INTRODUCTION:
Right from the conceptualizations of turbulence, instability of fluid flows is being regarded at its root. A detailed account of the theoretical and experimental study of the onset of thermal instability (Bénard Convection) in Newtonian fluids, under varying assumptions of hydrodynamics and hydromagnetics, has been given by Chandrasekhar1 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 Joseph2. 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. Stommel and Fedorov3 and Linden4 have remarked that the length scales characteristic of double-diffusive convecting layers in the ocean may be sufficiently large so 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 porous medium, and this distortion plays an important role in the extraction of energy in geothermal regions. The forced convection in a fluid saturated porous medium channel has been studied by Nield et al5. An extensive and updated account of convection in porous media has been given by Nield and Bejan6.
The effect of a magnetic field on the stability of such a flow is of interest in geophysics, particularly in the study of the earth’s core, where the earth’s mantle, which consists of conducting fluid, behaves like a porous medium that can become conductively unstable as result of differential diffusion. Another application of the results of flow through a porous medium in the presence of magnetic field is in the study of the stability of convective geothermal flow. A good account of the effect of rotation and magnetic field on the layer of fluid heated from below has been given in a treatise by Chandrasekhar1.
MHD finds vital applications in MHD generators, MHD flow-meters and pumps for pumping liquid metals in metallurgy, geophysics, MHD couplers and bearings, and physiological processes such magnetic therapy. With the growing importance of non-Newtonian fluids in modern technology and industries, investigations of such fluids are desirable. The presence of small amounts of additives in a lubricant can improve bearing performance by increasing the lubricant viscosity and thus producing an increase in the load capacity. These additives in a lubricant also reduce the coefficient of friction and increase the temperature range in which the bearing can operate.
Darcy’s law governs the
flow of a Newtonian fluid through an isotropic and homogeneous porous medium.
However, to be mathematically compatible and physically consistent with the Navier-Stokes
equations, Brinkman7 heuristically proposed the introduction of the
term
, (now known as Brinkman
term) in addition to the Darcian term 
. But the main effect is
through the Darcian term; Brinkman term contributes very little effect for flow
through a porous medium. Therefore, Darcy’s law is proposed heuristically to
govern the flow of this non-Newtonian couple-stress fluid through porous
medium. A number of theories of the micro continuum have been postulated and
applied (Stokes8; Lai et al9; Walicka10). The
theory due to Stokes8 allows for polar effects such as the presence
of couple stresses and body couples. Stokes’s8 theory has been
applied to the study of some simple lubrication problems (see e.g. Sinha et al11;
Bujurke and Jayaraman12; Lin13). According to the theory
of Stokes8, couple-stresses are found to appear in noticeable
magnitudes in fluids with very large molecules. Since the long chain hyaluronic
acid molecules are found as additives in synovial fluid, Walicki and Walicka14
modeled synovial fluid as couple stress fluid in human joints. The study is
motivated by a model of synovial fluid. The synovial fluid is natural lubricant
of joints of the vertebrates. The detailed description of the joints
lubrication has very important practical implications; practically all diseases
of joints are caused by or connected with a malfunction of the lubrication. The
external efficiency of the physiological joint lubrication is caused by more
mechanisms. The synovial fluid is caused by the content of the hyaluronic acid,
a fluid of high viscosity, near to a gel. A layer of such fluid heated from
below in a porous medium under the action of magnetic field and rotation may
find applications in physiological processes. MHD finds applications in
physiological processes such as magnetic therapy; rotation and heating may find
applications in physiotherapy. The use of magnetic field is being made for the
clinical purposes in detection and cure of certain diseases with the help of
magnetic field devices.
Sharma and Thakur15 have studied the thermal convection in couple-stress fluid in porous medium in hydromagnetics. Sharma and Sharma16 have studied the couple-stress fluid heated from below in porous medium. Kumar and Kumar17 have studied the combined effect of dust particles, magnetic field and rotation on couple-stress fluid heated from below and for the case of stationary convection, found that dust particles have destabilizing effect on the system, where as the rotation is found to have stabilizing effect on the system, however couple-stress and magnetic field are found to have both stabilizing and destabilizing effects under certain conditions. Sunil et al.18 have studied the global stability for thermal convection in a couple-stress fluid heated from below and found couple-stress fluids are thermally more stable than the ordinary viscous fluids.
Pellow and Southwell19 proved the validity of PES for the classical Rayleigh-Bénard convection problem. Banerjee et al20 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 Banerjee21 established a criterion on characterization of non-oscillatory motions in hydrodynamics which was further extended by Gupta et al.22. However no such result existed for non-Newtonian fluid configurations, in general and for couple-stress fluid configurations, in particular. Banyal23 have characterized the non-oscillatory motions in couple-stress fluid. Banyal and Singh24 have found the bounds for complex growth rate in the presence of uniform vertical rotation and Banyal and Khanna25 in the presence of uniform vertical magnetic field.
Keeping in mind the importance of non-Newtonian fluids, 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 couple-stress fluid in porous medium heated from below in the presence of uniform vertical rotation opposite to force field of gravity, when the bounding surfaces are of infinite horizontal extension, at the top and bottom of the fluid are rigid.
6. REFERENCES:
1. Chandrasekhar, S., Hydrodynamic and Hydromagnetic Stability, Dover Publications, New York 1981.
2. Joseph, D.D., Stability of fluid motions, vol. II, Springer-Verlag, berlin, 1976.
3. Stommel, H., and Fedorov, K.N., Small scale structure in temperature and salinity near Timor and Mindano, Tellus 1967, vol. 19, pp. 306-325
4. Linden, P.F.,Salt fingers in a steady shear flow, Geophys. Fluid Dynamics, 1974, vol.6, pp.1-27.
5. Nield, D.A., and Junqueira, S.L.M. and Lage,J.L., Fprced convection in a fluid saturated porous medium ahannel with isothermal or isoflux boundaries, J. fluid Mech. 1996, vol.322, pp. 201-214.
6. Nield, D.A., and Bejan, A., Convection in porous medium, Springer and Verlag, Newyark, 1999.
7. Brinkman, H.C., problems of fluid flow through swarms of particles and through macromolecules in solution, research(London) 1949,Vol. 2,p.190.
8. Stokes, V.K., Couple-stress in fluids, Phys. Fluids, 1966, Vol. 9, pp.1709-15.
9. Lai, W.M., Kuei,S.C., and Mow,V.C., Rheological equtions for synovial fluids, J. of Biomemechanical eng.1978, vol.100, pp. 169-186.
10. Walicka, A., Micropolar flow in a slot between rotating surfaces of revolution, Zielona Gora, TU Press, 1994.
11. Sinha, P., Singh, C., and Prasad, K.R., Couple-stresses in journal bearing lubricantsand the ffect of convection, Wear, vol.67, pp. 15-24, 1981.
12. Bujurke, N.M., and Jayaraman, G., The influence of couple-stresses in squeeze films Int. J. Mech. Sci. 1982, Vol. 24, pp.369-376.
13. Lin, J.R., Couple-stress effect on the squeeze film characteristics of hemispherical bearing with reference tosynovial joints, Appl. Mech.engg.1996, vol. 1, pp.317-332.
14. Walicki, E. and Walicka, A., Inertial effect in the squeeze film of couple-stress fluids in biological bearings, Int. J. Appl. Mech. Engg. 1999, Vol. 4, pp. 363-373.
15. Sharma, R.C. and Thakur, K. D., Couple stress-fluids heated from below in hydromagnetics, Czech. J. Phys. 2000, Vol. 50, pp. 753-58
16. Sharma, R.C. and Sharma S., On couple-stress fluid heated from below in porous medium, Indian J. Phys 2001, Vol. 75B, pp.59-61.
17. Kumar, V. and Kumar, S., On a couple-stress fluid heated from below in hydromagnetics, Appl. Appl. Math.2011, Vol. 05(10), pp. 1529-1542
18. Sunil, Devi, R. and Mahajan, A., Global stability for thermal convection in a couple stress- fluid, Int. comm.. Heat and Mass Transfer 2011, 38,pp. 938-942.
19. Pellow, A., and Southwell, R.V., On the maintained convective motion in a fluid heated from below. Proc. Roy. Soc. London A, 1940, 176, 312-43.
20. Banerjee, M.B., Katoch, D.C., Dube,G.S. and Banerjee, K., Bounds for growth rate of perturbation in thermohaline convection. Proc. R. Soc. A, 1981,378, 301-04
21. Banerjee, M. B., and Banerjee, B., A characterization of non-oscillatory motions in magnetohydronamics. Ind. J. Pure & Appl Maths., 1984, 15(4): 377-382
22. Gupta, J.R., Sood, S.K., and Bhardwaj, U.D., On the characterization of nonoscillatory motions in rotatory hydromagnetic thermohaline convection, Indian J. pure appl. Math. 1986,17(1) pp 100-107.
23. Banyal, A.S, The necessary condition for the onset of stationary convection in couple- stress fluid, Int. J. of Fluid Mech. Research, Vol. 38, No.5, 2011, pp. 450-457.
24. Banyal, A. S., and Singh, K., On The Region of Complex Growth Rate in Couple-Stress Fluid in the Presence of Rotation, J. of Pure Appld and Ind. Physics, Vol. 2(1), 2011, pp.75-83.
25. Banyal, A. S., and Khanna, M., Upper Limits to the Complex Growth Rate in Couple-Stress Fluid in the Presence of Magnetic Field, J. Comp. & Math. Sci., Vol. 3(2), 2012, pp.237-247.
26. Schultz, M.H., Spline Analysis, Prentice Hall, Englewood Cliffs, New Jersy, 1973.
27. Banerjee, M.B., Gupta, J.R. and Prakash, J. 1992, On thermohaline convection of Veronis type, J. Math. Anal. Appl., Vol.179, No. 2, pp. 327-334.
28. Banyal, A. S., Bounds for the Growth Rate of a Perturbation in a Couple-Stress Fluid in the Presence of Rotation in a Porous Medium, Int. J. Mathematical Archives, 3(5), 2012, pp.1-9.
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Received on 31.12.2014 Accepted on 28.02.2015 © EnggResearch.net All Right Reserved Int. J. Tech. 5(1): Jan.-June 2015; Page 41-49
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