IJCRR

IJCRR - 4(22), November, 2012

Pages: 35-46

Date of Publication: 24-Nov-2012

A CHARACTERIZATION OF THERMOSOLUTAL INSTABILITY IN RIVLIN-ERICKSEN ROTATING FLUID IN A POROUS MEDIUM

Author: Ajaib S. Banyal

Category: General Sciences

Abstract:Thermosolutal instability of Veronis (1965) type in Rivlin-Ericksen viscoelastic fluid in the presence of uniform vertical rotation in a porous medium is considered. The paper 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 a porous medium, are necessarily non-oscillatory, in the regime , where s R is the Thermosolutal Rayliegh number, A T is the Taylor number, 2 p is the magnetic Prandtl number, 3 p is the hermosolutal Prandtl number, l P 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 (1960) type of configuration.

Keywords: Thermal convection; Rivlin-Ericksen Fluid; Rotation; Rayleigh number; Taylor number.

Full Text:

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, and has been investigated by several authors (e.g., Bénard ?4?, Rayleigh ?13? , Jeffreys ?8? ) 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 ?6? in his celebrated monograph. 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. 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 ?19? . 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 ?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. Sharma ?16? 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 ?11? constitutive relations. Two such classes of fluids are Rivlin-Ericksen’s and Walter’s (model B’) fluids. Rivlin-Ericksen ?14? has proposed a theoretical model for such one class of elasticoviscous fluids. Sharma and kumar ?17? 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. ?9? 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 nonNewtonian 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 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? q t ' μ μ k 1 1 , where ? and ' ? are the viscosity and viscoelasticity of the RivlinEricksen fluid, 1 k is the medium permeability and q 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 nonNewtonian 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 ?10?. Pellow and Southwell ?12? proved the validity of PES for the classical Rayleigh-Bénard convection problem. Banerjee et al ?2? 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 igid or free boundaries and, Banerjee and Banerjee ?1? established a criterion on characterization of non-oscillatory motions in hydrodynamics which was further extended by Gupta et al ?7? . However no such result existed for non-Newtonian fluid configurations in general and in particular, for Rivlin-Ericksen viscoelastic fluid configurations. Banyal ?3? have characterized the oscillatory motions in RivlinEricksen fluid in the presence of magnetic field. Keeping in mind the importance of nonNewtonian fluids, as stated above, this article attempts to study Rivlin-Ericksen viscoelastic of Veronis and Stern type configuration in the presence of uniform vertical rotation 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 Rs , the Taylor number TA , the magnetic Prandtl number p2 , the thermosolutal Prandtl number 3 p , the medium permeability Pl , the porosity ? and the viscoelasticity parameter F satisfy the inequality 1 2 4 3 ' ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? l A s l l T P R E p P F P ? ? ? , 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, for all wave numbers and for any arbitrary combination of free and rigid boundaries at the top and bottom of the fluid. FORMULATION OF THE PROBLEM AND PERTURBATION EQUATIONS Here we Consider an infinite, horizontal, incompressible Rivlin-Ericksen viscoelastic fluid layer, of thickness d, heated from below so that, the temperature, density and solute concentrations at the bottom surface z = 0 are T0 , ? 0 and C0 at the upper surface z = d are Td , ? d and Cd respectively, and that a uniform adverse temperature gradient ? ? ? ? ? ? ? ? ? dz dT ? and a uniform solute gradient ? ? ? ? ? ? ? ? ? dz ' dC ? is maintained. The gravity field g? ?g? ? 0,0, and uniform vertical rotation ?? ?? ? 0,0, pervade on the system. This fluid layer is assumed to be flowing through an isotropic and homogeneous porous medium of porosity ? and medium permeability 1 k . Let p , ? , T, C ,? , ' ? , g and q?u,v,w? ? denote respectively the fluid pressure, fluid density temperature, solute concentration, thermal coefficient of expansion, an analogous solvent coefficient of expansion, gravitational acceleration and filter velocity of the fluid. Then the momentum balance, mass balance, and energy balance equation governing the flow of Rivlin-Ericksen fluid in the presence of uniform vertical vertical rotation (Rivlin and Ericksen ?14? ; Chandrasekhar ?6? and Sharma et al ?18? ) are given by

constant analogous to E but corresponding to solute rather than heat, while ? s , s c and ? 0 , i c , stands for the density and heat capacity of the solid (porous matrix) material and the fluid, respectively, ? is the medium porosity and r(x, y,z) ? . The equation of state is ?1 ? ? ( )? 0 ' ? ? ?0 ?? T ?T0 ?? C ?C , (5) Where the suffix zero refer to the values at the reference level z = 0. 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 kinematic viscosity ? , kinematic viscoelasticity ' ? , thermal diffusivity ? , the solute diffusivity ' ? and the coefficient of thermal expansion ? are all assumed to be constants. The steady state solution is ? ?0,0,0? ? q , (1 ) ' ' 0 ? ? ? ???z ?? ? z , T0 T ? ??z ? , 0 ' C ? ?? z ? C , (6) Here we use the linearized stability theory and the normal mode analysis method. Consider a small perturbations on the steady state solution, and let ?? ,?p , ? , ? and q?u,v,w? ? denote respectively the perturbations in density ? , pressure p, temperature T, solute concentration C and velocity (0,0,0) ? q . The change in density ?? , caused mainly by the perturbation ? and ? in temperature and concentration, is given by NORMAL MODE ANALYSIS Analyzing the disturbances into two-dimensional waves, and considering disturbances characterized by a particular wave number, we assume that the Perturbation quantities are of the form

CONCLUSIONS
Theorem 1 mathematically established that the onset of instability in a thermosolutal RivlinEricksen viscoelastic fluid configuration of Veronis (1965) type in the presence of uniform vertical rotation in a porous medium, cannot manifest itself as oscillatory motions of growing amplitude if the Thermosolutal Rayliegh number Rs , the Taylor number TA , the magnetic Prandtl number p2 , the thermosolutal Prandtl number 3 p , the medium permeability Pl , the porosity ? and the viscoelasticity parameter F satisfy
 

any arbitrary combination of free and rigid boundaries at the top and bottom of the fluid The essential content of the theorem 1, from the point of view of linear stability theory is that for the thermosolutal configuration of Veronis (1965) type of Rivlin-Ericksen viscoelastic fluid of infinite horizontal extension in the presence of uniform vertical rotation in a porous medium, for any arbitrary combination of free and rigid boundaries at the top and bottom of the fluid, an arbitrary neutral or unstable modes of the system are definitely non-oscillatory in character

particular PES is valid. The similar conclusions can be drawn for the thermosolutal configuration of Stern (1960) type of Rivlin-Ericksen viscoelastic fluid of infinite horizontal extension in the presence of uniform vertical rotation in a porous medium, for any arbitrary combination of free and rigid boundaries at the top and bottom of the fluid from Theorem 2.

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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