Effect of Rotation on Triple- Diffusive Convection in Walters’ (Model B´) Fluid in Porous Medium

Suresh Chand*

Department of Mathematics, SCVB Govt. College Palampur, HP-176061 India

*Corresponding Author: suresh-9@hotmail.com

ABSTRACT:

The effect of rotation on triple- diffusive convection in Walters’ (Model B´) fluid in porous medium is considered in the presence of uniform vertical rotation. For the case of stationary convection, the stable solute gradients and rotation have stabilizing effect on the system, whereas the medium permeability has a destabilizing (or stabilizing) effect on the system under certain conditions. A linear stability analysis theory and normal mode analysis method have been carried out to study the onset convection. The kinematic viscoelasticity has no effect on the stationary convection. The solute gradients, rotation, porosity and kinematic viscoelasticity introduce oscillatory modes in the system, which were non-existent in their absence. The sufficient conditions for the non-existence of overstability are also obtained.

KEY WORDS: Triple- diffusive convection; Walters’ (Model B´) fluid; Thermal convection; Solute gradients; Vertical magnetic field; Rotation.

INTRODUCTION:

The theoretical and experimental results of the onset of thermal instability (Bénard convection) in a fluid layer under varying assumptions of hydrodynamics have been treated by Chandrasekhar [1981] in his celebrated monograph. 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 to the stellar case in that helium acts like salt in raising the density and in diffusing more slowly than heat. The conditions under which convective motions are important in stellar atmospheres are usually far removed from consideration of a single component fluid and rigid boundaries, and therefore it is desirable to consider a fluid acted on by solute gradients and free boundaries. The problem of the onset of thermal instability in the presence of solute gradients 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 double-diffusive convection problems also arise in oceanography, limnology and engineering. With the growing importance of non-Newtonian fluids in modern technology and industries, the investigations on such fluids are desirable. The Walters’ [1962] fluid (Model B ) is one such fluid. In another study Sharma and Kumar [1995] have studied the steady flow and heat transfer of Walters’ (Model B ) fluid through a porous pipe of uniform circular cross- section with small suction.

Sharma and Kumar [1997] have studied the stability of the plane interface separating two viscoelastic Walters’ (Model B ) fluid of uniform densities and found that for stable configuration, the system is stable or unstable under certain conditions. Sharma et al. [1998, 1999] have studied the thermosolutal instability of Walters’ (Model B ) fluid in porous medium. In many astrophysical situations, the effect of rotation on thermosolutal convection in porous medium is also important.

In recent years, the investigation of flow of fluids through the porous media has become an important topic. A great number of applications in Geophysics may be found in the book written by Philips [9].When the fluid permeates through a porous material, the gross effect is represented by the law. As a result of this macroscopic law, the usual viscous term in the equation of Walters’ (Model B ) fluid motion is replaced by the resistance term )q ], where and are the viscosity and viscoelasticity of the Walters’ fluid, is the medium permeability and q is the Darcian (filter) velocity of the fluid. The problem of the thermosolutal convection in fluids in porous medium is of great importance in Geophysics, Soil Sciences, ground water Hydrology and Astrophysics. Generally, it is accepted that comets consists of a dust “snowball” made of mixture of frozen gases which, in the process of their journey, changes from solid to gas and vice-versa. The physical properties of comets, meteorites and interplanetary dust strongly suggest the importance of porosity in astrophysical context Mc Donnel [1978]. Out of large published work in pure fluid, the thermosolutal convection in porous medium has received only attention, because of its various engineering applications. A comprehensive review of the literature concerning thermosolutal convection in a fluid-saturated porous medium may be found in the book written by Nield and Bejan [1992]. The thermal convection of Walters’ (Model B ) fluid has been studied by many authors [1999b, 2000]. A recent review of numerical techniques and their applications may be found in O’Sullivan et al; [2000]. Oldenburg and Pruess [1998] have developed a model for convection in a Darcy’s porous medium, where the mechanism involves temperature, NaCl, CaCl₂ and KCl. Solar ponds are a particularly promising means of harnessing energy from the Sun by preventing convective overturning in a thermohaline system by salting from below. But we also appreciate the work of Bhattacharyya and Abbas [1985] and Qin and Kaloni [1992] , they have considered the effect of rotation in angular momentum equation.

In the standard Bénard problem, the instability is driven by a density difference caused by a temperature difference between the upper and lower planes bounding the fluid. If the fluid, additionally has salt dissolved in it , then there are potentially two destabilizing sources for the density difference, the temperature field and salt field. The solution behavior in the double-diffusive convection problem is more interesting than that of the single component situation in so much as new instability phenomena may occur which is not present in the classical Bénard problem. When temperature and two or more component agencies, or three different salts, are present then the physical and mathematical situation becomes increasingly richer. Very interesting results in triply diffusive convection have been obtained by Pearlstein et al., [1989]. The results of Pearlstein et al., are remarkable. They demonstrate that for triple diffusive convection linear instability can occur in discrete sections of the Rayleigh number domain with the fluid being linearly stable in a region in between the linear instability ones. This is because for certain parameters the neutral curve has a finite isolated oscillatory instability curve lying below the usual unbounded stationary convection one. Straughan and Walker [1997] derive the equations for non-Boussinesq convection in a multi- component fluid and investigate the situation analogous to that of Pearlstein et al., but allowing for a density non linear in the temperature field. Lopez et al., [1990] derive the equivalent problem with fixed boundary conditions and show that the effect of the boundary conditions breaks the perfect symmetry. In reality the density of a fluid is never a linear function of temperature, and so the work of Straughan and Walker applies to the general situation where the equation of state is one of the density quadratic in temperature. This is important, since they find that departure from the linear Boussinesq equation of state changes the perfect symmetry of the heart shaped neutral curve of Pearlstein et al.

In view of the recent increase in the number of non iso-thermal situations, we intend to extend our work to the problem of thermal convection in Walters’ (Model B ) fluid on triple-diffusive convection in the presence of rotation in porous medium.

5. REFERENCE:

1. A.J. Pearlstein, (1989). The onset of convective instability in a triply diffusive fluid layer. Journal of Fluid Mechanics, 202, pp. 443-465.

2. B. D. Straughan, W. Walker, (1997). Multi component diffusion and penetrative convection. Fluid Dynamics Research, 1997, pp. 77- 89.

3. C. Oldenberg, K. Pruess , (1998) . Layered thermohaline convection in hypersaline geothermal systems .Transport in Porous Media, 33, pp. 29-63.

4. D.A. Nield, A. Bejan ,(1992). Convection in porous medium, Springer.

5. G. Veronis, (1965). Thermohaline convection in a layer of fluid heated from below. J. Marine Res., 23, pp. 1-17.

6. J. A. M. Mcdonnel, Cosmic dust, (1978). John Willey and Sons, Toronto.

7. K. Waltetrs, (1962). Quart. Jour. Mech. Appl. Math. , 15 , pp. 63-76.

8. Lopez Amalia , (1990). Effect of rigid boundaries on the onset of convective instability in a triply diffusive fluid layer R. Phys. Fluids, 2 ,pp. 897-902.

9. O.M. Philips, (1991). Flow and reaction in permeable rocks, Cambridge University Press, Cambridge.

10. O’sullivan, M., Pruess, K., Lippmann, M., (2000). Geothermics, 30, pp. 395-429.

11. P.R. Sharma, H. Kumar, (1995).The steady flow and heat transfer of Walters’ (Model B ) fluid through a porous pipe of uniform circular cross- section with small suction . Proc. Nat. Acad. Sci. India, 65(A) ,pp. 75-88.

12. R.C. Sharma, P. Kumar, (1997).The stability of the plane interface separating two viscoelastic Walters’ (Model B ) fluid of uniform densities. Chech. J. Phys., 47,pp. 197-204.

13. R. C. Sharma, Sunil, Suresh Chand, (1998).The thermosolutal instability of Walters’ (Model B ) fluid in porous medium. Chech. J. Phys., 48 ,pp. 1-7.

14. R. C. Sharma, Sunil, Suresh Chand, (1999). Thermosolutal instability of Walters’ (model B ) rotating fluid in porous medium. Arch. Mech., 51(2), pp. 181-191.

15. R. C. Sharma, Sunil, Suresh Chand, (2000). Thermosolutal convection in Walters’ (model B ) fluid in porous medium in Hydrodynamics. Studia Geotechnica Mechanicas, 3-4 ,pp. 03-14.

16. R.C. Sharma, Sunil and Suresh Chand, (1999b).The instability of streaming Walters’ viscoelastic fluid B' in porous medium, Czechoslovak Journal of Physics, 49(2), pp. 189-195.

17. S. Chandrashekhar, Hydrodynamic and Hydromagnetic stability, (1981). Dover Publication, New York.

18. S.P. Bhattacharyya, M. Abbas, (1985). The effect of rotation in angular momentum equation. Int. J. Eng. Sci., 23, pp.371- 374.

19. Y. Qin and P.N.Kaloni,(1992) A thermal instability problem in arotating microplar fluid, International Journal of Engineering Science, Vol.30 , pp.1117-1126.

Received on 06.01.2013 Accepted on 10.02.2013

Research J. Science and Tech 5(1): Jan.-Mar.2013 page 184-188