A Characterization of Onset of Instability in Couple-Stress Fluid in the Presence of Magnetic Field

Kamal Singh¹ and Ajaib S. Banyal²

Research Scholar, Department of mathematics Singhania University Pacheri-Bari, Jhunjnu Rajasthan

²Department of Mathematics, Govt. College Nadaun, Dist. Hamirpur, (HP) INDIA 177033

*Corresponding Author E-mail: ajaibbanyal@rediffmail.com

ABSTRACT:

The thermal instability of a couple-stress fluid acted upon by uniform vertical magnetic field and heated from below is investigated. Following the linearized stability theory and normal mode analysis, the paper mathematically established the condition for characterizing the oscillatory motions which may be neutral or unstable for free and perfectly conducting boundaries at the top and bottom of the fluid. It is established that all nondecaying  slow motions starting from rest, in a couple-stress fluid of infinite horizontal extension and finite vertical depth, which is acted upon by uniform vertical magnetic field opposite to gravity and a constant vertical adverse temperature gradient, are necessarily nonoscillatory, in the regime

where R is the thermal Rayleigh number, F is the couple-stress parameter and  is the magnetic Prandtl number. The result is important since the exact solutions of the problem investigated in closed form, are not obtainable.

KEYWORDS: Thermal convection; Couple-Stress Fluid; PES; Magnetic Field; Chandrasekhar Number.

MSC 2000 No.: 76A05; 76E06; 76E15; 76U05

1.  INTRODUCTION:

Right from the conceptualizations of turbulence, instability of fluid flows is being regarded at its root. The thermal instability of a fluid layer with maintained adverse temperature gradient by heating the underside plays an important role in Geophysics, interiors of the Earth, Oceanography and Atmospheric Physics etc. A detailed account of the theoretical and experimental study of the onset of Bénard Convection in Newtonian fluids, under varying assumptions of hydrodynamics and hydromagnetics, has been given by Chandrasekhar (1981). The use of Boussinesq approximation has been made throughout, which states that the density changes are disregarded in all other terms in the equation of motion except the external force term. Sharma et al (1976) has considered the effect of suspended particles on the onset of Bénard convection in hydromagnetics. The fluid has been considered to be Newtonian in all above studies. With the growing importance of non-Newtonian fluids in modern technology and industries, the investigations on such fluids are desirable. Stokes (1966) proposed and postulated the theory of couple-stress fluid. One of the applications of couple-stress fluid is its use to the study of the mechanism of lubrication of synovial joints, which has become the object of scientific research. According to the theory of Stokes (1966), couple-stresses are found to appear in noticeable magnitude in fluids having very large molecules. Since the long chain hylauronic acid molecules are found as additives in synovial fluid, Walicki and Walicka (1999) modeled synovial fluid as couple-stress fluid in human joints. An electrically conducting couple-stress fluid heated from below in porous medium in the presence of uniform horizontal magnetic field has been studied by Sharma and Sharma (2001). Sharma and Thakur (2000) have studied the thermal convection in couple-stress fluid in porous medium in hydromagnetics. Sharma and Sharma (2004) and Kumar and Kumar (2011) have studied the effect of dust particles, magnetic field and rotation on couple-stress fluid heated from below and for the case of stationary convection, found that dust particles have destabilizing effect on the system, where as the rotation is found to have stabilizing effect on the system, however couple-stress and magnetic field are found to have both stabilizing and destabilizing effects under certain conditions, when both the boundaries are dynamically free..

Pellow and Southwell (1940) proved the validity of PES for the classical Rayleigh-Bénard convection problem. Banerjee et al (1981) gave a new scheme for combining the governing equations of thermohaline convection, which is shown to lead to the bounds for the complex growth rate of the arbitrary oscillatory perturbations, neutral or unstable for all combinations of dynamically rigid or free boundaries and, Banerjee and Banerjee (1984) established a criterion on characterization of non-oscillatory motions in hydrodynamics which was further extended by Gupta et al (1986). However no such result existed for non-Newtonian fluid configurations, in general and for couple-stress fluid configurations, in particular. Banyal (2011) have characterized the non-oscillatory motions in couple-stress fluid. Shivakumara et al (2011) have studied the effect of non-uniform temperature gradients on the onset of convection in a couple-stress fluid saturated porous medium.

Keeping in mind the importance of non-Newtonian fluids, the present paper is an attempt to characterize the onset of instability analytically, in a layer of incompressible couple-stress fluid heated from below in the presence of uniform vertical magnetic field opposite to force field of gravity, when the bounding surfaces of infinite horizontal extension, at the top and bottom of the fluid are free and the region outside the fluid is perfectly conducting. It is shown that for the configuration under consideration, if, then an

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Received on 04.01.2014    Accepted on 01.02.2014

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