On Rotatory Hydrodynamic Triply Diffusive Convection in Porous Medium

Jyoti Prakash*, Virender Singh, Shweta Manan

Department of Mathematics and Statistics. Himachal Pradesh University, Summer Hill, Shimla-171005, India.

*Corresponding Author Email: jpsmaths67@gmail.com, goputhakur1988@gmail.com

ABSTRACT:

Condition for characterizing nonoscillatory motions, which may be neutral or unstable, for rotatory hydrodynamic triply diffusive convection in a porous medium is derived. It is analytically proved that the principle of the exchange of stabilities, in rotatory triply diffusive convection in a porous medium, is valid in the regime , where and are the concentration Raleigh numbers, and and are the Lewis numbers for the two concentration components respectively, is the Taylor number, σ is the Prandtl number, is the Darcy number, and are constants.

KEYWORDS: Triply diffusive convection, Porous medium, Darcy-Brinkman model, the principle of the exchange of stabilities, Taylor number, Concentration Rayleigh number.

1. INTRODUCTION:

Research on convective fluid motion in porous media under the simultaneous action of a uniform vertical temperature gradient and a gravitationally opposite uniform vertical concentration gradient (known as double diffusive convection) has been an area of great activity due to its importance in the predication of ground water movement in aquifers, in assessing the effectiveness of fibrous materials, in engineering geology and in nuclear engineering. Double diffusive convection is now well known. For a broad view of the subject one may be referred to Nield and Bezan (2006), Murray and Chen (1989), Nield (1968), Taunton et al. (1972), Kuznetsov and Nield (2008), Lombardo and Mulone (2002).

All these researchers have considered double diffusive convection. However, it has been recognized later that there are many fluid systems, in which more than two components are present. For example, Degens et al (1973) reported that the saline waters of geothermally heated Lake kivu are strongly stratified by heat and a salinity which is the sum of comparable concentrations of many salts. Similarly the oceans contain many salts having concentrations less than a few percent of the sodium chloride concentration. Multi-component concentrations can also be found in magmas and substratum of water reservoirs. The subject with more than two components (in porous and non porous medium) has attached the attention of many researchers Grifiths (1979a, 1979b), Poulikakos (1985), Pearlstein et al. (1989), Terrones and Pearlstein (1989), Rudraiah and Vortmeyer (1982), Lopez et al (1990), Tracey (1996, 1998), Terrones (1993), Straughan and Tracey (1999), Ryzhkov and Shevtsova (2009)and Rionero (2013a, 2013b). The essence of the works of these researchers is that small salinity of a third component with a smaller mass diffusivity can have a significant effect upon the nature of convection; and ‘oscillatory’ and direct ‘salt finger’ modes are simultaneous possible under a wide range of conditions, when the density gradients due to components with greatest and smallest diffusivity are of same signs.

Double or triply diffusive convection in a rotating fluid layer saturating a porous medium is an interesting topic due to its applications in chemical process industry, food processing industry, solidification and centrifugal costing of metals and rotating machinery, petroleum industry, geophysics and biomechanics. Several studies are available in which phenomena related to the onset of single diffusive (Benard Problem) and double diffusive convection in a rotating porous medium have been investigated. For a detailed review of the subject one may be referred to Vadasz ( 1998), Nield and Bezan (2006), Sengupta and Gupta (1971), Rionero (2013 a, b). To the authors knowledge no such significant work has been done so far in rotatory hydrodynamic triply diffusive convection.

The validity of the principle of the exchange of stabilities (PES) (i.e. nonoccurrence of oscillatory motions) in stability problems removes the unsteady terms from the linear perturbation equations which results in notable mathematical simplicity since the transition from stability to instability occurs via a marginal state which is defined by the vanishing of both real and imaginary parts of the complex time eigen value associated with the perturbation. Pellew and Southwell (1940) proved the validity of PES for Rayleigh-Benard problem. However no such result exists for other more complex hydrodynamic configurations. Banerjee et al. (1985) derived a sufficient condition for the validity of PES for hydromagnetic Rayleigh-Benard problem. Gupta et al. (1986) extended Banerjee et al’s (1985) criterion to rotatory hydromagnetic thermohaline convection problem. Recently Prakash et al. (2014) derived a sufficient for the validity of PES for rotatory hydrodynamic triply diffusive convection. To the author’s knowledge no such result exists for triply diffusive convection in a rotating porous medium. Thus the present paper which provides a sufficient condition for the validity of PES rotatory hydrodynamic in triply diffusive convection in porous medium may be regarded as a first step in this scheme of extended investigations. The following result is obtained in this direction:

For rotatory hydrodynamic triply diffusive convection in porous medium, if , then an arbitrary neutral or unstable mode of system is definitely nonoscillatory in character and in particular PES is valid where and are the concentration Raleigh numbers, and and are the Lewis numbers for the two concentration components respectively, is the Taylor number, σ is the Prandtl number, is the Darcy number, and are constants. It is further proved that this result is uniformly valid for all combinations of rigid and dynamically free boundaries and the results for Rayleigh-Benard convection in porous medium and double diffusive convection (with and without rotation) in porous medium follow as a consequence.

MATHEMATICAL FORMULATION AND ANALYSIS

A viscous finitely heat conducting Boussinesq fluid layer, saturating a porous medium, of infinite horizontal extension is statically confined between two horizontal boundaries (rotating with uniform angular velocity) which are respectively maintained at uniform temperatures and uniform concentrations (as shown in Fig.1). It is assumed that the saturating fluid and the porous layer are incompressible and that the porous medium is a constant porosity medium. It is further assumed that the cross-diffusion effects of the stratifying agencies can be neglected. The Brinkman extended Darcy model has been used to investigate the triply diffusive convection in porous medium.

Fig.1

The nondimensional equations which govern the rotatory hydrodynamic triply diffusive convection in porous medium are given by [Prakash et al. (2016)]

CONCLUSIONS:

Linear stability theory is used to derive a sufficient condition for the validity of ‘the principle of the exchange of stabilities’ in rotatory hydrodynamic triply diffusive convection in porous medium. It is further proved that these results are uniformly valid for any combination of rigid and / or free boundaries.

REFERENCES:

1. Banerjee, M. B., Katoch, D. C. and Dube, G. S., Bounds for growth rate of perturbation in thermohaline convection, Proc Roy Soc London Ser A, 378 (1981) 301 – 304.

2. Banerjee, M. B., Gupta, J. R., Shandil, R. G., Sood, S. K., Banerjee, B. and Banerjee, K., On the principle of exchange of stabilities in the magnetohydrodynamic simple Benard problem, J. Math. Anal. Appln., 108 (1) (1985) 216 – 222.

3. Degens, E. T., Herzen, R. P., Wong, H. K., Deuser, W. G. and Jannasch, H. W., Lake Kivu: structure, chemistry and biology of an East African rift lake. Geologische Rundschau, 62 (1973) 245-277.

4. Griffiths, The influence of a third diffusing component upon the onset of convection J Fluid Mech 92 (1979) 659-670.

5. Griffiths, R. W., A note on the formation of “salt finger” and ‘diffusive’ interfaces in three component systems, Int. J. Heat Mass Transf., 22 (1979b) 1687-1693.

6. Gupta, J. R., Sood, S. K. and Bhardwaj, U. D., On the characterization of non oscillatory motions in a rotatory hydromagnetic thermohaline convection, Indian J. Pure Appl. Math., 17(1) (1986) 100 – 107.

7. Kuznetsov A. V. and Nield, D. A., The effects of combined horizontal and vertical heterogencity on the onset of convection in a porous medium: double diffusive case. Trans. Porous. Med., 72 (2008) 157-170.

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

9. Lombardo, S. and Mulone, G., Necessary and sufficient conditions of global nonlinear stability for rotating double-diffusive convection in porous medium, Continuum Mech. Thermodynamic 14 (2002) 524-540.

10. Murray, B. T. and Chen, G. F., Double-diffusive convection in porous media, J. Fluid Mech., 201 (1989) 147-166.

11. Nield, D. A. and Bezan, A., Convection in porous media, third Ed., Springer-Verlag, New York (2006).

12. Nield, D. A., Onset of thermohaline convection in porous medium, Water, Resour. Res., 4 (1968) 553-560.

13. Pellew, A. and Southwell, R. V., On the maintained convective motion in a fluid heated from below, Proc. Roy. Soc. London A., 176 (1940) 312-343.

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

15. Poulikakos, D., The effect of a third diffusing component on the onset of convection in a horizontal porous layer, Phys. Fluids, 28 (10) (1985) 3172-3174.

16. Prakash, J., Bala, R. and Vaid, K., On The Principle of the Exchange of Stabilities in Rotatory Triply Diffusive Convection, Proc. Nat. Acad. Sc., Ind., (2014) In Press.

17. Prakash, J., Singh, V. and Manan, S., On limitations of linear growth rates in triply diffusive convection in porous medium, J. Assoc. Arab Univ. Basic and Appl. Sc. (2016) In press.

18. Ryzhkov, I. and Shevtsova, V. M., Long-wave instability of a multicomponent fluid layer with the Soret effect, Phys. Fluids, 21 (2009) 014102 (1-14).

19. Rudraiah, M. and Vortmeyer, D., The influence of permeability and of a third diffusing component upon the onset of convection in a porous medium, Int. J. Heat Mass Trans., 25(4) (1982) 457-464.

20. Rionero, S., Triply diffusive convection in porous media, Acta Mech. 224 (2013a) 447-458.

21. Rionero, S., Multicomponent diffusive convection fluid motions in porous layers ultimately boundedness, absence of subcritical instabilities, and global nonlinear stability for any number of salts, Phys. Fluids, 25 (2013b) 054104 (1-23).

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

23. Straugham, B. and Tracey, J., Multi-component convection diffusion with internal heating or cooling, Acta Mech.133 (1999) 219-239.

24. Tracey, J., Multi-component convection-diffusion in a porous medium, Continuum Mechanics and Thermodynamics. 8 (1996), 361-381.

25. Tracey, J., Penetrative convection and Multi-component diffusion in a porous medium, Adv. Water Res. 22 (1998) 399-412.

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

27. Terrones and Pearlstein, The onset of convection in a multicomponent fluid layer, Phys. Fluids A 1 (5) (1989) 845-853.

28. Taunton, J. W., Lightfoot E. N. and Green, T., Thermohaline instability and salt fingers in porous medium, Phys. Fluids, 15 (1972) 748-753.

29. Vadasz, P., Coriolis effect on gravity-driven convection in a rotating porous layer heated from below, Journal of Fluid Mechanics, 276 (1998) 351-375.

Received on 20.08.2016 Accepted on 05.09.2016

Int. J. Tech. 2016; 6(2): 113-117.

DOI: 10.5958/2231-3915.2016.00018.3