On Triply Diffusive Convection Analogous to Stern type with Variable Viscosity

Jyoti Prakash¹, Rajeev Kuma¹, Prakash Chopra²

¹Department of Mathematics and Statistics, Himachal Pradesh University, Shimla, India.

²J.N. Government Engineering College, Sunder Nagar (H.P.), India.

*Corresponding Author E-mail: rajeevkumar2012math@gmail.com

ABSTRACT:

The paper mathematically establishes that triply diffusive convection (analogous to Stern type), with variable viscosity and with one of the components as heat, cannot manifest itself as oscillatory motions of growing amplitude in an initially bottom heavy configuration if the thermal Rayleigh number, the Lewis number for the second concentration component, (the minimum value of viscosity in the closed interval ) and the Prandtl number satisfy the inequality
provided is positive everywhere. It is further proved that this result is uniformly valid for the quite general nature of the bounding surfaces.

KEYWORDS: Triply diffusive convection, variable viscosity, concentration Rayleigh number, oscillatory motion, initially bottom heavy configuration.

Mathematical Subject Classification Number: 76E06, 76E20.

1. INTRODUCTION:

The hydrodynamic instability that manifest under appropriate conditions in a static horizontal initially homogeneous viscous and Boussinesq liquid layer of infinite horizontal extension and finite vertical depth which is kept under the simultaneous action of a uniform vertical temperature gradient and a gravitationally opposite uniform vertical concentration gradient in the force field of gravity is known as thermosolutal convection or more generally double diffusive convection. Double diffusive convection is now well known. For a broad view of the subject one may be referred to Turner [20], Brandt and Fernando [4] and Radko [14]. Only the case of two component systems has been considered by these scientists. However, it has been recognized later on [5, 8, 9, 18, 21] that there are many situations wherein more than two components are present. Examples of such multiple diffusive convection fluid systems include the solidification of molten alloys, sea water, Earth core, geothermally heated lakes, magmas and their laboratory models etc. The presence of more than one salt in fluid mixtures is very often requested for describing natural phenomena such as acid rain effects, contaminant transport, warming of the stratosphere, underground water flow. Further a number of technologically important alloys such as nickel- based alloys [8] used in turbine blades and another high-strength applications, containing significant mass fraction of as many as seven metallic elements.

The recently established characterization theorem of Banerjee et al. [3], which states that oscillatory motions (neutral or unstable) of growing amplitude cannot manifest in an initially bottom heavy thermohaline convection whenever the concentration Rayleigh numbers is less than a critical value, has brought a fresh outlook to the subject matter of double diffusive convection and paved the way for further theoretical and experimental investigations in this field of enquiry. The summary of Banerjee et al.’s [3] characterization theorem is that it provides a classification of the neutral and unstable double diffusive convection into two classes namely the bottom heavy class and the top heavy class and strikes a distinction between them by means of characterization theorems which disallow the existence of oscillatory motions in the former class. Recently Prakash et al. [12] extended the results of Banerjee et al. [3] to triply diffusive convection. For the field of applications of the Prakash et al.’s [12] theorem in flows that are of interest in certain fields like geophysics, oceanography, astrophysics etc. it is necessary to extend the classical analysis where in the fluid viscosity is a function of temperature and /or depth because the effects of viscosity variation play an important role in several physical situations in these fields [2, 6, 7, 17, 19]. Prakash et al. [11] further extended their work [12] to incorporate the effect of viscosity variations. In the present paper we consider the triply diffusive convection analogous to Stern type (wherein temperature gradient is stabilizing and the two concentration components are stabilizing) and derive a similar characterization theorem. Since the variation of viscosity of liquids with temperature is extremely rapid, the inclusion of variation effects certainly extends the domain of validity of the existing results in the literature.

CONCLUSION:

Linear stability analysis of triply diffusive convection (analogous to Stern type) with variable viscosity effects included has been performed. A classification of the neutral or unstable triply diffusive convection configuration (analogous to Stern type) with variable viscosity has been made into two classes namely, the bottom heavy class and top heavy class and then a distinction between these classes is made by means of a characterization theorem which disallow the existence of oscillatory motions in the former class. The work done in the present paper will certainly pave the way for further theoretical and experimental investigation in this field of enquiry.

REFERENCES:

1. Banerjee, M. B., Gupta, J. R. and Prakash, J. (1992). Upper limits to the complex growth rate in thermohaline instability, J. Math. Anal. Appl., 167 (1), 66-73.

2. Banerjee, M. B., Gupta, J. R. and Shandil, R. G. (1977). Generalized thermal convection with viscosity variations, J. Math. Phys. Soc., 11, 421-442.

3. Banerjee, M. B., Gupta, J. R. and Prakash, J. (1993). On thermohaline convection of the Veronis type, J. Math. Anal. Appl., 179, 327-334.

4. Brandt, A. and Fernando, H. J. S. (1996), Double Diffusive Convection, Am. Geophys. Union. Washington. DC.

5. Griffiths, R. W. (1979). The influence of a third diffusing component upon the onset of Convection, J. Fluid Mech., 92, 659-670.

6. Hooman, K. and Gurgenci, H. (2008). Effects of temperature dependent viscosity on benard convection in a porous medium using a non-Darcy model, Int. J. Heat Mass Transfer, 51(5-6), 1139-1149.

7. Korenga, J. and Jordhan, T. H. (2002). Onset of convection with temperature and depth dependent Viscosity, Geophysical research Letters, 29(19), 1923doi:10.1029 /2002 GL 015672.

8. Lopez, A. R., Romero, L. A. and Pearlstein, A. J. (1990). Effect of rigid boundaries on the onset of convective instability in a triply diffusive fluid layer, Physics of fluids A., 2 (6), 897-902.

9. Pearlstein, A. J., Harris, R. M. and Terrones, G. (1989). The onset of convective instability in a triply diffusive fluid layer, J. Fluid Mech., 202, 443-465.

10. Prakash J. (1995). A mathematical theorem for thermohaline convection of the Veronis type with viscosity variations, Indian J. Pure Appl. Math., 26(8), 813-821.

11. Prakash, J., Kumar, R. and Kumari, K. (2015). A Characterization Theorem in Triply Diffusive Convection with Viscosity Variations, Journal of Rajasthan Academy of Physical Sciences, 14 (3 & 4), 249-259.

12. Prakash J., Vaid K., Manan S. and Kumar R. (2015). On characterization of triply diffusive convection, Int. J. Ext. Res., 3, 87-92.

13. Prakash, J., Kumar, R. and Kumari, K. (2015). Linear triply diffusive convection with viscosity variations, International Journal of Physical and Mathematical Sciences., 5(1), 186-196.

14. Radko, T. (2013), Double Diffusive Convection, Cambridge University Press.

15. Schultz, M. H. (1973), Spline Analysis, Prentice-Hall Inc., Englewood Cliffs., NJ.

16. Stengel, K. C., Oliver, D. S. and Booker, J. R. (1982). Onset of convection in variable viscosity Fluid, J. Fluid Mech., 120, 411-431.

17. Sunil and Choudhary, S. (2013). Global stability for thermal convection in a couple stress fluid with temperature and pressure dependent viscosity, Studia Geotechanica et Mechanica, 35(3) doi:10.2478/sgem-2013-0032.

18. Terrones, G. (1993). Cross-diffusion effects on the stability criteria in a triply diffusive system, Phys. Fluids A., 5(9), 2172-2182.

19. Torrance, K. E. and Turcotte, D. L. (1971). Thermal convection with large viscosity variations, J. Fluid. Mech., 47, 113-125.

20. Turner, J. S. (1974). Double-diffusive phenomena, Ann. Rev. Fluid Mech., 6, 37-56.

21. Turner, J. S. (1985). Multicomponent convection, Ann. Rev. Fluid Mech., 17, 11-44.

Received on 16.11.2016 Modified on 30.11.2016

DOI: 10.5958/2349-2988.2017.00016.X

Research J. Science and Tech. 2017; 9(1):111-114.