Upper Limits to the Complex Growth Rates in Triply Diffusive Convection in Porous Medium

 

Jyoti Prakash*, Virender Singh, Shweta Manan and Vinod Kumar**

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

** Department of Physics, MLSM College, Sundernagar, H.P. (India).

*Corresponding Author E-mail: jpsmaths67@gmail.com

 

ABSTRACT:

The paper mathematically establishes that the complex growth rate () of an arbitrary neutral or unstable oscillatory perturbation of growing amplitude, in a triply diffusive fluid layer in porous medium (Darcy Model) with one of the components  as heat with diffusivity  , must lie inside a semicircle in the right- half of the ()-plane whose centre is origin and radius equals , where  and  are the Raleigh numbers for  the  two concentration components with diffusivities   and  (with no loss of generality,  ) and σ is the Prandtl number. Further, it is proved that above result is uniformly valid for quite general nature of the bounding surfaces.

 

KEYWORDS: Triply Diffusive convection, Oscillatory motions, complex growth rate, Porous Medium.

 

INTRODUCTION:

The hydrodynamic instability that manifests 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 thermohaline convection or more generally double diffusive convection. The thermohaline convection problem has been extensively studied in the recent past on account of its interesting complexities as a double diffusive phenomenon  as well as its importance  in many problems of practical interest in the fields of oceanography, astrophysics, limnology and chemical engineering etc. Double diffusive convection is now well known. For a broad view of the subject one may be referred to Stern[1], Veronis[2], Nield[3], Baines and Gill[4], Turner[5] and Brandt and Fernando[6] etc.

                                                                                                                                                                                                                All these researchers have considered the case of two component systems. However, it has been recognized previously (Griffiths[7]) 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, Earth core, geothermally heated lakes, and magmas and their laboratory models and seawater etc. Griffiths[7], Lopez et al.[8] etc. have theoretically studied the onset of convection in a triply diffusive fluid layer (where density depends on three independently diffusing agencies with different diffusivities). The essence of the investigations of these researchers is that small concentrations of a third component with a smaller diffusivity can have a significant effect upon the nature of diffusive instabilities and ‘oscillatory’ and direct ‘salt finger’ modes are simultaneously unstable under a wide range of conditions when the density gradients due to components with the greatest and smallest diffusivity are of same signs.

 

The problem of obtaining bounds for the complex growth rate of an arbitrary oscillatory perturbation of growing amplitude in various hydrodynamical stability problems is an important feature of fluid dynamics, especially when both the boundaries are not dynamically free so that exact solutions in the closed form are not obtainable.Banerjee et al.[9] formulated a novel way of combining a governing equations and boundary conditions for double diffusive convection problem so that a semicircle theorem is derivable and which in turn yields the desired bounds. Since the inability of finding the exact solutions in the closed form also exist for the case of three component systems when both the boundaries are not dynamically free, the bounds for the complex growth rate of an arbitrary oscillatory perturbation of growing amplitude in triply diffusive convection in porous media must be found.

 

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. Lapwood[10], Wooding[11], Nield and Bezan[12], Straughan[13].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[12].

In the present communication Banerjee et al’s[9] technique has been used to obtain the bounds for the complex growth rates in triply diffusive convection in porous medium(Darcy model). The following result is obtained in this direction:

 

The complex growth rate () of an arbitrary neutral or unstable oscillatory perturbation of growing amplitude, in a triply diffusive fluid layer in porous medium (Darcy Model) with one of the components  as heat with diffusivity  , must lie inside a semicircle in the right- half of the ()-plane whose centre is origin and radius equals  where  and  are the Raleigh numbers for  the  two concentration components with diffusivities   and  (with no loss of generality,  ) and σ is the Prandtl number. Further, it is proved that above result is uniformly valid for quite general nature of the bounding surfaces.

 

Mathematical Formulation of the problem

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 (< T0) and uniform concentrations S10, S20 and S11 (< S10), S21 (< S20). It is further assumed that the saturating fluid and the porous layer are incompressible and that the porous medium is a constant porosity medium.(as shown in Fig.1)

 

Fig.1 Physical configuration

 

This establishes the theorem:

The above theorem may be stated in an equivalent form as: the complex growth rate () of an arbitrary neutral or unstable oscillatory perturbation of growing amplitude, in a triply diffusive fluid layer in porous medium (Darcy Model) with one of the components  as heat with diffusivity  , must lie inside a semicircle in the right- half of the ()-plane whose centre is origin and radius equals  where  and  are the Raleigh numbers for  the  two concentration components with diffusivities   and  (with no loss of generality,  ) and σ is the Prandtl number. Further, it is proved that above result is uniformly valid for quite general nature of the bounding surfaces.

 

REFERENCES                                          

[1]     Stern, M. E.: The ‘salt fountain’ and ‘thermohaline convection’. Tellus 12, 172-175(1960)

[2]     Veronis, G.: On finite amplitude instability in thermohaline convection. J. Mar. Res. 23, 1-17(1965)

[3]     Nield, D. A.: The thermohaline Rayleigh- Jeffrey’s problem. J. Fluid Mech. 29, 545-558 (1967)

[4]     Baines, P. G.: Gill, A. E.: On thermohaline convection with linear gradient. J. Fluid Mech. 37, 289 – 306(1969)

[5]     Turner, J. S.: The behaviour of a stable salinity gradient heated from below. J. Fluid Mech. 33, 183 – 200(1968)

[6]     Brandt, A.: Fernando H. J. S.: Double diffusive convection. Am. Geophys. Union, Washington, DC, (1996)

[7]     Griffiths, R. W.: The influence of a third diffusing component upon the onset of convection. J. Fluid Mech. 92, 659 – 670(1979)

[8]     Lopez, A. R. Romero, L. A. Pearlstein, A. J.: Effect of rigid boundaries on the onset of convective instability in a triply diffusive fluid layer. Physics of fluids. A 260, 897 – 902 (1990)

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

[10]   Lapwood E. R.: Convection of fluid in a porous medium. Proc. Camb. Phil. Soc. 44, 508(1948).

[11]   Wooding R. A.:  Rayleigh instability of a thermal boundary layer in flow through a porous medium. J. Fluid Mech. 9 183(1960).

[12]   Nield D. A and Bejan A.: Convection in porous Media, Springer (2006).

[13]   Straughan B.: Stability and wave motion in porous media, Springer (2008).

[14]   N.Rudraiah and D.Vortmeyer.: Int. J. Heat Mass Transfer. Vol. 25, No 4, 457- 464 (1982).

 

 

 

Received on 03.01.2014    Accepted on 29.01.2014

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