On Double-Diffusive Convection in a Binary Viscoelastic Fluid Saturated Anisotropic Porous Layer

Jyoti Prakash*, Kultaran Kumari

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

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

ABSTRACT:

In the present paper it is mathematically established that the linear growth rate of an arbitrary neutral or unstable oscillatory perturbation of growing amplitude in a double diffusive binary viscoelastic fluid saturated anisotropic porous layer heated from below must lie inside a semicircle in the right half of the - plane whose centre is at the origin and radius equals where and are the Darcy -Rayleigh number and the solute Rayleigh number respectively. Further, it is proved that this result is uniformly valid for quite general nature of the bounding surfaces.

KEYWORDS Double-diffusive convection, viscoelastic fluid, porous medium, complex growth rate, solute Rayleigh number.

Introduction:

Thermosolutal instability problem or more generally known as double diffusive convection problem in porous medium has attracted physicists and mathematicians during recent past due to its wide range of applications in many fields of interest which includes geophysical system, electrochemistry, solidification of binary mixtures, migration of moisture through air contained in fibrous solutions (Malashetty et al. (2009)). For the broad study of the subject of double diffusive convection in porous medium one may be referred to Ingham and Pop(2005), Nield and Bejan(2006), Vafai(2005) and Vadasz(2008). The study of the flow of viscoelastic fluids is of great importance because of its wide range of applications in various fields such as petroleum, oil reservoir modelling, chemical and nuclear industries, geothermal energy utilization, bioengineering, building thermal insulation and carbon dioxide geologic sequestration (Gaikwad and Dhanraj(2014)). Although the problem of thermal convection has been extensively studied for Newtonian fluids, comparatively less attention has been given to thermal convection of non-Newtonian fluids (Gaikwad and Kamble(2016)). The investigation of thermal convection in non-Newtonian fluid is important from a rheological point of view since the observation of manifestation of instability provides useful methods to study the suitability of a constitutive model adopted for a certain viscoelastic fluid. Further, convection in viscoelastic fluids may manifest in the form of oscillatory motions which is not observed in thermal convection of Newtonian flow. The published work on thermal convection of viscoelastic fluids in porous media is fairly limited. Rudraih et al. (1990) investigated the stability of viscoelastic fluid saturated porous layer by using Darcy and Brinkman models. Kim et al. (2003) analysed theoretically thermal instability of viscoelastic fluids in porous media. Mardones et al. (2003) investigated thermal convection in binary fluids with Oldroyd viscoelastic properties. Wang and Tan analysed the stability of double-diffusive convection of Maxwell fluid in a porous medium heated from below. Malashetty et al. (2009) investigated the double –diffusive convection in a viscoelastic binary fluid saturated porous layer. Kumar and Shivakumara (2014) studied the effects of quadratic drag and vertical through flow on the onset of double-diffusive convection in a non-Newtonian fluid saturated horizontal porous layer by using modified Forchheimer- extended Darcy- model. The problem of obtaining bounds for the complex growth rate of an arbitrary oscillatory perturbation of growing amplitude in double diffusive convection 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. (1981) formulated a noble way of combining the governing equations and boundary conditions to obtain such bounds. We used their technique to derive the bounds for the present problem. Further, it is proved that this result is uniformly valid for quite general nature of the bounding surfaces

CONCLUSION:

A linear stability analysis is used to derive the upper bounds for complex growth rates in double diffusive convection problem. These bounds are important especially when both the boundaries are not dynamically free so that exact solutions in the closed form are not obtainable. Further, the results so obtained are uniformly valid for all the combinations of rigid and free boundaries.

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Received on 17.11.2016 Modified on 23.11.2016

DOI: 10.5958/2349-2988.2017.00019.5

Research J. Science and Tech. 2017; 9(1):123-126.