Hall Effect on Magneto-Thermal Stability of Rivlin-Ericksen Ferromagnetic Fluid Saturating A Porous Medium

Veena Sharma, Kavita, Sumit Gupta

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

Department of Mathematics, Govt. Degree College Diggal, Distt. Solan (H.P)

*Corresponding Author E-mail: veena_math_hpu@yahoo.com, kavi6991@gmail.com

ABSTRACT:

This paper deals with the electrically conducting and numerical investigation of the effect of Hall currents on the thermal stability of a ferromagnetic viscoelastic fluid heated from below saturating porous medium. The rheology of the fluid in described by the Rivlin- Ericksenian. The boundaries are considered to be stress-free. The eigen-value problem obtained using linear stability theory and normal mode technique is solved numerically using the Galerkin technique and the software MATHEMATICA by assuming the trial functions satisfying the boundary conditions. A dispersion relation governing the effects of medium permeability, a uniform horizontal magnetic field, magnetization and Hall currents is derived. For the case of stationary convection, it is found that the magnetic field and magnetization have a stabilizing effect on the system, as such their effect is to postpone the onset of thermal instability, whereas the Hall currents are found to hasten the onset of convection under certain conditions.

KEYWORD:

INTRODUCTION:

Ferro fluids are magnetic fluids formed by a stable colloidal suspension of magnetic nanoparticle dispersed in a carrier liquid such as kerosene, heptanes or water. These deal with the mechanics of fluids motions influenced by strong forces of magnetic polarization. Ferro hydrodynamics concerns usually non- conducting liquids with magnetic properties and constitutes an entire field of physics close to magneto hydrodynamics but still different. An authoritative introduction to this fascinating subject has been discussed in detail in the celebrated monograph by Rosensweig [1985]. This monograph reviews several applications of heat transfer through ferromagnetic fluids. One such phenomenon is enhanced convective cooling having a temperature dependent 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 convection [1981]. In our analysis, we assume that the magnetization is aligned with the magnetic field. Convection instability of a ferromagnetic fluid for a fluid layer heated from below in the presence of uniform vertical ferromagnetic fluid has been considered by Finlayson [1970]. He explained the concept of thermo mechanical interaction in ferromagnetic fluids. Thermo convective stability of ferromagnetic fluids without considering buoyancy effects has been investgated by Lalas and Carmi [1971], whereas Shliomis [1974] analyzed the linearized relation for magnetized perturbed quantities at the limit of instability. The enard convection in ferromagnetic fluids has been considered by Siddheswar [1993], Siddheswar [1995], Venkatasubramaniam [1994], Aggarwal [2009], Sunil [2006], Sunil [2007]. 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 convection flow. The stability of flow of a fluid through a porous medium taking into account the Darcy resistance was considered by Lapwood [1948], Wooding [1962], Sunil et al.[2008]. A macroscopic equation describing incompressible flow of a fluid of viscosity, through a macroscopically homogeneous and isotropic porous medium of permeability k₁, 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. With the growing importance of non-Newtonian fluids in modern technology and industries, the investigation of such fluids is desirable. There are many elasto-viscous fluids that cannot be characterized by Maxwell’s or Oldroyd’s constitutive relations. One such class of the viscoelastic fluids is the Rivlin– Ericksen fluid. Rivlin and Ericksen (1955) have proposed a theoretical model for such viscoelastic fluid. This and other class of polymers is used in the manufacture of parts of space–crafts, aeroplanes, tyres, belt conveyers, ropes cushions, seats, foams, engineering equipments etc. Recently, polymers are also used in agriculture, communication appliances and in biomedical appliances. 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 in the literature 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 various parameters on the thermal instability in non- Newtonian ferrofluids have been studied by Raghavachar [1998], Sharma [1993], Gupta [1967], Sharma [2000],Sunil [2005],Gupta [2011], Gupta [2012]. Motivated by the various application of rheology, medium porosity, Hall currents and medium permeability, an attempt has been made to study the thermal stability of a layer of ferromagnetic viscoelastic fluid heated from below saturating a porous medium in the presence of Hall currents (and hence magnetic field) in this paper, which is an extension of the research work by Sharma and Thakur [2016].

NUMERICAL RESULTS AND DISCUSSION:

The equations (29) and (34) have been examined numerically using the software Mathematica version- 5.2. The variation of Rayleigh number with respect to wave number for the stationary case for the fixed permissible vales of the dimensionless parameters M=100, Q=10, =1.2, , , have been plotted.

Figure 2: The variation of Rayleigh number versus wavenumber a for different values of Hall currents M(=80, 90, 100, 110.

Figure 3: The variation of Rayleigh number versus wavenumber a for different values of Hall current (=1.2, 1.4, 1.6).=

Figure 4: The variation of Rayleigh number versus wave number a for different values of Magnetic field Q(=20, 25, 30).

Figure 5: The variation of Rayleigh number versus wave number a for different values of Magnetization parameter M₀(=10, 50, 100, 150).

Figure 2 represents the variation of Rayleigh number versus wave number a for the stationary convection for various values of M Hall currents. It is evident from the graphs that the Rayleigh number increases with the increase in the parameter M showing thereby the stabilizing effect, whereas Figure 3 represents the variation of Rayleigh number versus wave number a for the stationary convection for various values of Medium permeability. The Rayleigh number decreases with the increase in the parameter showing thereby the destabilizing effect on the system.

Figure 6: The variation of Rayleigh number versus wavenumber a for different values of the medium porosity (=0.4, 0.5).

Figure 4 represents the variation of Rayleigh number versus wave number a for the stationary convection for various values of Q magnetic field. The Rayleigh number decreases with the increase in the parameter Q for the wave number showing thereby the destabilizing effect and stabilizing effect on the system for a >1.8. Figure 5 represents the variation of Rayleigh number versus wave number a for the stationary convection for various values of magnetization parameter M₀. The Rayleigh number increases with the increase in the parameter M₀ showing thereby the stabilizing effect. Figure 6 represents the variation of Rayleigh number versus wave number a for the stationary convection for various values of medium porosity (=0.4, 0.5). The Rayleigh number increases with the increase in the medium porosity for the wave number a 2.8 showing there by the stabilizing effect and destabilizing effect for a > 2.8.

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 whereas medium permeability shows destabilizing effect on the system. Using the Galerkin weighted Residuals method the effect of various parameters such as magnetic field, Hall currents and medium permeability has been investigated numerically. The principal from the analysis are as follows:

· It is found that Hall currents, magnetic field and magnetization have a stabilizing effect on the system. Figures 2, 4 and 5 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.

REFERENCES:

1. Odenbach, S., Magneto viscous Effects in Ferrfluids, Springer-Verlag, Berlin, Heidelberg, 2002

2. Rosenwieg, R. E., Ferrohydrodynamics, Cambridge University Press, Cambridge, 1985

3. Chandrasekhar, S., Hydrodynamic and Hydromagnetic Stability, Dover publications, New York, USA, 1981

4. Finlayson, B. A., Convective Instability of Ferromagnetic Fluids, J. Fluid Mech., 40 (1970), 4, pp. 753-767

5. Lalas D. P., Carmi, S., Thermo convective Stability of Ferrofluids, Phys. Fluids, 14 (1971), 1, pp. 436-438

6. Shliomis, M. I., Magnetic Fluids. Soviet Phys. Uspekhi (Engl. Transl.), 17 (1974), 153, pp. 153 - 169

7. Siddheswar, P. G., Rayleigh-Bėnard Convection in a Ferromagnetic Fluid with Second Sound, Japan Soc. Mag. Fluids, 25 (1993), 1, pp. 32-36

8. Siddheswar, P. G., Convective Instability of a Ferromagnetic Fluids Bounded by Fluid Permeable Magnetic Boundaries, J. Magnetism and Magnetic Materials, 49 (1995), 1-2, pp. 148-150

9. Venkatasubramaniam, S., Kaloni, P. N., Effect of Rotation on the Thermo convective Instability of a Horizontal Layer of Ferrofluids, Int. J. Engg. Sci., 32 (1994), 2, pp. 237-256

10. Aggarwal, A. K., Prakash, K., Effect of Suspended Particles and Rotation on Thermal Instability of Ferrofluids, Int. J. of Applied Mechanics and Engineering, 14 (2009), 1, pp. 55-66

11. Sunil, Sharma, A. Shandil, R. C., Effect of Rotation on a Ferromagnetic Fluid Heated and Soluted from Below in the Presence of Dust Particles, Applied Mathematics and Computation, 177 (2006), 2, pp. 614- 628

12. Sunil, Kumar, P., Sharma, D., Thermal Convection in Ferrofluid in a Porous Medium, Studia Geotechnica et Mechanica, 29 (2007), 3-4, pp. 143-157

13. Lap wood, E. R., Convection of a Fluid in a Porous Medium, Math. Proc. Cambridge Phil. Soc., 44 (1948), 4, pp. 508-521

14. Wooding, R. A., Rayleigh Instability of a Thermal Boundary Layer in Flow Through a Porous Medium, J. Fluid Mech., 9 (1960), 2, pp. 183-192

15. Sunil, Sharma, A., et al., Effect of Magnetic Field Dependent Viscosity on Thermal Convection in a Ferromagnetic Fluid, Chemical Engineering Communications, 195 (2008), 5, pp. 571-583

16. Vaidyanathan, G., Sekar, R. et al., Ferro convective Instability of Fluids Saturating a Porous Medium, Int. J. Engg. Sci., 29 (1991), 10, pp. 1259-1267

17. Raghavachar, M. R., Gothandaraman, V. S., Hydro magnetic Convection in a Rotating Fluid Layer in the Presence of Hall Current, Geophys. Astro. Fluid Dyn., 45 (1988), 3-4, pp. 199-211

18. Sharma, R. C., Gupta, U., Thermal Instability of Compressible Fluids with Hall Currents and Suspended Particles in Porous Medium, Int. Journal of Engg. Sci., 31 (1993), 7, pp. 1053-1060

19. Gupta, A. S., Hall Effects on Thermal Instability, Rev. Roum. Math. Pure Appl., 12 (1967), pp. 665-677

20. Sharma, R. C., Sunil, Chand, S., Hall Effect on Thermal Instability of Rivlin-Ericksen Fluid, Indian J. Pure Appl .Math. 31 (2000), 1, pp. 49-59

21. Sunil, Sharma, Y. D., et al., Thermosolutal Instability of Compressible Rivlin-Ericksen Fluid with Hall Currents, Int. J. Applied Mechanics and Engineering, 10 (2005), 2, pp. 329-343

22. Gupta, U., Aggarwal, P., Thermal Instability of Compressible Walters’ (Model B’) Fluid in the Presence of Hall Currents and Suspended Particles, Thermal Science, 15 (2011), 2, pp. 487-500

23. Gupta, U., et al., Thermal Convection of Dusty Compressible Rivlin-Ericksen Fluid with Hall Currents, Thermal Science, 16 (2012), 1, pp. 177-191

24. Kumar, P., et al., Thermal Instability of Walters B’ Viscoelastic Fluid Permeated with Suspended Particles in Hydromagnetics in Porous Medium, Thermal Science, 8 (2004), 1, pp. 51-61

25. Penfield, P., Haus, H. A., Electrodynamics of Moving Media, Institute of Technology Press, Cambridge, Mass., USA, 1967

26. Cowley, M.D. and Rosenweig, R.E., The interfacial stability of a ferromagnetic fluid, J. Fluid Mech., 1967, 30(4), 671-688

27. Rivlin R.S, Ericksen, J.L. J. Rat. Mech. Anal .1955 ,4, pp323-425

28. Sharma, V., Thakur, A., Effect of Hall currents on the stability of Ferromagnetic Fluid heated from below in the presence of a magnetic field saturating porous medium, Int.J.Tech.,2016,6,2, pp 239- 247.

Received on 21.11.2016 Modified on 25.11.2016

DOI: 10.5958/2349-2988.2017.00026.2

Research J. Science and Tech. 2017; 9(1):160-166.