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Effect of Thermal Non-Equilibrium on Convective Instability in a Ferromagnetic Fluid-Saturated Porous Medium

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Abstract

The effect of local thermal non-equilibrium (LTNE) on the onset of thermomagnetic convection in a ferromagnetic fluid-saturated horizontal porous layer in the presence of a uniform vertical magnetic field is investigated. A modified Forchheimer-extended Darcy equation is employed to describe the flow in the porous medium, and a two-field model is used for temperature representing the solid and fluid phases separately. It is found that both the critical Darcy–Rayleigh number and the corresponding wave number are modified by the LTNE effects. Asymptotic solutions for both small and large values of scaled interphase heat transfer coefficient H t are presented and compared with those computed numerically. An excellent agreement is obtained between the asymptotic and the numerical results. Besides, the influence of magnetic parameters on the instability of the system is also discussed. The available results in the literature are recovered as particular cases from the present study.

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Abbreviations

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

Overall horizontal wavenumber

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

Magnetic induction

c :

Specific heat

c F :

Dimensionless form drag constant

d :

Thickness of the porous layer

D = d/dz:

Differential operator

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

Acceleration due to gravity

h :

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 :

Permeability of the porous medium

k t :

Thermal conductivity

\({K=-(\partial M/\partial T)_{H_0}{,}_{T_0}}\) :

Pyromagnetic co-efficient

ℓ, m:

Wave numbers in the x and y directions

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

Magnetization

M0 = M(H0, T0):

Constant mean value of magnetization

\({M_1 =\mu_0 K^{2}\beta /(1+\chi )\alpha_{\rm t}\rho_0 g}\) :

Magnetic number

M3 = (1 + M0/H0)/(1 + χ):

Non-linearity of magnetization parameter

p :

Pressure

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

Velocity vector

\({R= \rho_0 \alpha_{\rm t} g\beta k d^{2}/\varepsilon \mu_f \kappa_{\rm f}}\) :

Darcy–Rayleigh number

Rm = RM1:

Magnetic 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

\({\chi =(\partial M/\partial H)_{H_0}{,}{\kern 1pt}_{T_0}}\) :

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

ε :

Porosity of the porous medium

\({\kappa_{\rm f} =k_{\rm tf} /(\rho_0 c)_f}\) :

Thermal diffusivity of the fluid

μ f :

Dynamic viscosity

μ 0 :

Free space magnetic permeability of vacuum

φ :

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

Θ:

Amplitude of temperature

b :

Basic state

f :

Fluid

s :

Solid

References

  • Alexiou C., Arnold W., Hulin P., Klein R., Schmidt A., Bergemann C., Parak F.G.: Therapeutic efficacy of ferrofluid bound anticancer agent. Magnetohydrodynamics 37, 318–322 (2001)

    Google Scholar 

  • Banu N., Rees D.A.S.: Onset of Darcy–Benard convection using a thermal non-equilibrium model. Int. J. Heat Mass Transf. 45, 2221–2228 (2002)

    Article  Google Scholar 

  • Berkovsky B.M., Medvedev V.F., Krakov M.S.: Magnetic Fluids, Engineering Applications. Oxford University Press, New York (1993)

    Google Scholar 

  • Blums E., Cebers A., Maiorov M.M.: Magnetic Fluids. de Gruyter, New York (1997)

    Google Scholar 

  • Borglin S.E., Mordis J., Oldenburg C.M.: Experimental studies of the flow of ferrofluid in porous media. Transp. Porous Med. 41, 61–80 (2000)

    Article  Google Scholar 

  • Finlayson B.A.: Convective instability of ferromagnetic fluids. J. Fluid Mech. 40, 753–767 (1970)

    Article  Google Scholar 

  • Hergt R., Andrä W., Ambly C.G., Hilger I., Kaiser W.A., Richter U., Schmidt H.G.: Physical limitations of hypothermia using magnetite fine particles. IEEE Trans. Magn. 34, 3745–3754 (1998)

    Article  Google Scholar 

  • Kuznetsov A.V.: Thermal non-equilibrium forced convection in porous media. In: Ingham, D.B., Pop, I. (eds) Transport Phenomena in Porous Media, pp. 103–130. Pergamon, Oxford (1998)

    Chapter  Google Scholar 

  • Malashetty M.S., Shivakumara I.S., Sridhar K.: The onset of Lapwood–Brinkman convection using a thermal non-equilibrium model. Int. J. Heat Mass Transf. 48, 1155–1163 (2005a)

    Article  Google Scholar 

  • Malashetty M.S., Shivakumara I.S., Sridhar K.: The onset of convection in an anisotropic porous layer using a thermal non-equilibrium model. Transp. Porous Med. 60, 199–215 (2005b)

    Article  Google Scholar 

  • Malashetty M.S., Shivakumara I.S., Sridhar K., Mahantesh S.: Convective instability of Oldroyd-B fluid saturated porous layer heated from below using a thermal non-equilibrium model. Transp. Porous Med. 64, 123–139 (2006)

    Article  Google Scholar 

  • Matura P., Lucke M.: Thermomagnetic convection in a ferrofluid layer exposed to a time-periodic magnetic field. Phys. Rev. E 80, 026314–026322 (2009)

    Article  Google Scholar 

  • Nanjundappa C.E., Shivakumara I.S.: Effect of velocity and temperature boundary conditions on convective instability in a ferrofluid layer. ASME J. Heat Transf. 130, 104502-1–104502-5 (2008)

    Article  Google Scholar 

  • Nanjundappa C.E., Shivakumara I.S., Arunkumar C.: Benard–Marangoni ferroconvection with magnetic field dependent viscosity. J. Magn. Magn. Mat 322, 2256–2263 (2010)

    Article  Google Scholar 

  • Nield D.A., Bejan A.: Convection in Porous Media, 3rd edn. Springer, New York (2006)

    Google Scholar 

  • Nouri-Borujerdi A., Noghrehabadi A.R., Rees D.A.S.: Onset of convection in a horizontal porous channel with uniform heat generation using a thermal non-equilibrium model. Transp. Porous Med. 69, 343–357 (2007a)

    Article  Google Scholar 

  • Nouri-Borujerdi A., Noghrehabadi A.R., Rees D.A.S.: The effect of local thermal non-equilibrium on impulsive conduction in porous media. Int. J. Heat Mass Transf. 50, 3244–3249 (2007b)

    Article  Google Scholar 

  • Odenbach S.: Recent progress in magnetic fluid research. J. Phys. Condens. Mat. 16, R1135–R1150 (2004)

    Article  Google Scholar 

  • Postelnicu A., Rees D.A.S.: The onset of Darcy–Brinkman convection in a porous medium using a thermal non-equilibrium model. Part I: stress-free boundaries. Int. J. Energy Res. 27, 961–973 (2003)

    Article  Google Scholar 

  • Postelnicu A.: The onset of a Darcy–Brinkman convection using a thermal non-equilibrium model. Part II. Int. J. Therm. Sci. 47, 1587–1594 (2008)

    Article  Google Scholar 

  • Qin Y., Chadam J.: A non-linear stability problem for ferromagnetic fluids in a porous medium. Appl. Math. Lett. 8(2), 25–29 (1995)

    Article  Google Scholar 

  • Rees D.A.S., Pop I.: Free convective stagnation-point flow in a porous medium using a thermal non-equilibrium model. Int. Commun. Heat Mass Transf. 26, 945–954 (1999)

    Article  Google Scholar 

  • Rees D.A.S., Pop I.: Vertical free convective boundary layer flow in a porous medium using a thermal non-equilibrium model. J. Porous Med. 3, 31–44 (2000)

    Google Scholar 

  • Rees D.A.S., Pop I.: Vertical free convective boundary-layer flow in a porous medium using a thermal non-equilibrium model: elliptic effects. J. Appl. Math. Phys. (ZAMP) 53, 1–12 (2002)

    Article  Google Scholar 

  • Rees D.A.S., Pop I.: Local thermal non-equilibrium in porous medium convection. In: Ingham, D.B., Pop, I. (eds) Transport Phenomena in Porous Media, vol. III, pp. 147–173. Elsevier, Oxford (2005)

    Chapter  Google Scholar 

  • Rosensweig R.E.: Ferrohydrodynamics. Cambridge University Press, Cambridge, London (1985)

    Google Scholar 

  • Rosensweig R.E., Zahn M., Volger T.: Stabilization of fluid penetration through a porous medium using magnetizable fluids. In: Berkovsky, B. (eds) Thermomechanics of magnetic fluids, pp. 195–211. Hemisphere, Washington, DC (1978)

    Google Scholar 

  • Shivakumara I.S., Malashetty M.S., Chavaraddi K.B.: Onset of convection in a viscoelastic fluid-saturated sparsely packed porous layer using a thermal non-equilibrium model. Can. J. Phys. 84, 973–990 (2006)

    Article  Google Scholar 

  • Shivakumara I.S., Nanjundappa C.E., Ravisha M.: Thermomagnetic convection in a magnetic nanofluids fluid saturated porous medium. Int. J. Appl. Math. Eng. Sci. 2(2), 157–170 (2008)

    Google Scholar 

  • Shivakumara I.S., Nanjundappa C.E., Ravisha M.: Effect of boundary conditions on the onset of thermomagnetic convection in a ferrofluid saturated porous medium. ASME J. Heat Transf. 131, 101003-1-9 (2009)

    Article  Google Scholar 

  • Shivakumara, I.S., Mamatha, A.L., Ravisha, M.: Effects of variable viscosity and density maximum on the onset of Darcy–Benard convection using a thermal non-equilibrium model. J. Porous Med. (2010a). (to appear)

  • Shivakumara, I.S., Mamatha, A.L., Ravisha, M.: Boundary and thermal non-equilibrium effects on the onset of Darcy–Brinkman convection in a porous layer. J. Eng. Math. (2010b). doi:10.1007/s10665-010-9362-3

  • Singh J., Bajaj R.: Temperature modulation in ferrofluid convection. Phys. Fluids 21, 064105-1–064105-12 (2009)

    Google Scholar 

  • Straughan B.: Global nonlinear stability in porous convection with a thermal non-equilibrium model. Proc. R. Soc A 462, 409–418 (2006)

    Article  Google Scholar 

  • Sunil , Mahajan A.: A non-linear stability analysis for magnetized ferrofluid heated from below. Proc. R. Soc. Lond. A 464, 88–94 (2008)

    Google Scholar 

  • Sunil , Mahajan A.: Nonlinear stability analysis for thermoconvective magnetized ferrofluid saturating a porous medium. Transp. Porous Med. 76, 327–343 (2009)

    Article  Google Scholar 

  • Vafai K., Amiri A.: Non-Darcian effects in combined forced convective flows. In: Ingham, D.B., Pop, I. (eds) Transport Phenomena in Porous Media, pp. 313–329. Pergamon, Oxford (1998)

    Chapter  Google Scholar 

  • Vaidaynathan G., Sekar R., Balasubramanian R.: Ferroconvective instability of fluids saturating a porous medium. Int. J. Eng. Sci. 29, 1259–1267 (1991)

    Article  Google Scholar 

  • Zhan M., Rosensweig R.E.: Stability of magnetic fluid penetration through a porous medium with uniform magnetic field oblique to the interface. IEEE Trans. Magn. 16, 275–282 (1980)

    Article  Google Scholar 

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Authors and Affiliations

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

    Jinho Lee & I. S. Shivakumara

  2. Department of Mathematics, UGC-Centre for Advanced Studies in Fluid Mechanics, Bangalore University, Bangalore, 560001, India

    I. S. Shivakumara

  3. Department of Mathematics, Smt. Rukmini Shedthi Memorial National Government First Grade College, Udupi District, Barkur, 576210, India

    M. Ravisha

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  1. Jinho Lee
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  2. I. S. Shivakumara
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  3. M. Ravisha
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Correspondence to I. S. Shivakumara.

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Lee, J., Shivakumara, I.S. & Ravisha, M. Effect of Thermal Non-Equilibrium on Convective Instability in a Ferromagnetic Fluid-Saturated Porous Medium. Transp Porous Med 86, 103–124 (2011). https://doi.org/10.1007/s11242-010-9608-6

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