Onset of Triply-Diffusive Convection in a Fluid Layer with Suspended Particles and Temperature Dependent Viscosity

Joginder Singh Dhiman and Nivedita Sharma

Department of Mathematics, Himachal Pradesh University, Summerhill, Shimla-171005, INDIA.

*Corresponding Author E-mail: js.dhiman@hpuniv.ac.in, sharmanivedita84@gmail.com

ABSTRACT:

The present paper mathematically investigates the triply-diffusive convection problem with suspended particles by considering the viscosity to be temperature dependent. The temperature gradient is considered to be destabilizing whereas the solute gradients may be stabilizing or destabilizing. A sufficient condition for the validity of principle of exchange of stabilities (PES) is obtained and a bound for the complex growth rate of an arbitrary oscillatory perturbation, which may be neutral or unstable, is derived for this general problem. Various consequences of the above results are discussed and the analogous results under the individual effect of suspended particles, solute gradients and viscosity variations are also deduced.

KEYWORDS: triply-diffusive convection, principle of exchange of stabilities, viscosity variations, suspended particles.

1. INTRODUCTION:

The hydrodynamic instability that manifests under the appropriate conditions in a static horizontal, viscous and Boussinesq fluid layer of infinite horizontal extension and finite vertical depth, which is kept under the action of uniform vertical adverse temperature gradient, in the force of gravity is known as thermal convection. The earliest experiments to demonstrate in a definitive manner the onset of thermal convection in fluids are those of Bénard [1]. He found that when the temperature of the lower surface was gradually increased, at a certain instant, the layer became reticulated and revealed its dissection into equal, hexagonal and properly aligned cells. This phenomenon is also known as Bénard convection, as a mark of tribute to him. Bénard experiments were repeated by many others including Schmidt and Milverton [2] and Koshmieder [3]. Further, Rayleigh [4] and Jeffreys [5] studied theoretically the problem of thermal convection in fluids and obtained the critical conditions for the onset of instability. A detailed account of problems of onset of thermal convection in a fluid layer has been discussed by Chandrasekhar [6] and Drazin and Reid [7] in their respective monographs.

Wollkind and Zhang [8] observed that the experimental determination of the onset of convection agreed with the theoretical prediction in liquid layers regardless of the measuring technique employed, but in gas layers there was a discrepancy when the visual method of adding an aerosol to the gas was used, as the convection occur at much lower gradient then predicted. Chandra [9] performed the experiment in an air layer mixed with smoke and was first to observe this phenomenon. He found that the instability depends on the depth of the layer. He further noticed that a Bénard - type cellular convection with the fluid descending at the cell centre was observed when the predicted gradients were imposed for layers deeper than 10 mm. A convection which was different in character from that in deeper layers occurred at much lower gradients than predicted, if the layer depth was less than 7 mm it was called columnar instability. Motivated by the interest in fluid-particle mixtures and columnar instability, Scanlon and Segel [10] studied the effect of suspended particles on the onset of Bénard convection and found that the critical Rayleigh number was reduced, solely because the heat capacity of the pure gas was supplemented by that of the particles.

Griffiths [11] recognized that there are many situations wherein more than two diffusing components are present. The multiple diffusive convective fluid systems include the solidification of molten alloys, magmas and sea water. The onset of convective motions in a fluid layer with two or more destabilizing sources for the density difference, the temperature field and salt fields are fundamentally different from single diffusive case as new instability phenomena may occur which is not present in the classical Bénard problem. Griffiths [11], Pearlstein et al [12], Lopez [13] and many other authors have studied theoretically the onset of convection in a horizontal layer of a triply diffusive fluid and found some fundamental differences between the double and triply diffusive convection.

It is well known fact that the viscosity is the only property which may have considerable effect on the heat transfer and hence it is not realistic to consider the viscosity as a constant in flow problems. In view of the importance of the variation of viscosity with temperature, Nield [14], Straughan [15] and Palm [16] have investigated the onset of convection for the ordinary fluids with strongly temperature-dependent viscosity. Recently, Dhiman and Kumar [17] have studied the stability of Rayleigh-Bénard convection with temperature dependent viscosity for general boundary conditions using Galerkin method.

Motivated by above discussions and the importance of multi diffusive components and the variable viscosity in thermal convection problems, the aim of the present paper is to mathematically investigates the effect of suspended particles and viscosity variation on the triply-diffusive convection problem for general boundary conditions.

2. PHYSICAL CONFIGURATION AND EIGEN VALUE PROBLEM

Consider an infinite horizontal layer of viscous, quasi-incompressible (Boussinesq) fluid of continuously varying density and viscosity statically confined between two horizontal boundaries and , maintained respectively at constant temperature and and uniform concentrations and in the presence of suspended particles. Our object here is to investigate the stability of the above configuration by taking account of variations in viscosity ( due to temperature effects and the conflicting tendencies of the various diffusing agents. Here, is the value of at and is the non dimensional temperature dependent viscosity variation function.

3. DISCUSSION AND CONCLUSIONS:

In the present analysis, we have discussed the stability of the triply diffusive convection problem in the presence of suspended particles and temperature dependent viscosity, using linear stability analysis method. A mathematical condition for the validity of principle of exchange of stabilities (PES) is derived using Pellew and Southwell method. It is also seen when the compliment of this condition holds good, the oscillatory motions of growing amplitude may exist and therefore the upper bound for the complex growth rate of an arbitrary oscillatory perturbation which may be neutral or unstable is also derived, which is uniformly valid for all cases of boundary conditions. The derivation of the bound is also important especially when both the boundaries are not dynamically free as the exact solutions in the closed form are not obtainable then. Further, it is important to note from the above theorems that for the validity of the results the condition in is a necessary condition. Which implies that temperature dependent viscosity function is an increasing function of vertical coordinate z. This condition is obviously satisfied for the case of present geometry of the problem, since we are dealing with fluid heated from below in which due to temperature variation viscosity increases vertically upward along the z coordinate. Though the condition is a curious, however this is always true for the cases of linear and exponential variations of viscosity. Thus, the obtained results are of wider generality.

It is to mention that Scanlon and Segel [10] have pointed out that the effect of suspended particles can be presented in terms of the relaxation time . As is proportional to the square of the particle radius, if is large the particles are coarse and if is small, the particles are fine. In view of this, one can observe from Theorem 2 that growth rate of oscillations increases with increasing value of and vice versa. Hence, for large values of the system becomes stable. We also observe from the above result that by introducing the third stabilizing diffusive component the radius of growth rate increases and hence has stabilizing effect on the system. Further, we can easily observe from corollary 1 the effects of destabilizing diffusive salt gradient.

4. REFERENCES:

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Received on 10.01.2014 Accepted on 30.01.2014

Int. J. Tech. 4(1): Jan.-June. 2014; Page 112-116