IJCRR - 4(15), August, 2012
Pages: 12-20
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A SEMI-CIRCLE THEOREM IN RIVLIN-ERICKSEN VISCOELASTIC FLUID IN THE PRESENCE OF MAGNETIC FIELD
Author: Ajaib S. Banyal
Category: General Sciences
Abstract:A layer of Rivlin-Ericksen viscoelastic fluid heated from below is considered in the presence of uniform vertical magnetic field. Following the linearized stability theory and normal mode analysis, the paper through mathematical analysis of the governing equations of Rivlin-Ericksen viscoelastic fluid convection with a uniform vertical magnetic field, for any combination of perfectly conducting free and rigid boundaries of infinite horizontal extension at the top and bottom of the fluid, established 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 r i ? ? -plane, where Q is the Chandrasekhar number and F is the viscoelastic parameter of the Rivlin-Ericksen fluid, which prescribes the upper limits to the complex growth rate of arbitrary oscillatory motions of growing amplitude in the couple-stress fluid heated from below in the presence of uniform vertical magnetic field. The result is important since the result hold for all wave numbers and the exact solutions of the problem investigated in closed form, are not obtainable for any arbitrary combinations of perfectly conducting dynamically free and rigid boundaries.
Keywords: Thermal convection; Rivlin-Ericksen Fluid; Magnetic field; PES; Rayleigh number; Chandrasekhar number.
Full Text:
INTRODUCTION
Stability of a dynamical system is closest to real life, in the sense that realization of a dynamical system depends upon its stability. 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, and has been investigated by several authors (e.g., Bénard ?5? , Rayleigh ?15? , Jeffreys ?11? ) under different conditions. 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 ?8? . 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. Bhatia and Steiner ?6? 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. The thermal instability of a Maxwell fluid in hydromagnetics has been studied by Bhatia and Steiner ?7? . They have found that the magnetic field stabilizes a viscoelastic (Maxwell) fluid just as the Newtonian fluid. Sharma ?17? 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. In another study Sharma ?18? has studied the stability of a layer of an electrically conducting Oldroyd fluid ?13? in the presence of magnetic field and has found that the magnetic field has a stabilizing influence.
There are many elastic-viscous fluids that cannot be characterized by Maxwell‘s constitutive relations or Oldroyd‘s ?13? constitutive relations. Two such classes of fluids are Rivlin-Ericksen‘s and Walter‘s (model B‘) fluids. RivlinEricksen ?9? have proposed a theoretical model for such one class of elastic-viscous fluids. Sharma and kumar ?19? have studied the effect of rotation on thermal instability in Rivlin-Ericksen elasticoviscous fluid and found that rotation has a stabilizing effect and introduces oscillatory modes in the system. Kumar et al. ?14? considered effect of rotation and magnetic field, with free boundaries only, on Rivlin-Ericksen elasticoviscous 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.
Pellow and Southwell ?14? proved the validity of PES for the classical Rayleigh-Bénard convection problem. Banerjee et al ?1? 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 ?2? established a criterion on characterization of nonoscillatory motions in hydrodynamics which was further extended by Gupta et al. ?10? . However no such result existed for non-Newtonian fluid configurations, in general and for Rivlin-Ericksen viscoelastic fluid configurations, in particular. Banyal ?4? have characterized the non-oscillatory motions in couple-stress fluid.
Keeping in mind the importance of nonNewtonian fluids and magnetic field, as stated above, 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 RivlinEricksen fluid heated from below, in the presence of uniform vertical magnetic field, opposite to force field of gravity, when the bounding surfaces are of infinite horizontal extension, at the top and bottom of the fluid and are perfectly conducting with any combination of dynamically free and rigid boundaries. The result is important since the exact solutions of the problem investigated in closed form, are not obtainable, for any arbitrary combination of perfectly conducting dynamically free and rigid boundaries
Where the suffix zero refer to the values at the reference level z = 0. Here g? ?g? ? 0,0, is acceleration due to gravity and ? is the coefficient of thermal expansion. In writing the equation (1), we made use of the Boussinesq approximation, which states that the density variations are ignored in all terms in the equation of motion except the external force term. The magnetic permeability ? e , thermal diffusivity ? , and electrical resistivity ? , are all assumed to be constant. The initial state is one in which the velocity, density, pressure, and temperature at any point in the fluid are, respectively, given by
The essential content of the theorem, from the point of view of linear stability theory is that for the configuration of Rivlin-Ericksen viscoelastic fluid of infinite horizontal extension heated form below, having top and bottom bounding surfaces of infinite horizontal extension, at the top and bottom of the fluid and are perfectly conducting with any arbitrary combination of dynamically free and rigid boundaries, in the presence of uniform vertical magnetic field parallel to the force field of gravity, the complex growth rate of an arbitrary oscillatory motions of growing amplitude, lies inside a semi-circle in the right half of the ? r ? i - plane whose centre is at the origin
the Chandrasekhar number and F is the viscoelasticity parameter of the Rivlin-Ericksen fluid. The result is important since the exact solutions of the problem investigated in closed form, are not obtainable, for any arbitrary combinations of perfectly conducting dynamically free and rigid boundaries
ACKNOWLEDGEMENT
Author acknowledges the immense help received from the scholars whose articles are cited and included in references of this manuscript. The authors are also grateful to authors / editors /publishers of all those articles, journals and books from where the literature for this article has been reviewed and discussed. The author is highly thankful to the referees for their very constructive, valuable suggestions and useful technical comments, which led to a significant improvement of the paper.
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