Effect of dust particles on a ferromagnetic fluid heated and soluted from below saturating a porous medium

https://doi.org/10.1016/j.amc.2004.09.083Get rights and content

Abstract

A layer of a ferromagnetic fluid permeated with dust particles heated and soluted from below saturating a porous medium is considered in the presence of a transverse uniform magnetic field. Using linearized stability theory and normal mode analysis, an exact solution is obtained for the case of two free boundaries. For the case of stationary convection, medium permeability, dust particles and non-buoyancy magnetization have a destabilizing effect whereas stable solute gradient has a stabilizing effect on the onset of instability. The critical wave number and critical magnetic thermal Rayleigh number for the onset of instability are also determined numerically for sufficiently large values of buoyancy magnetic parameter M1 and results are depicted graphically. It is observed that the critical magnetic thermal Rayleigh number is reduced because of the specific heat of the dust particles. The principle of exchange of stabilities is found to hold true for a ferromagnetic fluid saturating a porous medium heated from below in the absence of dust particles and stable solute gradient. The oscillatory modes are introduced due to the presence of the dust particles and stable solute gradient, which were non-existent in their absence. The sufficient conditions for the non-existence of overstability are also obtained. The paper also reaffirms the qualitative findings of earlier investigations which are, in fact, limiting cases of the present study.

Introduction

Ferromagnetic fluids are obtained by suspending submicron sized particles of magnetite in a carrier such as kerosene, heptane or water. These fluids not found in nature, behave as a homogeneous medium and exhibit interesting phenomena. In the last millennium, the investigation on ferrofluids attracted researchers because of the increase of applications in area such as instrumentation, lubrication, vacuum technology, vibration damping, metals recovery, acoustics; its commercial usage includes vacuum feed-throughs for semiconductor manufacturing and related uses, pressure seals for compressors and blowers, engineering, medicine, chemical reactor and high-speed silent printers, etc. During the last half century, research on magnetic liquids has been very productive in many fields. The major perspective are connected with a massive shocks and oscillation damping (earthquake, airbags), but the contemporary application concerned mostly seals and cooling of loudspeakers. Strong efforts have been undertaken to synthesize stable suspensions of magnetic particles with different performances in magnetism, fluid mechanics or physical chemistry. Many research workers have paid their attention towards the study of applications of ferrofluid see [1], [2], [3], [4], [5], [6].

Experimental and theoretical physicists and engineers gave significant contributions to ferrohydrodynamics and its applications [7]. An authoritative introduction to the research on magnetic liquid has been discussed in detail in the celebrated monograph by Rosensweig [8]. This monograph reviews several applications of heat transfer through ferrofluids. One such phenomenon is enhanced convective cooling having a temperature dependent magnetic moment due to magnetization of the fluid. This magnetization, in general, is function of the magnetic field, temperature and density of the fluid. Any variation of these quantities can induce a change of body force distribution in the fluid. This leads to convection in ferrofluids in the presence of magnetic field gradient. This mechanism is known as ferroconvection, which is similar to Bénard convection [9]. The convective instability of a ferromagnetic fluid for a fluid layer heated from below in the presence of uniform vertical magnetic field has been considered by Finlayson [10]. He explained the concept of thermo-mechanical interaction on ferrofluids. Thermoconvective stability of ferrofluids without considering buoyancy effects has been investigated by Lalas and Carmi [11], whereas Shliomis [12] analyzed the linearized relation for magnetized perturbed quantities at the limit of instability. The stability of a static ferrofluid under the action of an external pressure drop has been studied by Polevikov [13] whereas the thermal convection in a ferrofluid has been considered by Zebib [14]. The thermal convection in a layer of magnetic fluid confined in a two-dimensional cylindrical geometry has been studied by Lange [15]. Schwab et al. [16] investigated experimentally the Finlayson’s problem in the case of a strong magnetic field and detected the onset of convection by plotting the Nusselt number versus the Rayleigh number. Then, the critical Rayleigh number corresponds to a discontinuity in the slope. Later, Stiles and Kagan [17] examined the experimental problem reported by Schwab et al. [16] and generalized the Finlayson’s model assuming that under a strong magnetic field, the rotational viscosity augments the shear viscosity.

The Bénard convection in ferromagnetic fluids has been considered by many authors [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28]. The ferromagnetic fluid has been considered to be clean in all the above studies. In many situations, the fluid is often not pure but contains suspended/dust particles. Saffman [29] has considered the stability of laminar flow of a dusty gas. Scanlon and Segel [30] have considered the effects of suspended particles on the onset of Bénard convection, whereas Sharma et al. [31] have studied the effect of suspended particles on the onset of Bénard convection in hydromagnetics and found that the critical Rayleigh number is reduced because of the heat capacity of the particles. The separate effects of suspended particles, rotation and solute gradient on thermal instability of fluids through a porous medium have been discussed by Sharma and Sharma [32]. The suspended particles were thus found to destabilize the layer. Palaniswamy and Purushotham [33] have studied the stability of shear flow of stratified fluids with fine dust and found the effects of fine dust to increase the region of instability. On the other hand, the multiphase fluid systems are concerned with the motion of a liquid or gas containing immiscible inert identical particles. Of all multiphase fluid systems observed in nature, blood flows in arteries, flow in rocket tubes, dust in gas cooling systems to enhance the heat transfer processes, movement of inert solid particles in atmosphere, sand or other particles in sea or ocean beaches are the most common examples of multiphase fluid systems. Naturally studies of these systems are mathematically interesting and physically useful for various good reasons. The effect of dust particles on non-magnetic fluids has been investigated by many authors [34], [35], [36], [37]. The main result from all these studies is that dust particles have destabilizing effect on the system and specific heat of fluid is greater than the specific heat of particles is the sufficient condition for the non-existence of overstability.

There has been a lot of interest, in recent years, in the study of the breakdown of the stability of a layer of a fluid subjected to a vertical temperature gradient in a porous medium and the possibility of convective flow. The stability of flow of a fluid through a porous medium taking into account the Darcy resistance was considered by Lapwood [38] and Wooding [39]. A porous medium is a solid with holes in it, and is characterized by the manner in which the holes are imbedded, how they are interconnected, and the description of their location, shape and interconnection. However, the flow of a fluid through a homogeneous and isotropic porous medium is governed by Darcy’s law. A macroscopic equation describing incompressible flow of a fluid of viscosity, μ, through a macroscopically homogeneous and isotropic porous medium of permeability, k1, is the well-known Darcy’s equation, in which the usual viscous term in the equations of fluid motion is replaced by the resistance term -(μk1)q, where q is the filter velocity of the fluid.

A porous medium of very low permeability allows us to use the generalized Darcy’s model including the inertial forces. This is because for a medium of very large stable particle suspension, the permeability tends to be small justifying the use of the generalized Darcy’s model including the inertial forces. This is also because the viscous drag force is negligibly small in comparison with the Darcy resistance due to the presence of a large particle suspension. Ferrofluids are mostly organic solvent carriers having a ferromagnetic salt acting as solute. The effect of temperature, rotation and porous medium on ferromagnetic fluids as a single-component fluid has been studied by Sekar et al. [40], Sekar and Vaidyanathan [41], Gupta and Gupta [42] and Finlayson [10]. A layer of ferromagnetic fluid heated and soluted from below in a porous medium has relevance and importance in chemical technology, geophysics and bio-mechanics. More recently, Sunil et al. [27] have considered the effect of rotation on ferromagnetic fluid heated and soluted from below saturating a porous medium.

In view of the above investigations and keeping in mind the importance of ferrofluids, it is attempted to discuss the effect of dust particles on thermosolutal convection in ferromagnetic fluid in homogeneous and isotropic porous medium of very low permeability, subjected to a vertical magnetic field, using generalized Darcy’s model including the inertial forces. The present study can serve as a theoretical support for an experimental investigation. This problem, to the best of our knowledge, has not been investigated yet.

Section snippets

Mathematical formulation of the problem

We consider an infinite, horizontal layer of thickness d of an electrically non-conducting incompressible ferromagnetic fluid embedded in suspended particles in porous medium heated and soluted from below. A uniform magnetic field H0 acts along the vertical direction which is taken as z-axis. The temperature and solute concentration at the bottom and top surfaces z=12d are T0, T1 and C0, C1, respectively, and that a uniform temperature gradient β (=|dTdz|) and a uniform solute gradient β′ (=|dC

The perturbation equations

We shall analyze the stability of the basic state by introducing the following perturbations:q=qb+q,qd=(qd)b+q1,p=pb(z)+p,ρ=ρb+ρ,T=Tb(z)+θ,C=Cb(z)+γ,H=Hb(z)+H,M=Mb(z)+MandN=Nb(z)+N,where q=(u,v,w), q1=(,r,s), p, ρ, θ, γ, H, M′ and N′ are perturbations in ferromagnetic fluid velocity, particle velocity, pressure, density, temperature, concentration, magnetic field, magnetization and suspended particle number density. These perturbations are assumed to be small and then the

Exact solution for free boundaries

Here we consider the case where both boundaries are free as well as perfect conductors of heat. The case of two free boundaries is of little physical interest, but it is mathematically important because one can derive an exact solution, whose properties guide our analysis. Here we consider the case of an infinite magnetic susceptibility χ and we neglect the deformability of the horizontal surfaces. Thus the exact solution of the system (27), (28), (29), (30) subject to the boundary conditionsW=

The case of stationary convection

When the instability sets in as stationary convection (M2  0 and M20), the marginal state will be characterized by σ1 = 0, then the Rayleigh number is given byR1=(1+x)2(1+xM3)xh1P{(1+xM3)+xM3M1(1-M5)}+S1h1(1+xM3)+xM3M11M5-1h1{(1+xM3)+xM3M1(1-M5)}which expresses the modified Rayleigh number R1 as a function of the dimensionless wave number x, non-buoyancy magnetization parameter M3, dust particles parameter h1, medium permeability parameter P (Darcy number), solute gradient parameter S1 and the

The case of oscillatory modes

Here we examine the possibility of oscillatory modes, if any, on stability problem due to the presence of dust particles, stable solute parameter, magnetization parameter and medium permeability. Equating the imaginary parts of Eq. (39), we obtainσ1(1+x)2L0(1+x)L0τ1P+1ε+fε+1P(L2+L3)-xR1L4[τ1(1-M2)(1+x)L0+{h+(1-M2)}L3]+xS1L5[τ1(1-M2)(1+x)L0+{h+(1-M2)}L2]-σ12(1+x)L2L3τ1P+1ε+fε+τ1ε(1+x)2L0(L2+L3)=0.It is evident from Eq. (52) that σ1 may be either zero or non-zero, meaning that the modes may be

The case of overstability

The present section is devoted to find the possibility that the observed instability may really be overstability. Since we wish to determine the Rayleigh number for the onset of instability via a state of pure oscillations, it suffices to find conditions for which (39) will admit of solutions with σ1 real.

Equating real and imaginary parts of (39) and eliminating R1 between them, we obtainA2c12+A1c1+A0=0,where c1=σ12, b = 1 + x,A2=τ1bL32τ1ε(1-M2)L0b+L2τ1P(1-M2)+1ε{f(1-M2)-h},A1=τ12εL03(1-M2)b4+τ1L02L

Discussion of results and conclusions

In this paper, we studied the effect of dust particles on a ferromagnetic fluid heated and soluted from below saturating a porous medium in the presence of uniform vertical magnetic field. We have investigated the effects of medium permeability, non-buoyancy magnetization, stable solute gradient and dust particles on the onset of convection. The principal conclusions from the analysis of this paper are as under:

  • (i)

    For the case of stationary convection, the medium permeability, non-buoyancy

Acknowledgment

Financial assistance to Dr. Sunil in the form of a Research and Development Project [No. 25(0129)/02/EMR-II] and to Mrs. Divya in the form of a Senior Research Fellowship (SRF) of the Council of Scientific and Industrial Research (CSIR), New Delhi is gratefully acknowledged.

References (46)

  • R.L. Bailey

    J. Magn. Magn. Mater.

    (1983)
  • K. Raj et al.

    J. Magn. Magn. Mater.

    (1995)
  • A. Lange

    Magn. Magn. Mater.

    (2002)
  • L. Schwab et al.

    Magnetic Bénard convection

    J. Magn. Magn. Mater.

    (1983)
  • P.J. Stiles et al.

    J. Magn. Magn. Mater.

    (1990)
  • S. Aniss et al.

    J. Magn. Magn. Mater.

    (1993)
  • P.G. Siddheshwar

    J. Mag. Matls.

    (1995)
  • M. Souhar et al.

    Int. J. Heat Mass Transfer

    (1999)
  • R.C. Sharma et al.

    Arch. Mech.

    (2002)
  • R. Sekar et al.

    Int. J. Eng. Sci.

    (1993)
  • R. Sekar et al.

    Int. J. Eng. Sci.

    (1993)
  • G. Vaidyanathan et al.

    J. Magn. Magn. Mater.

    (1995)
  • G. Vaidyanathan et al.

    J. Magn. Magn. Mater.

    (1997)
  • R. Moskowitz

    ASLE Trans.

    (1975)
  • D.B. Hathaway

    dB-Sound Eng. Mag.

    (1979)
  • J.A. Barclay

    J. Appl. Phys.

    (1982)
  • Y. Morimoto et al.

    Chem. Pharm. Bull.

    (1982)
  • S. Odenbach

    Magnetoviscous Effects in Ferrofluids

    (2002)
  • R.E. Rosensweig

    Ferrohydrodynamics

    (1985)
  • S. Chandrasekhar

    Hydrodynamic and Hydromagnetic Stability

    (1981)
  • B.A. Finlayson

    J. Fluid Mech.

    (1970)
  • D.P. Lalas et al.

    Phys. Fluids

    (1971)
  • M.I. Shliomis

    Soviet Phys. Uspekhi (Engl. Transl.)

    (1974)
  • Cited by (5)

    • Convection in porous media

      2017, Convection in Porous Media
    • Convection in porous media: Fourth edition

      2012, Convection in Porous Media: Fourth Edition
    View full text