INTRODUCTION

Magneto-Thermosolutal Convection in Rivlin-Ericksen Viscoelastic Fluid in a Porous Medium

Pawan Kumar*, Ajaib S. Banyal² , Nirmal Singh³

¹Department of Mathematics, NSCBM GC Hamirpur, (HP) INDIA 177005

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

³Department of Mathematics, NSCBM GC Hamirpur, (HP) INDIA 177005

*Corresponding Author E-mail: pawanmaths78@gmail.com, ajaibbanyal@rediffmail.com

ABSTRACT:

Thermosolutal instability of Veronis type in Rivlin-Ericksen viscoelastic fluid in the presence of uniform vertical magnetic field in a porous medium is considered. 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 any arbitrary combination of free and rigid boundaries at the top and bottom of the fluid. It is established that all non-decaying slow motions starting from rest, in a Rivlin-Ericksen viscoelastic fluid of infinite horizontal extension and finite vertical depth in the presence of uniform vertical magnetic field in a porous medium, are necessarily non-oscillatory, in the regime , where is the Thermosolutal Rayliegh number, Q is the Chandrasekhar number, is the magnetic Prandtl number, is the thermosolutal Prandtl number, is the medium permeability, is the porosity and F is the viscoelasticity parameter. The result is important since it hold for all wave numbers and for any arbitrary combination of free and rigid boundaries at the top and bottom of the fluid. A similar characterization theorem is also proved for Stern type of configuration.

KEYWORDS: Thermosolutal convection; Rivlin-Ericksen Fluid; Magnetic Field; PES; Rayleigh number; Chandrasekhar number.

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

1.INTRODUCTION

Stability of a dynamical system is closest to real life, in the sense that realization of a dynamical system depends upon its stability. A comprehensive 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) in his celebrated monograph. The use of Bossiness 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. The problem of thermohaline convection in a layer of fluid heated from below and subjected to a stable salinity gradient has been considered by Veronis (1965). The physics is quite similar in the stellar case, in that helium acts like in raising the density and in diffusing more slowly than heat. The condition under which convective motions are important in stellar atmospheres are usually far removed from consideration of single component fluid and rigid boundaries and therefore it is desirable to consider a fluid acted upon by a solute gradient with free or rigid boundaries. The problem is of great importance because of its applications to atmospheric physics and astrophysics, especially in the case of the ionosphere and the outer layer of the atmosphere. The thermosolutal convection problems also arise in oceanography, limnology and engineering. Bhatia and Steiner (1972) have considered the effect of uniform rotation on the thermal instability of a viscoelastic (Maxwell) fluid and found that rotation has a destabilizing influence in contrast to the stabilizing effect on Newtonian fluid. Sharma (1976) has studied the thermal instability of a layer of viscoelastic (Oldroydian) fluid acted upon by a uniform rotation and found that rotation has destabilizing as well as stabilizing effects under certain conditions in contrast to that of a Maxwell fluid where it has a destabilizing effect. There are many elastico-viscous fluids that cannot be characterized by Maxwell’s constitutive relations or Oldroyd’s (1958) constitutive relations. Two such classes of fluids are Rivlin-Ericksen’s and Walter’s (model B’) fluids. Rivlin-Ericksen (1955) has proposed a theoretical model for such one class of elastico-viscous fluids. Sharma and kumar (1996) have studied the effect of rotation on thermal instability in Rivlin-Ericksen elastico-viscous fluid and found that rotation has a stabilizing effect and introduces oscillatory modes in the system. Kumar et al. (2006) considered effect of rotation and magnetic field on Rivlin-Ericksen elastico-viscous fluid and found that rotation has stabilizing effect; where as magnetic field has both stabilizing and destabilizing effects. A layer of such fluid heated from below or under the action of magnetic field or rotation or both may find applications in geophysics, interior of the Earth, Oceanography, and the atmospheric physics. With the growing importance of non-Newtonian fluids in modern technology and industries, the investigations on such fluids are desirable. In all above studies, the medium has been considered to be non-porous with free boundaries only, in general. In recent years, the investigation of flow of fluids through porous media has become an important topic due to the recovery of crude oil from the pores of reservoir rocks. When a fluid permeates a porous material, the gross effect is represented by the Darcy’s law. As a result of this macroscopic law, the usual viscous term in the equation of Rivlin-Ericksen fluid motion is replaced by the resistance term, where and are the viscosity and viscoelasticity of the Rivlin-Ericksen fluid, is the medium permeability and is the Darcian (filter) velocity of the fluid. The problem of thermosolutal convection in fluids in a porous medium is of great importance in geophysics, soil sciences, ground water hydrology and astrophysics. Generally, it is accepted that comets consist of a dusty ‘snowball’ of a mixture of frozen gases which, in the process of their journey, changes from solid to gas and vice-versa. The physical properties of the comets, meteorites and interplanetary dust strongly suggest the importance of non-Newtonian fluids in chemical technology, industry and geophysical fluid dynamics. Thermal convection in porous medium is also of interest in geophysical system, electrochemistry and metallurgy. A comprehensive review of the literature concerning thermal convection in a fluid-saturated porous medium may be found in the book by Nield and Bejan (1992). Pellow and South well (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 in particular, for Rivlin-Ericksen viscoelastic fluid configurations. Banyal (2012) have characterized the oscillatory motions in Rivlin-Ericksen fluid. Keeping in mind the importance of non-Newtonian fluids, as stated above, this article attempts to study Rivlin-Ericksen viscoelastic of Veronis and Stern type configuration in the presence of uniform vertical magnetic field in a porous medium, and it has been established that the onset of instability in a Rivlin-Ericksen viscoelastic fluid heated from below in a porous medium Veronis type configuration, cannot manifest itself as oscillatory motions of growing amplitude if the Thermosolutal Rayliegh number, the Chandrasekhar number Q, the magnetic Prandtl number, the thermosolutal Prandtl number, the medium permeability, the porosity and the viscoelasticity parameter F satisfy the inequality, for all wave numbers and for any arbitrary combination of free and rigid boundaries at the top and bottom of the fluid. A similar characterization theorem is also proved for Stern type of configuration in the presence of uniform vertical magnetic field in a porous medium, for all wave numbers and for any arbitrary combination of free and rigid boundaries at the top and bottom of the fluid.

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Received on 10.11.2016 Modified on 24.11.2016

DOI: 10.5958/2349-2988.2017.00015.8

Research J. Science and Tech. 2017; 9(1):101-110.