplitude of perturbed magnetic potential

\{\gamma =\varepsilon k { m tf} -\left {1-\varepsilon } ight k { m ts}}\Porosity modified conductivity ratio

ρfFluid density

ρ0Reference density at Ta

\{\Theta}\Amplitude of temperature

SubscriptsbBasic state

fFluid

sSolid

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Author: I. S. Shivakumara - Chiu-On Ng - M. Ravisha

Source: https://link.springer.com/



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