Thermal Instability of Ferromagnetic Fluid in the Presence of Hall Effect and Suspended Particle under Varying Gravity Field
S. K. Kango1, Vikram Singh2 and Virender Singh3
1Department of Mathematics, Govt. College, Haripur (Manali) (H,P.) – 175136 (INDIA)
2Department of Mathematics, Jwalaji Degree College, Jwalamukhi (H,P.) – 176031 (INDIA)
3Department of Mathematics, Jwalaji Degree College, Jwalamukhi (H,P.) – 176031 (INDIA)
*Corresponding Author E-mail: skkango72@gmail.com; vsmanhas23@gmail.com; virenderbharoli@gmail.com
ABSTRACT:
In this paper we consider theoretical
investigation of the effect hall current and the suspended particle under
varying gravity field on the thermal instability of ferromagnetic fluid heated
from below. For a fluid layer between two free boundaries and exact solution
is obtained using a linearized stability theory and normal mode analysis . A
dispersion relation governing the effect of hall current and the suspended
particle is obtained. For the case of stationary convection it is found that
the magnetic field has a stabilizing the system when the gravity is increasing
upward i.e. (
), where as the hall current and suspended particle are found to
have the destabilizing effect on the system when the gravity is increasing
upward i.e. (). The critical Rayleigh numbers and wave numbers of the
associated disturbances for the onset of stability as stationary convection are
obtained. The principle of exchange of stabilities is not valid for the problem
under consideration, whereas in the absence of Hall currents
hence magnetic field, it is valid under certain conditions.
KEYWORDS: Hall currents, suspended particles, ferromagnetic fluid and varying gravity field.
1. INTRODUCTION:
Ferrohydrodynamics deals with the mechanics of fluid motions influenced with the mechanics of fluid motions influenced by strong forces of magnetic polarization. The Ferrohydrodynamics concerns usually with the interaction of magnetic field on conducting as well as non-conducting ferromagnetic fluids. The polarization force and the body couple are the two main features that distinguish a ferromagnetic fluid from an ordinary fluid. Ferromagnetic fluids are electrically non conducting colloidal suspensions of solid ferromagnetic particles in a
non –electrically conducting carrier fluid like water, kerosene, hydrocarbon etc. A Ferromagnetic fluid contains 1023 particles per cubic meter. These fluids behave as a homogeneous continuum and exhibit a variety of interesting phenomena. Ferromagnetic fluids are not found in nature but are artificially synthesized.
A detailed introduction to this fascinating subject has been given in the celebrated monograph by Rosensweig (1985). This monograph reviews several applications of heat transfer through magnetic fluids. One such phenomenon is enhanced convicted cooling having a temperature dependent magnetic moment due to magnetization of the fluid. This magnetization, in general, is a function of the magnetic field, temperature and density of the fluid. A variation of any of these causes a change of body force. This leads to convection in Ferromagnetic fluids in the presence of the magnetic field gradient. This mechanism is known as ferro convection, which is similar to Benard convection (Chandrasekhar, 1981). In our analysis, we assume that magnetization is aligned with the magnetic field. Convective stability of a Ferromagnetic fluid for a fluid layer heated from below in the presence of uniform vertical magnetic field was considered by Finlayson (1970). He explained the concept of thermo-mechanical interaction in Ferromagnetic fluids. Thermoconvetive stability of a Ferromagnetic fluid without considering buoyancy effect was investigated by Lalas and Carmi (1971) whereas Shliomis (1974) analyzed the linealized relation for magnetized perturbed quantities at the limit of stability. In geophysical situation, the fluid is often not pure but contains suspended particles. Scanlon and Segel (1973) investigated some of the continuum effects of particles on Benard convection and found that a critical Rayleigh number was reduced solely because the heat capacity of the pure was supplemented by that of the particles. The effect of the suspended particles was thus found to destabilized the layer. The effect of the suspended particles on non magnetic fluids has been investigated by many authors (Sharma et al., 1976; Sharma and Aggarwal, 2006). The main results from these studies are that suspended particles have a destabilizing effect on the system and the fact that the specific heat of a fluid is greater than the specific heat of the particles is the sufficient condition for the non-existence of over stability. The effect of suspended particles and rotation o thermal stability of Ferromagnetic fluids is studied by Aggarwal and Prakash (2009) .
In the presence of a strong electric field, 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 the electric and magnetic field is known as ‘Hall effect’. The Hall current is likely to be important in many geophysical and astrophysical situations as well as in flows of laboratory plasmas. The effect of Hall current on thermal stability has also been studied by many other authors as Raghavachar and Gothandaraman (1988), Sharma and Gupta (1993), Gupta (1967).
Soon after the development of the method of formation of Ferromagnetic fluids in the early or mid 1960s, the importance of Ferrohydrodynamics was realized. Due to the vide ranges of application of Ferromagnetic fluids to instrumentation, lubrication, printing, vacuum technology, vibration damping, metal recovery, acoustics and medicine, their commercial usage includes vacuum feed through for semiconductor manufacturing and related uses (Moskowitz, 1975), pressure seals for compressors and blowers (Roseasweig, 1979). They are also used in liquid cooled loudspeakers that involve small bulk quantities of Ferromegnetic fluids to conduct heat away from the speaker coil (Hathaway, 1979). This innovation increases the amplifying power of the coil, and hence it leads the loudspeakers to produce high fidelity sound. In order to bring 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 (Morimoto et al., 1982). The novel zero leakage rotating shaft seal are used in computer disc drives (Bailey, 1983). Experimental and theoretical physicists and engineers contributed significantly to Ferrohydrodynamics and its applications (Odenbach, 2002). During the last half century, research on magnetic liquid has been very productive in many fields. Strong efforts have been undertaken to synthesized stable suspensions of magnetic particles with different performances in magnetism, fluid mechanics or physical chemistry.
Keeping in view the above mentioned significance, the present problem, therefore deals with the thermal instability of a ferromagnetic fluid heated from below in the presence of uniform magnetic field under the influence of Hall currents and suspended particles under varying gravity field. In this problem we have extended the earlier results by Sharma and Aggarwal (2006), Aggarwal and Makhija (2012) under varying gravity field for a incompressible ferromagnetic fluid.
8. CONCLUSION:
In the present paper, the following observations have been derived from the study conducted the combined effect of suspended particles, magnetic field and hall currents on thermal stability of a ferromagnetic fluid under varying gravity field has been considered. The effect of various parameters such as the magnetic field, hall currents and suspended particles have been investigated analytically. The principle conclusions are:-
I. In order to investigate the effects of the magnetic field suspended particles and Hall current, we examined the behavior of dR1/dQ, dR1/dH1 and dR1/dM analytically.
II. It is observed that magnetic field has stabilizes the system when the gravity is increasing upward and destabilizes the system when the gravity is decreasing upward. It is also shows that suspended particles and hall current have a stabilizing or destabilizing effects on thermal convection as gravity decreasing or increasing upwards.
III. The principle of exchange of stability is not valid in the problem whereas in the absence of magnetic field it is valid under certain condition if
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Received on 14.01.2014 Accepted on 31.01.2014 © EnggResearch.net All Right Reserved Int. J. Tech. 4(1): Jan.-June. 2014; Page 91-99 |