On Triply Diffusive Convection in Porous Medium: Darcy Brinkman Model

Jyoti Prakash¹*,Shweta Manan², Vinod Kumar³

¹Department of Mathematics and Statistics, Himachal Pradesh University, Summer Hill, Shimla-171005, India

²Department of Physics, MLSM College, Sundernagar, H. P., India

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

ABSTRACT:

The present paper deals with the problem of triply diffusive convection analogous to Stern type in porous medium using Darcy-Brinkman model. Bounds are obtained for the complex growth rate of an arbitrary oscillatory perturbation of growing amplitude, neutral or unstable for this configuration which is uniformly valid for any combination of bounding surfaces.

KEYWORDS: Triply diffusive convection, Concentration Rayleigh number, Porous medium, Darcy-Brinkman model.

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 thermosolutal convection or more generally double diffusive convection) has been an area of great interest due to its importance in assessing the effectiveness of fibrous materials, in the predication of ground water movement in aquifers, in engineering geology and in nuclear engineering. Double diffusive convection in porous medium 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), Vafai (2000) and Kellner and Tilgner (2014). All these scientists have considered two component systems. However, it has been recognized later that there are many fluid systems, in which more than two components are present. 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 area with more than two components (in porous and non porous medium) has attracted the attention of many researchers (Griffiths (1979a, b), Poulikakos (1985), Pearlstein et al. (1989), Terrones and Pearlstein (1989), Rudraiah and Vortmeyer (1982), Lopez et al. (1990), Tracey (1996, 1998), Straughan and Tracey (1999), Rionero (2010), Prakash et al. (2015a, b)). The summary of the works of these researchers is that small concentration 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 oscillatory motions of neutral or growing amplitude exists, then it is important to derive upper bounds for the complex growth rate of such motions when atleast one of the bounding surfaces is rigid so that exact solutions of the problem in closed form are not obtainable. Banerjee et al. (1981) derived upper bounds for double diffusive convection problem in the form of a semicircle. Prakash et al. (2014a, b, 2015c) further extended this problem to triply diffusive convection configuration. In the present paper we extended this problem to triply diffusive convection analogous to Stern type. The following result is obtained in this direction: The complex growth rate of an arbitrary neutral or unstable oscillatory disturbance of growing amplitude, in a triply diffusive fluid layer saturating a porous medium (Darcy Brinkman Model) with one of the components as heat, must lie inside a semicircle in the right- half of the -plane whose centre is origin and radius equals, where is the Rayleigh number, is the Prandtl number and is constant. Further, it is shown that above result is uniformly valid for all combinations of rigid and / or dynamically free boundaries.

2. MATHEMATICAL FORMULATION:

A viscous finitely heat conducting Bossiness fluid layer, saturating a porous medium, of infinite horizontal extension is statically confined between two horizontal boundaries which are respectively maintained at uniform temperatures and uniform concentrations (see 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 triple diffusive convection in porous medium.

Fig.1 Geometrical Configuration

3. CONCLUSION:

Linear stability theory is used to study triply diffusive convection analogous to Stern type in porous medium using Darcy-Brinkman model. Upper bounds for the complex growth rate of an arbitrary neutral or unstable oscillatory disturbance of growing amplitude are derived. Further, the result for double diffusive convection of Stern (1960) type in porous medium is also obtained as a consequence.

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Received on 16.11.2016 Modified on 22.11.2016

DOI: 10.5958/2349-2988.2017.00020.1

Research J. Science and Tech. 2017; 9(1):127-130.