Hydrodynamic and Hydromagnetic Stability of Stratified Elastico-Viscous Walters’ (model B') Fluid in the Presence of Suspended Particles in Porous Medium

S. K. Kango*

Department of Mathematics, Govt. College, Haripur (Manali) Himachal Pradesh – 175136 (INDIA)

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

ABSTRACT:

In this paper, the stability of stratified elastico-viscous Walters’ (model B') elastico-viscous fluid in the presence of suspended particles and uniform horizontal magnetic field in porous medium is considered. By applying normal mode analysis method, the dispersion relation has been derived and solved numerically. The system is found to be unstable for stable stratifications and unstable for unstable stratifications, which is in contrast to the Newtonian fluid where the system is always stable for stable stratifications. The uniform horizontal magnetic field stabilizes the system for certain wave number range. It has been found that the growth rates decrease with the increase in kinematic viscosity, kinematic viscoelasticity, suspended particles number density and magnetic field.

KEY WORDS: Walters’ B' Fluid; Viscoelasticity; Permeability; Suspended Particles; Porous Medium; Stratification.

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]. Reid [2] investigated the effect of surface tension and viscosity on the stability of two superposed fluids The character of equilibrium of an inviscid, incompressible fluid having variable density in the vertical direction was investigated by Rayleigh [1883]. Sharma and Gupta [12] have also studied the stability of stratified elastico-viscous Walters’ B' fluid in the presence of horizontal magnetic field and rotation.

The medium has been considered to be non-porous in all the above studies. Sharma [6] has studied thermal instability of a viscoelastic fluid in hydromagnetics whereas Sharma et al. [11] have studied the stability of stratified elastico-viscous Walters’ B' fluid in the presence of horizontal magnetic field and rotation in porous medium. Kango and Singh [14] have stability of stratified elastico-viscous Rivlin-Ericksen rotating fluid in the presence of uniform horizontal magnetic field in porous medium. There are many elastico-viscous fluids that cannot be characterized by Maxwell’s constitutive relations or Oldroyd’s constitutive relations. Rivlin-Ericksen [3] and Walters’ B' [15] fluids are two such classes of elastico-viscous fluids. Walters has proposed the constitutive equations of such elastico-viscous fluids.

The flow through porous medium has been of considerable interest in recent years, particularly in geophysical fluid dynamics. 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 which states that the usual viscous term in the equations of stratified Walters’ (model B') fluid motion is replaced by the resistance term, where and are the viscosity and viscoelasticity of the incompressible Rivlin-Ericksen fluid, is the medium permeability and is the Darcian (filter) velocity of the fluid. The stability of flow of a single component fluid through porous medium, taking into account the Darcy’s resistance, has been studied by Lapwood [1948] and Wooding [13]. Sharma [7] has studied the effect of uniform magnetic field and uniform rotation on the stability of two superposed fluids in porous medium. Sharma [8] has also studied the stability of stratified fluid in porous medium in the presence of horizontal magnetic field and rotation. Sharma and Kumar [4] have studied the steady flow and heat transfer of Walters’ B' fluid through porous pipe of uniform circular cross-section with small suction. Sharma and Kango [10] have studied the thermal convection in Rivlin-Ericksen elastico-viscous fluid in porous medium in hydromagnetics. Sharma and Kango [9] have studied the stability of two superposed Walters’ (model B’) viscoelastic fluids of suspended particles and variable magnetic field in porous medium.

Another application of the result of flow through porous medium in the presence of magnetic field is in the geothermal region. The rotation of the Earth distorts the boundaries of a hexagonal convection cells in fluid through a porous medium and the distortion plays an important role in the extraction of energy in the geothermal regions. Keeping in mind the growing importance of non-Newtonian fluids in modern technology, chemical technology, industry and dynamics of geophysical fluids and considering the conflicted tendencies of magnetic field and rotation while acting together, our interest, in the present paper is to study the hydrodynamic and hydromagnetic stability of stratified elastico-viscous Walters’ (model B') fluid in the presence of suspended particles in porous medium.

6. REFERENCE:

1. E. R. Lapwood, Convection of a fluid in a porous medium, Proc. Camb. Phil. Soc. 44 (1948), 508-519.

2. K. Walters, The motion of an elastico-viscous liquid contained between coaxial Cylinders, Quart. J. Mech. Apple. Math., Vol. 13 (1960), 444-461.

3. Lord Rayleigh, Investigation of the character of an incompressible heavy fluid of variable density, Proc. London Math. Soc. Vol. 14 (1883), 170-177.

4. P. R. Sharma and H. Kumar, Steady flow and heat transfer of Walters’ B' fluid through porous pipe of uniform circular cross-section with small suction, Proc. Nat. Acad. Sci. India, 65(A) I (1995), 75.

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6. R. C. Sharma, Thermal instability of a viscoelastic fluid in hydromagnetics, Acta Physics Hungrica, 38 (1975), 293.

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12. S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Dover Publication, New York, 1981.

13. S. K. Kango and V. Singh, Stability of stratified elastico-viscous Rivlin-Ericksen rotating fluid in the presence of uniform horizontal magnetic field in porous medium, Studia Geotechnica et Mechanica, Vol. XXXIII (2011), 33-40.

14. Sharma et al, Stability of stratified elastico-viscous Walters’ B' fluid in the presence of horizontal magnetic field and rotation in porous medium, Arch. Mech. 58 (2006), 187-197.

15. V. Sharma and U. Gupta, Stability of stratified elastico-viscous Walters’ B' fluid in the presence of horizontal magnetic field and rotation, Studia Geotechnica et Mechanica, Vol. XXXII (2010), 41-53.

16. W. H. Reid, The effect of surface tension and viscosity on the stability of two superposed fluids, Proc. Camb. Phil. Soc., 57 (1961), 415-424.

Received on 10.12.2011 Accepted on 12.02.2012

Research J. Engineering and Tech. 3(2): April-June 2012 page146-151