Effect of Hall Currents on Thermosolutal Instability in Non-Newtonian Fluid in Non-Porous Medium

S. K. Kango¹, Sanjay Sharma²

¹Jawaharlal Nehru Government College, Haripur (Manali), HP – 175136 (INDIA)

²Vallabh Government PG College, Mandi, Himachal Pradesh, INDIA

*Corresponding Author E-mail: skkango72@gmail.com

ABSTRACT:

The thermosolutal instability of a layer of Rivlin-Ericksen elastico-viscous fluid (a Non-Newtonian fluid) is considered in the presence of uniform horizontal magnetic field to include the Hall currents in non-porous medium. For the case of stationary convection, the Hall currents hasten the onset of convection, the magnetic field postpones the onset of convection, whereas the kinematic viscoelasticity has no effect on the onset of convection. The Hall currents, kinematic viscoelasticity, magnetic field and the solute parameter introduce oscillatory modes in the system, which were non-existent in their absence. The case of overstability is also considered wherein the sufficient conditions for the non-existence of overstability are obtained. The results are also shown graphically.

KEYWORDS: Thermosolutal Instability; Hall Currents; Rivlin-Ericksen Fluid; Viscoelasticity;

Overstability.

AMC 2000: 76A05; 76A10; 76E06; 76E25.

1. INTRODUCTION:

A detailed account of the theoretical and experimental results of the thermal instability (Benard convection) in an incompressible, viscous(Newtonian) fluid layer, under varying assumptions of hydrodynamics and hydromagnetics, has been given in the celebrated monograph by Chandrasekhar [1]. If an electric field is applied at right angles to the magnetic field, the whole current will not flow along the electric field. This tendency of the electric current of flow across an electric field in the presence of a magnetic field is called Hall Effect. The Hall Effect is likely to be important in many geophysical and astrophysical situations as well as in flows of laboratory plasma. Sherman and Sutton [2] have considered the effect of Hall currents on the efficiency of a magneto-fluid-dynamic generator while Sato [3] and Tani [4] have studied the incompressible viscous flow of an ionized gas with tensor conductivity in channels. The effect of Hall currents on the thermal instability of electrically conducting fluid in the presence of a uniform vertical magnetic field has been studied by Gupta [5]. Sharma and Sunil [6] have considered the effect of Hall currents and finite Larmor radius on the thermal instability of a compressible plasma in porous medium. The fluid is considered to be Newtonian in all the above studies.

There is growing importance of non-Newtonian fluids in geophysical fluid dynamics, chemical technology and petroleum industry. Bhatia and Steiner [7] have studied the problem of thermal instability of Maxwellian viscoelastic fluid in the presence of rotation and have found that rotation has a destabilizing influence in contrast to the stabilizing effect on an ordinary viscous (Newtonian) fluid. An experimental demonstration by Toms and Strawbridge [8] has revealed that a dilute solution of methyle methacrylate in n-butyle acetate agrees well with the theoretical model of Oldroyd [9]. There are many elastico-viscous fluids that cannot be characterized by Maxwell’s constitutive relations or Oldroyd’s constitutive relations. Two such classes of fluids are Rivlin-Ericksen [10] and Walters’ B' [11] fluids. Walters has proposed the constitutive equations for such elastico-viscous fluids. Walters’ B' [12] reported that the mixture of polymethyl methacrylate and pyridine at 25˚C containing 30.5g of polymer per litre behaves very nearly as the Walters’ (model B') elatico-viscous fluid. Rivlin and Ericksen have proposed a theoretical model for such another elastico-viscous fluid. Such and other polymers are used in the manufacture of space crafts, aeroplane parts, tyres, belt conveyers, ropes, cushions, seats, foams, plastics, adhesives, engineering equipments, contact lens etc. Recently, polymers are also used in agriculture, communication appliances and in bio-medical applications. A good account of such fluids has been given by Fredricksen [13]. Sharma et al. [14] have studied the Hall Effect on thermal instability of Walters’ (model B') fluid.

The main large-scale engineering application of thermosolutal concepts is to solar ponds, shallow artificial lakes that are density stratified. A direct analogue of heat/salt diffusive convection has been used to explain the properties of large stars with a helium-rich core, which is heated from below and thus convecting. Another example of thermosolutal convection is when metals solidify since as metals solidify, undesirable in homogeneities on the microscopic scale can be produced by several mechanisms, among which is thermosolutal convection.

The thermosolutal convection in the presence of Hall currents is likely to be important in many geophysical situations and in industry (e.g. MHD generator). The Rivlin-Ericksen fluid has relevance and importance in geophysical fluid dynamics, chemical technology and industry. The present paper, therefore, deals with the effect of Hall currents on thermosolutal instability in Rivlin-Ericksen fluid in non-porous medium.

7. CONCLUSION:

The inclusion of Hall currents give rise to a cross- flow i.e. a flow at right angles to the primary flow in a channel in the presence of a transverse magnetic field, has been shown by Sato [3] and Tani [4] has found that Hall effect produces a cross- flow of double-swirl pattern in incompressible flow through a straight channel with arbitrary cross-section. This breakdown of the primary flow and formation of a secondary flow may be attributed to the inherent instability of the primary flow in the presence of Hall current. Sato [3] has pointed out that even if the distribution of the primary flow velocity be stable to external disturbances, the whole layer may become turbulent if the distribution of the cross-flow velocity is unstable. A similar situation occurs on the three dimensional boundary layer along a swept- back wing. Gupta [5] has found that the presence of Hall current induces a vertical component of vorticity and this may well be the reason for the destabilizing influence.

The Hall current, therefore, has a destabilizing influence for the stationary convection. Also medium permeability has stabilizing as well as destabilizing effects on the thermosolutal instability of Rivlin-Ericksen elastico-viscous fluid in non-porous medium. The Hall current, kinematic viscoelasticity, stable solute gradient and magnetic field introduce the oscillatory modes in the system which were non-existent in their absence. The sufficient condition for the non-existence of overstability for thermosolutal instability of Rivlin-Ericksen elastico-viscous fluid with stable solute gradient, Hall current, magnetic field, viscoelasticity and medium permeability are

8. REFERENCES:

[1] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Dover Publication, New York,

1981.

[2] A. Sherman and G. W. Sutton, Magnetohydrodynamics, Northwestern University, Press, Evanston,

Illinois, 1962.

[3] H. Sato, The Hall effects in the viscous flow of ionized gas between parallel plates under transverse

magnetic field, Phys. Soc. Japan, 16 (1961) 1427-1433.

[4] I. Tani, Steady flow of conducting fluids in channels under transverse magnetic ...consideration of

Hall effects, J. Aerospace Sci., 29 (1962) 297-305.

[5] A. S. Gupta, Hall effect on thermal instability of a horizontal layer of a conducting fluid, Rev.

Roumaine Math pure. Appl., 12 (1967) 665-677.

[6] R. C. Sharma and Sunil,Thermal instability of compressible finite−Larmor radius Hall plasma in a

porous medium, J. Plasma Phys., 55(1) (1996) 35-45.

[7] P. K. Bhatia and J. M. Steiner, Convective instability in a rotating viscoelastic fluid layer, Z.

Angew. Math. mech., 52 (1972) 321.

[8] B. A. Toms and D. J. Strawbridge, Elastic and Viscous properties of dilute solutions of polymethyl

methacrylate in organic liquids, Trans. Faraday Soc., 49(1953) 1225-1232.

[9] J. G. Oldroyd, Non-Newtonian effects in steady motion of some idealized elastico- viscous liquids,

Proc. Roy. Soc. (London), A 245 (1958) 278-297.

[10] R. S. Rivlin and J. L. Ericksen, Stress-deformation relations for isotropic materials, J. Rational

Mech. Anal. Vol. 4, (1955) 323-329.

[11] K. Walters,The motion of an elastico-viscous liquid contained between coaxial Cylinders, Quart. J.

Mech. Apple. Math., 13 (1960) 444-461.

[12] K. Walters, J. Mecanique 1, 479 (1962).

[13] A. G., Fredricksen, Principles and applications of Rheology, Prentice-Hall Inc. New Jersey, USA,

1964.

[14] Sharma et al, Hall effect on thermal instability of Walters’ (model B') fluid, Indian J. of Theo.

Phys., 48(1) (2000) 81-92.

[15] E. A. Spiegel, Convective instability in a compressible atmosphere, Astrophys. J. 141(1965) 068-

1090.

[16] A. K., Aggarwal, Effect of rotation on thermosolutal instability of Walters’ (model B') fluid

permeated with suspended particles in porous medium, Adv. Theor. Appl. Mech., 3 (2010) 177-

188.

Received on 13.01.2014 Accepted on 30.01.2014

Int. J. Tech. 4(1): Jan.-June. 2014; Page 240-247