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Characterization of magnetorotatory thermohaline instability in porous medium: Darcy Brinkman model

Jyoti Prakash1, Kanu Vaid2
  1. Associate Professor, Department of Mathematics and Statistics, Himachal Pradesh University, Summer Hill, Shimla-171005, India
  2. Research Scholar, Department of Mathematics and Statistics, Himachal Pradesh University, Summer Hill, Shimla-171005, India
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Abstract

The present paper prescribes upper bounds for oscillatory motions of neutral or growing amplitude in magnetorotatory thermohaline configurations of Veronis (Veronis, G., J. Mar. Res., 23(1965)1) and Stern types (Stern, M.E., Tellus 12(1960)172) in porous medium(Darcy-Brinkman model) in such a way that also result in sufficient conditions of stability for an initially bottom-heavy as well as initially top-heavy configuration.

Keywords
Thermohaline instability, oscillatory motions, initially bottom-heavy configuration, initially top-heavy configuration, Porous medium.

I.INTRODUCTION
The convective phenomenon which is driven by the differential diffusion of two properties such as heat and salt is known as thermohaline convection or, more generally, double diffusive convection. The onset of motion in thermohaline convection is fundamentally different from Rayleigh – Benard convection and is of direct relevance to the fields of astrophysics, oceanography, limnology and chemical engineering etc. For a broad view of the subject one may be referred to Turner [1] and Brandt and Fernando [2]. Two fundamental configurations have been studied in the context of thermohaline convection problem, one by Veronis [3], wherein the temperature gradient is destabilizing and the concentration gradient is stabilizing and another by Stern [4] wherein the temperature gradient is stabilizing and the concentration gradient is destabilizing. The main results derived by Veronis and Stern for their respective problems are that instability might occur in the configuration through a stationary pattern of motions or oscillatory motions provided the destabilizing temperature gradient or the concentration gradient is sufficiently large but compatible with the conditions that the total density field is gravitationally stable. Thus thermohaline configurations of the Veronis and the Stern type can be classified into the following two classes:
1. The first class, in which thermohaline instability manifests itself when the total density field is initially bottom heavy, and
2. The second class, in which thermohaline instability manifests itself when the total density field is initially top heavy. Banerjee et al. [5] derived a characterization theorem for thermohaline convection of the Veronis type that disallow the existence of oscillatory motions of neutral or growing amplitude in an initially bottom heavy configuration for the certain parameter regime.
The problem of thermohaline instability in porous media has attracted considerable interest during the past few decades because of its wide range of applications including the ground water contamination, disposal of waste material, food processing, prediction of ground water movement in aquifers, the energy extraction process from the geothermal reservoirs, assessing the effectiveness of fibrous insulations etc.([6],[7]).The thermohaline instability problem in porous media has been extensively investigated and the growing volume of work devoted to this area is well documented by Nield and Bezan[6] and Vafai [8].Prakash and Vinod[9] have proved the nonexistence of nonoscillatory motions in thermohaline convection of Stern type in porous medium. In the early researches most of the researchers have studied double diffusive convection in porous medium by considering the Darcy flow model which is relevant to densely packed, low permeability porous medium. However, experiments conducted with several combinations of solids and fluids covering wide ranges of governing parameters indicate that most of the experimental data do not agree with the theoretical predictions based on the Darcy flow model. Hence, non-Darcy effects on double diffusive convection in porous media have received a great deal of attention in recent years. Poulikakos [10] has used the Brinkman extended Darcy flow model for the problem to investigate the linear stability analysis. Recently, Givler and Altobelli [11] have demonstrated that for high permeability porous media the effective viscosity is about ten times the fluid viscosity. Therefore, the effect of viscosities on the stability analysis is of practical interest. Thus in the present paper the Brinkman extended Darcy model has been used to investigate the thermohaline convection in porous medium. The work of Banerjee et al.[5] have been extended to magnetorotatory thermohaline instability problem in a porous medium by Prakash et al. [12] in the form of characterization theorems for magnetorotatory thermohaline convection of Veronis type and Stern type that disapprove the existence of oscillatory motions of growing amplitude in initially bottom-heavy configurations of the two types respectively and left open the possibility for the derivation of the analogous theorems for an initially top-heavy configuration of the Veronis type and an initially top-heavy configuration of the Stern type.
Moreover, when compliment of the sufficient conditions contained in the characterization theorems of Prakash et al. [12] holds good, oscillatory motions of growing amplitude can exist, and thus it is important to derive bounds for the complex growth rate of such motions when both the boundaries are not dynamically free, so that exact solutions in the closed form are not obtained. Thus present communication, which prescribes upper bounds for the oscillatory motions of neutral or growing amplitude in magnetorotatory thermohaline configurations of Veronis and Stern types in porous medium in such a manner that also results in sufficient conditions for stability for an initially top heavy or initially bottom configuration, may be regarded as a further step in this scheme of extended investigations.

II. MATHEMATICAL FORMULATION AND ANALYSIS
An infinite horizontal porous layer filled with a viscous fluid is statically confined between two horizontal boundaries z = 0 and z = d maintained at constant temperatures T₀ and T₁ and solute concentrations S₀ and S₁ at the lower and upper boundaries respectively, where T₁ < T₀ and S₁ < S₀(as shown in Fig.1) in the presence of uniform vertical magnetic field. The layer is rotating about its vertical axis with constant angular velocity . It is further assumed that the saturating fluid and the porous layer are incompressible and that the porous medium is a constant porosity medium. Darcy-Brinkman model has been used to investigate the present problem. Let the origin be taken on the lower boundary z=0 with z-axis perpendicular to it.
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References
  1. Turner, J. S., “Double diffusive phenomena”, Ann. Rev. Fluid mech.,Vol. 6, pp.37-54, 1974.
  2. Brandt, A., and Fernando, H. J. S., “Double Diffusive Convection”, Am. Geophys. union, Washington,1996.
  3. Veronis, G., “On finite amplitude instability in thermohaline convection”, J. Mar. Res.,Vol. 23, pp.1-17, 1965.
  4. Stern, M. E., “The salt fountain and thermohaline convection”, Tellus,Vol. 12, pp.172-175, 1960.
  5. Banerjee, M.B., Gupta, J.R., and Prakash, J., “On thermohaline convection of the Veronis type”, J. Math. Anal. Appl.,Vol. 179, pp.327-334, 1993.
  6. Nield, D. A., and Bejan, A., “Convection in porous Media”, Springer, New York,2006.
  7. Straughan, B., “Stability and wave motion in porous media”, Springer, 2008.
  8. Vafai, K., “Handbook of porous medium”, Taylor and Francis,2006.
  9. Prakash, J., and Kumar, V., “On Nonexistence of Oscillatory Motions in ther mohaline Convection of Stern Type in Porous Medium”, J. Rajasthan Acad. Phy. Sc. Vol.10(4), pp.331-338, 2011.
  10. Poulikakos, D., “Double-diffusive convection in a horizontally sparsely packed porous layer”, Int. Comm. Heat Mass Transfer,Vol. 13,pp.587 – 598, 1986.
  11. Givler, R. C., and Altobelli, S. A., “A determination of the effective viscosity for the Brinkman-Forchheimer flow model”, J. Fluid Mech.,Vol. 258,pp.355 – 370, 1994.
  12. Prakash, J., and Gupta, S.K., “A mathematical theorem in magnetorotatory thermohaline Convection in Porous Medium”, Int. J..Math.Arch, Vol. 3(12), pp.4997-5005, 2012.
  13. Schultz, M.H., “Spline Analysis”, Prentice Hall, Englewood Cliffs, NJ,1973.