Effect of Hall currents on the stability of Ferromagnetic Fluid heated from below in the presence of a magnetic field saturating porous media

 

Veena Sharma*, Anukampa Thakur, Abhishek Sharma

Department of Mathematics and Statistics, Himachal Pradesh University, Shimla-5

*Corresponding Author Email: veena_math_hpu@yahoomail.com

 

ABSTRACT:

In this study, the effect of Hall currents on the onset of stability of saturating porous media an electrically conducting ferromagnetic fluid heated from below using linear stability analysis is investigated. Using the Darcy law to modify the ferromagnetic fluid the momentum equations for a porous medium. The employed model incorporates the effects of polarization force and body couple. The coupled partial differential equations governing the physical problem are reduced to a set of ordinary differential equations using normal mode technique. These equations are solved analytically for stress- free boundaries and numerical results are computed by obtaining approximate solutions using Galerkin method using the software Mathematica for the case of stationary convection. It is found that the magnetic field and magnetization have a stabilizing effect as such their effect is to postpone the onset of thermal instability; whereas Hall currents are found to hasten the same. The medium permeability prepones the onset of convection under certain conditions implying thereby the destabilizing effect.

 

KEYWORDS:

 

INTRODUCTION:

Ferrohydrodynamics deals with the mechanics of fluid motions influenced by strong forces of magnetic polarization. Ferromagnetic fluids are electrically non-conducting colloidal suspensions of solid ferromagnetic particles in a non-electrically conducting carrier fluid like water, kerosene, hydrocarbon, etc. These fluids behave as a homogeneous continuum and exhibit a variety of interesting phenomena. The polarization force and the body couple are the two main features that distinguish ferromagnetic fluid from ordinary fluid. Ferromagnetic fluids are not found in nature but are artificially synthesized. Soon after the method of formation of ferromagnetic fluids in the early or mid 1960s, the importance of ferrohydrodynamics was realized. Due to the wide ranges of applications of ferromagnetic fluids to instrumentation, lubrication, printing, vacuum technology, vibration damping, metal recovery, acoustics and medicine, its commercial usage includes vacuum feed throughs for semiconductor manufacturing and related uses [1], pressure seals for compressors and blowers[2]. These are also used in liquid cooled loudspeakers that involve small bulk quantities of the ferromagnetic fluid to conduct heat away from the speaker coils [3]. This innovation increases the amplifying power of the coil, and hence it leads the loudspeakers to produce high fidelity sound. In order to bring the drugs to a target site in a human body, a magnetic field can pilot the path of a drop of the ferromagnetic fluid in the human body [4]. The novel zero leakage rotating shaft seals are used in computer disk drives [5]. Experimental and theoretical physicists and engineers gave significant contributions to ferrohydrodynamics and its applications [6]. During the last half century, research on magnetic liquids has been very productive in many fields. Strong efforts have been undertaken to synthesize stable suspensions of magnetic particles with different performances in magnetism, fluid mechanics or physical chemistry. An authoritative introduction to this fascinating subject has been discussed in detail in the celebrated monograph by Rosensweig[7]. This monograph reviews several applications of heat transfer through ferromagnetic fluids. One such phenomenon is enhanced convective cooling having a temperature dependent magnetic moment due to magnetization of the fluid. This magnetization, in general, is a function of magnetic field, temperature and density of the fluid. The variation of anyone of these causes a change of body force. This leads to convection in ferromagnetic fluids in the presence of magnetic field gradient. This mechanism is known as ferroconvection, which is similar to Bėnard convection [8]. In this analysis, it is assumed that the magnetization is aligned with the magnetic field. 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 [9]. He explained the concept of thermomechanical interaction in ferromagnetic fluids. Thermo-convective stability of ferromagnetic fluids without considering buoyancy effects has been investigated by Lalas and Carmi [10], whereas Shliomis [11] analyzed the linearized relation for magnetized perturbed quantities at the limit of instability.

 

The Bėnard convection in ferromagnetic fluids has been considered by many authors [12-17]. The medium has been considered to be non-porous in all the above studies. There has been a lot of interest, in recent years, in the study of the breakdown of the stability of a fluid layer subjected to a vertical temperature gradient in a porous medium and the possibility of the convective flow. The stability of flow of a fluid through a porous medium taking into account the Darcy resistance was considered by Lapwood [18], Wooding [19], Sunil et al. [20] and many others. 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 , is well-known Darcy’s equation, in which the usual viscous term in the equations of fluid motion is replaced by the resistance term , where q is the filter velocity of the fluid. The thermoconvective instability in a ferromagnetic fluid saturating a porous medium of very large permeability subjected to a vertical magnetic field has been studied using the Brinkman model by Vaidyanathan et al. [21], and indicated that only stationary convection can exist. In the presence of strong electric field, the electric conductivity is affected by the magnetic field. Consequently, the conductivity parallel to the electric field is reduced. Hence, the current is reduced in the direction normal to both electric and magnetic field. This phenomenon is known as Hall effect. The Hall current is likely to be important in flows of laboratory plasmas as well as in many geophysical and astrophysical situations. The effect of Hall current on thermal instability has also been studied by several authors [22-28].

 

In the present paper, the effect of Hall currents on thermal stability of ferromagnetic fluid heated from below saturating a porous medium in the presence of horizontal magnetic field has been investigated numerically, which is an  extension of  the results reported by Kumar et al. [29] to include the effect of Hall currents for ferromagnetic fluids in porous medium

 

MATHEMATICAL FORMULATION OF THE PROBLEM:

An infinite, incompressible, electrically non-conducting thin ferromagnetic fluid, bounded by the planes z=0 and z=d saturating a porous medium is considered to include the effect of Hall currents. This layer is heated from below so that uniform temperature gradient  is maintained. A uniform horizontal magnetic field and gravity force g (0, 0, - g) pervade the system. This fluid layer is flowing through an isotropic and homogeneous porous medium of porosity  and medium permeability k1.

 

NUMERICAL RESULTS AND DISCUSSION

To investigate the complete parameter effects on the stability problem, equation (35) for stationary convection has been analysed numerically using the MATHEMATICA Software. Figure 2 and 3 represent the plots of the thermal Rayleigh number (), versus medium permeability (P) and magnetic field (Q) for the various values of the non-dimensional wave numbers a =2, 4, 6, 8, 10 respectively.  Figure 2 illustrates that the Rayleigh number is uninfluenced with the increase in medium permeability implying thereby that the medium permeability has no effect on the stability of the physical system. It is clear from the figure 3 that the Chandersekhar number (Q) accounting for the magnetic field has a large enough stabilizing effect as the Rayleigh number increases with the increase in Q, which is similar to the case of regular fluids. In figure 4,  is plotted against a for M>1 and it is depicted very slightly from the graphs that  increases with the increase in medium permeability implying thereby the stabilizing (slight) effect of medium  permeability on the system. From the figure 5, it is evident that the Hall currents parameter M has again uninfluenced effect in the stability of the physical system.

 

Figure 2.Variation of R1 and P for fixed M = 0.1, Q1 = 10, θ = 45°, and P (= 1, 2,…, 6)

 

Figure 3.Variation of R1 with Q1 for fixed P = 50, θ = 45°, M = 10, Q1 (= 10, 20,…, 60)

 

Figure 4.Variation of R1 and a for a fixed M = 100, Q1 = 10, θ = 45°, for a (= 0.1, 0.5, 1,…, 4)

 

Figure 5.Variation of R1 with Q1 for fixed P = 50, θ = 45°, M = 10, Q1 (= 10, 20,…, 60)

 

Figure 6.Variation of R1 with Mo for fixed P = 10, θ = 45°, M = 10,Q= 10, Mo (= 10, 20,…, 50)

 

CONCLUSIONS:

The combined effect of medium permeability, horizontal magnetic field, Hall currents, and magnetization has been considered on the thermal stability of a ferromagnetic fluid. Using the first approximation of Galerkin method the effect of various parameters such as magnetic field, Hall currents and medium permeability has been investigated  numerically. The main results from the analysis o are as follows.

·        In order to investigate the effects of magnetic field, Hall currents and Medium permeability, we examine the behavior of dRf /dQ, dRf /dM, /and dRf /dP analytically.

·        It is found that Hall currents have a destabilizing effect whereas magnetic field and magnetization have a  stabilizing effect on the system. Figures 3, 5 and 6 support the analytic results graphically. These are valid for second-order fluids as well.

·        The medium permeability always hastens the onset of convection for all wave numbers as the Rayleigh number decreases with an increase in medium permeability parameter  whereas for M > 1, the medium permeability hastens the onset of convection for small wave numbers as the Rayleigh number decreases with an increase in medium permeability parameter and postpones the onset of convection for higher wave numbers as the Rayleigh number increases with an increase in medium permeability parameter.

 

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Received on 28.08.2016            Accepted on 15.09.2016           

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Int. J. Tech. 2016; 6(2): 239-247.

DOI: 10.5958/2231-3915.2016.00037.7