Onset of Thermosolutal Convection in Couple-Stress Fluid in a Porous Media in the Presence of Magnetic Field

Ajaib S. Banyal¹, Kamal Singh²

¹Department of Mathematics, Sidharth Govt. College Nadaun, Distt. Hamirpur, (HP) INDIA 177033

²Department of Mathematics, Govt. College Indora, Distt. Kangra, (HP) INDIA 176401

*Corresponding Author E-mail: ajaibbanyal@rediffmail.com, singh_kamal1979@rediffmail.com

ABSTRACT:

Thermosolutal instability of Veronis (1965) type in a couple-stress 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 proved analytically that all non-decaying slow motions starting from rest, in a couple-stress fluid of infinite horizontal extension and finite vertical depth in a porous medium, which is acted upon by uniform vertical magnetic field opposite to force field of gravity and a constant vertical adverse temperature gradient, are necessarily non-oscillatory, in the regime established, the result is important since the exact solutions of the problem investigated are not obtainable in closed form, for any arbitrary combination of free and rigid boundaries. A similar characterization theorem is also established for Stern (1960) type of configuration.

KEYWORDS: Thermosolutal convection; couple-stress Fluid; Magnetic Field; Rayleigh number; Chandrasekhar number.

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

1. INTRODUCTION:

A detailed account of the theoretical and experimental study of the onset of thermal instability in Newtonian fluids, under varying assumptions of hydrodynamics and hydromagnetics, has been given by Chandrasekhar and the Boussinesq approximation has been used throughout, which states that the density changes are disregarded in all other terms in the equation of motion, except in the external force term. The formation and derivation of the basic equations of a layer of fluid heated from below in a porous medium, using the Boussinesq approximation, has been given in a treatise by Joseph. When a fluid permeates through an isotropic and homogeneous porous medium, the gross effect is represented by Darcy’s law. The study of layer of fluid heated from below in porous media is motivated both theoretically and by its practical applications in engineering. Among the applications in engineering disciplines one can name the food processing industry, the chemical processing industry, solidification, and the centrifugal casting of metals. The development of geothermal power resources has increased general interest in the properties of convection in a porous medium. 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. Double-diffusive convection problems arise in oceanography (salt fingers occur in the ocean when hot saline water overlies cooler fresher water which believed to play an important role in the mixing of properties in several regions of the ocean), limnology and engineering. The migration of moisture in fibrous insulation, bio/chemical contaminants transport in environment, underground disposal of nuclear wastes, magmas, groundwater, high quality crystal production and production of pure medication are some examples where double-diffusive convection is involved. Examples of particular interest are provided by ponds built to trap solar heat Tabor and Matz and some Antarctic lakes Shirtcliffe. The physics is quite similar in 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 a solute gradient and free boundaries. The flow through porous media is of considerable interest for petroleum engineers, for geophysical fluid dynamists and has importance in chemical technology and industry. An example in the geophysical context is the recovery of crude oil from the pores of reservoir rocks. Among the application in engineering disciplines one can find the food processing industry, chemical processing industry, solidification and centrifugal casting of metals. Such flows has shown their great importance in petroleum engineering to study the movement of natural gas, oil and water through the oil reservoirs; in chemical engineering for filtration and purification processes and in the field of agriculture engineering to study the underground water resources, seepage of water in river beds. The problem of thermosolutal convection in fluids in a porous medium is of importance in geophysics, soil sciences, ground water hydrology and astrophysics. The study of thermosolutal convection in fluid saturated porous media has diverse practical applications, including that related to the materials processing technology, in particular, the melting and solidification of binary alloys. The development of geothermal power resources has increased general interest in the properties of convection in porous media. The scientific importance of the field has also increased because hydrothermal circulation is the dominant heat-transfer mechanism in young oceanic crust Lister. Generally it is accepted that comets consists 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 comets, meteorites and interplanetary dust strongly suggest the importance of porosity in the astrophysical context Mc Donnel. The effect of a magnetic field on the stability of such a flow is of interest in geophysics, particularly in the study of Earth’s core where the Earth’s mantle, which consists of conducting fluid, behaves like a porous medium which can become convectively unstable as a result of differential diffusion. The other application of the results of flow through a porous medium in the presence of a magnetic field is in the study of the stability of a convective flow in the geothermal region. Also the magnetic field in double-diffusive convection has its importance in the fields of engineering, for example, MHD generators and astrophysics particularly in explaining the properties of large stars with a helium rich core. Stommel and Fedorov and Linden have remarked that the length scales characteristics of double-diffusive convective layers in the ocean may be sufficiently large that the Earth’s rotation might be important in their formation. Moreover, the rotation of the Earth distorts the boundaries of a hexagonal convection cell in a fluid through a porous medium and the distortion plays an important role in the extraction of energy in the geothermal regions. Brakke explained a double - diffusive instability that occurs when a solution of a slowly diffusing protein is layered over a denser solution of more rapidly diffusing sucrose. Nason et al. found that this instability, which is deleterious to certain biochemical separations, can be suppressed by rotation in the ultracentrifuge. The theory of couple-stress fluid has been formulated by Stokes. One of the applications of couple-stress fluid is its use to the study of the mechanisms of lubrications of synovial joints, which has become the object of scientific research. A human joint is a dynamically loaded bearing which has articular cartilage as the bearing and synovial fluid as the lubricant. When a fluid film is generated, squeeze - film action is capable of providing considerable protection to the cartilage surface. The shoulder, ankle, knee and hip joints are the loaded – bearing synovial joints of the human body and these joints have a low friction coefficient and negligible wear. Normal synovial fluid is a viscous, non-Newtonian fluid and is clear or yellowish. According to the theory of Stokes, couple-stresses appear in noticeable magnitudes in fluids with very large molecules. Since the long chain hyaluronic acid molecules are found as additives in synovial fluids, Walicki and Walicka modelled the synovial fluid as a couple-stress fluid. The synovial fluid is the natural lubricant of joints of the vertebrates. The detailed description of the joint lubrication has very important practical implications. Practically all diseases of joints are caused by or connected with malfunction of the lubrication. The efficiency of the physiological joint lubrication is caused by several mechanisms. The synovial fluid is due to its content of the hyaluronic acid, a fluid of high viscosity, near to gel. Goel et al. have studied the hydro magnetic stability of an unbounded couple-stress binary fluid mixture under rotation with vertical temperature and concentration gradients. Sharma et al. have considered a couple - stress fluid with suspended particles heated from below. In another study, Sunil et al. have considered a couple- stress fluid heated from below in a porous medium in the presence of a magnetic field and rotation. Kumar et al. have considered the thermal instability of a layer of couple-stress fluid acted on by a uniform rotation, and have found that for stationary convection the rotation has a stabilizing effect whereas couple-stress has both stabilizing and destabilizing effects.

REFERENCES

1. Chandresekhar S (1961) Hydrodynamic and Hydro magnetic Stability, Clarendon Press, Oxford, UK.

2. Joseph DD (1976) Stability of fluid motions. vol. II, Springer-Verlag, berlin.

3. Veronis G (1965) On finite amplitude instability in thermohaline convection. J. Marine Res., 23: 1-17.

4. Tabor H, Matz R (1965) Solar pond project. Solar Energy, 9: 177-182.

5. Shirtcliffe TGL (1967), Thermosolutal convection: observation od an over stable mode. Nature (London), 213: 489-490.

6. Lister PE (1972) On the thermal balance of a mid-ocean ridge. Geophys. J. Roy. Astr. Soc., 26: 515-535.

7. McDonnel JAM (1978) Cosmic Dust. Johan Wiley and Sons. Toronto, p. 330.

8. Stommel H, Fedorov KN (1967) Small scale structure in temperature and salinity near Timor and Mindano, Tellus, 19: 306.

9. Linden PF (1974) Salt fingers in a steady shear flow. Geophys. Fluid Dynamics, 6: 1.

10. Brakke MK (1955) Zone electrophoresis of dyes, proteins and viruses in density- gradient columns of sucrose solutions. Arch. Biochem. Biophys., 55: 175-190.

11. Nason P, Schumaker V, Halsalt B, Schwedes J (1969) Formation of a streaming convective disturbance which may occur at one gravity during preparation of samples for zone centrifugation, Biopolymers, 7: 241-249.

12. Stokes VK (1966) Couple–stresses in fluids, Phys. Fluids, 9: 1709-1715.

13. Walicki E, Walicka A (1999) Inertia effects in the squeeze film of a couple stress fluid in biological bearing, Appl. Mech. Engng., 4: 363-373.

14. Goel AK, Agarwal SC, Agarwal GS (1999) Hydro magnetic stability of an unbounded couple-stress binary fluid mixture having vertical temperature and concentration gradients with rotation, Indian J. Pure Appl. Math., 30: 991-1001.

15. Sharma RC, Sunil, Sharma YD, Chandel RS (2002) On Couple-stress fluid permeated with suspended particles heated from below, Arch. Mech. (Poland), 54: 287-298.

16. Sunil, Sharma RC, Pal M (2002) On couple – stress fluid heated from below in porous medium in presence of magnetic field and rotation. J. Porous Media, 5: 149-158.

17. Kumar P, Lal R, Sharma P (2004) Effect of rotation on thermal instability in couple-stress elastico-viscous fluid. Z. Naturforsch., 59a: 407-411.

18. Pellow A, and South well RV (1940) On the maintained convective motion in a fluid heated from below, Proc. Roy. Soc. London A, 176: 312-43.

19. Banerjee MB, Katoch DC, Dube GS, Banerjee K (1981) Bounds for growth rate of perturbation in thermohaline convection. Proc. R. Soc. A, 378: 301-304

20. Banerjee MB, Banerjee B (1984) A characterization of non-oscillatory motions in magnetohydronamics. Ind. J. Pure & Appl Maths., 15(4): 377-382

21. Gupta JR, Sood SK, Bhardwaj UD (1986) On the characterization of non-oscillatory motions in rotatory hydro magnetic thermohaline convection, Indian J. pure appl. Math., 17(1): 100-107.

22. Banyal AS (2011) the necessary condition for the onset of stationary convection in couple-stress fluid, Int. J. of Fluid Mech. Research, 38(5): 450-457.

23. Stern ME (1960) the salt fountain and thermohaline convection, Tellus 12: 172-175.

24. Schultz MH (1973) Spline Analysis, Prentice Hall, Englewood Cliffs, New Jersey.

25. Banerjee MB, Gupta JR, Prakash, J (1992) On thermohaline convection of Veronis type, J. Math. Anal. Appl., 179(2): 327-334.

Received on 21.11.2016 Modified on 28.11.2016

DOI: 10.5958/2349-2988.2017.00013.4

Research J. Science and Tech. 2017; 9(1):81-92.