INTRODUCTION

Bounds for the Complex Growth Rate in Rivlin-Ericksen Viscoelastic Fluid in the Presence of Rotation in a Porous Medium

Daleep K. Sharma¹*, Ajaib S. Banyal²

¹Department of Mathematics, G. C.Arki, Distt. Solan (HP), INDIA

²Department of Mathematics, Govt. College Nadaun, Dist. Hamirpur, (HP) INDIA 177033

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

ABSTRACT:

The thermal instability of a Rivlin-Ericksen viscoelastic fluid acted upon by uniform vertical rotation and heated from below in a porous medium is investigated. 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 rotation, for the case of rigid boundaries shows that the complex growth rate of oscillatory perturbations, neutral or unstable for all wave numbers, must lie inside the right half of the semi-circle

KEY WORDS: Thermal convection; Rivlin-Ericksen Fluid; Rotation; PES; Rayleigh number; Taylor number.

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

1. 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 and 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 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. There is growing importance of non-Newtonian fluids in geophysical fluid dynamics, chemical technology and petroleum industry. Bhatia and Steiner 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. In another study Sharma has studied the stability of a layer of an electrically conducting Oldroyd fluid in the presence of magnetic field and has found that the magnetic field has a stabilizing influence. There are many elastico-viscous fluids that cannot be characterized by Maxwell’s constitutive relations or Oldroyd’s constitutive relations. Two such classes of fluids are Rivlin-Ericksen’s and Walter’s (model B’) fluids. Rivlin-Ericksen has proposed a theoretical model for such one class of elastico-viscous fluids. Kumar et al. 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 non-Newtonian fluids in modern technology and industries, the investigations on such fluids are desirable.

6. REFERENCES:

1. Chandrasekhar, S. Hydrodynamic and Hydromagnetic Stability, 1981, Dover Publication, New York.

2. Bhatia, P.K. and Steiner, J.M., Convective instability in a rotating viscoelastic fluid layer, Zeitschrift fur Angewandte Mathematik and Mechanik 52 (1972), 321-327.

3. Sharma, R.C., Thermal instability in a viscoelastic fluid in hydromagnetics, Acta Physica Hungarica 38 (1975), 293-298.

4. Oldroyd, J.G., Non-Newtonian effects in steady motion of some idealized elastic-viscous liquids, Proceedings of the Royal Society of London A245 (1958), 278-297.

5. Rivlin, R.S. and Ericksen, J.L., Stress deformation relations for isotropic materials, J. Rat. Mech. Anal. 4 (1955), 323.

6. Kumar, P., Mohan, H. and Lal, R., Effect of magnetic field on thermal instability of a rotating Rivlin-Ericksen viscoelastic fluid, Int. J. of Maths. Math. Scs., Vol-2006 article ID 28042, pp. 1-10.

7. Nield D. A. and Bejan, A., Convection in porous medium, springer, 1992.

8. Sharma, R.C., Sunil and Pal, M., Thermosolutal convection in Rivlin-Ericksen rotating fluid in porous medium in hydromagnetics, indian J. pure appl. Math., 32(1) (2001), pp. 143-156.

9. 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.

10. 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

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

12. 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.

13. 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.

Received on 16.11.2016 Modified on 24.11.2016

DOI: 10.5958/2349-2988.2017.00005.5

Research J. Science and Tech. 2017; 9(1):29-34.