INTRODUCTION

Limitations to the Growth Rate of Perturbation in Rotatory Convection in Couple-Stress Fluid in the presence of Magnetic Field

Monika Khanna¹, Ajaib S. Banyal²

¹Department of Mathematics, Govt. College Amb, Dist. Una, (HP) INDIA

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

*Corresponding Author E-mail: khannamonika1501@gmail.com, ajaibbanyal@rediffmail.com,

ABSTRACT:

The thermal instability of a couple-stress fluid acted upon by uniform vertical magnetic field and rotation heated from below is investigated. Following the linearized stability theory and normal mode analysis, the paper through mathematical analysis of the governing equations of couple-stress fluid convection with a uniform vertical magnetic field and rotation, for the case of rigid boundaries shows that the complex growth rate of oscillatory perturbations, neutral or unstable for all wave numbers, must lie inside a semi-circle , in the right half of a complex plane, where Q is the Chandrasekhar number, is the Taylor number, and F is the couple-stress parameter, which prescribes the upper limits to the complex growth rate of arbitrary oscillatory motions of growing amplitude in a rotatory couple-stress fluid in porous medium heated from below.

KEYWORDS: Thermal convection; Couple-Stress Fluid; Rotation; Magnetic Field; PES; Chandrasekhar number; Taylor number.

MSC 2000 No.: 76A05, 76E06, 76E15; 76E07.

1. INTRODUCTION:

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. 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 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 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, 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 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. Sharma and Thakur have studied the thermal convection in couple-stress fluid in porous medium in hydromagnetics. Sharma and Sharma and Kumar and Kumar 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. However, in all above studies the case of two free boundaries which is a little bit artificial except the stellar atmospheric case is considered. Banerjee et al 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 established a criterion on characterization of non-oscillatory motions in hydrodynamics which was further extended by Gupta et al.. However no such result existed for non-Newtonian fluid configurations, in general and for couple-stress fluid configurations, in particular. Banyal and Banyal and Singh have characterized the non-oscillatory motions in couple-stress fluid. Keeping in mind the importance of non-Newtonian fluids, the present paper is an attempt to prescribe the upper limits to the complex growth rate of arbitrary oscillatory motions of growing amplitude, in a layer of incompressible couple-stress fluid heated from below in the presence of uniform vertical magnetic field and rotation opposite to force field of gravity, when the bounding surfaces are of infinite horizontal extension, at the top and bottom of the fluid are rigid.

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Received on 21.11.2016 Modified on 25.11.2016

DOI: 10.5958/2349-2988.2017.00007.9

Research J. Science and Tech. 2017; 9(1):41-47.