On the onset of stationary convection in double-diffusive binary viscoelastic fluid saturated anisotropic porous layer

Jyoti Prakash^1*, Kultaran Kumari¹, Vinod Kumar²

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

²Department of Physics, M.L.S.M. College Sunder- Nagar, Distt. Mandi (H.P.)India.

*Corresponding Author Email: jpsmaths67@gmail.com

ABSTRACT:

Linear stability of double diffusive convection in a binary viscoelastic fluid saturated anisotropic porous layer has been studied analytically. A sufficient condition for the occurrence of stationary convection has been derived in terms of the parameters of the system alone. It is further proved the above result is uniformly valid for any combination of the bounding surfaces.

KEY WORDS:

Double-diffusive convection, viscoelastic fluid, the principle of the exchange of stabilities, porous medium.

INTRODUCTION:

Thermohaline instability problem or more generally known as double diffusive convection problem in porous medium has attracted considerable attention during recent past due to its wide range of applications in many fields of interest which includes electrochemistry, geophysical system, solidification at binary mixtures, migration of moisture through air contained in fibrous solutions (Malashetty et al. (2009)). For the broad and recent view 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 oil reservoir modelling, petroleum, 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 viscoelastic fluids (Gaikwad and Kamble(2016)). The investigation of thermal convection in viscoelastic 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 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) studied thermal convection in binary fluids with Oldroyd viscoelastic properties. Wang and Tan studied 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 throughflow on the onset of double-diffusive convection in a non-Newtonian fluid saturated horizontal porous layer by using modified Forchheimer- extended Darcy- model.

The establishment of the nonoccurrence of any slow oscillatory motions which may be neutral or unstable implies the validity of the principle of the exchange of stabilities (PES). The validity of this principle in stability problems eliminates the unsteady terms from the linearized perturbation equations which results in notable mathematical simplicity since the transition from stability to instability occurs via a marginal state which is characterized 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 (i.e. occurrence of stationary convection) for the classical Rayleigh-Benard instability problem. Recently, Prakash et al. (2014a,b,c) established such criteria for triply diffusive convection problems. To the authors’ knowledge no such result exists for double diffusive convection problem of binary viscoelastic fluid in anisotropic porous medium. Hence, as a first step a sufficient condition for the validity of PES is derived for double diffusive binary viscoelastic fluid saturating an anisotropic porous medium.

Mathematical Formulation and Analysis:

Consider an infinite horizontal anisotropic porous layer saturated with a Boussinesq viscoelastic fluid mixture heated and salted underside confined between two horizontal boundaries and which are respectively maintained at uniform temperatures and and uniform concentrations and (see Fig.1). The modified Darcy law for the viscoelastic fluid of Oldroyd type is used to model the momentum equation Malashetty et al. (2006).

Fig.1. Physical configuration of the problem.

CONCLUSION:

Linear stability theory is used to derive a sufficient condition for the validity of ‘the principle of the exchange of stabilities’ in double- diffusive binary viscoelastic fluid in an anisotropic porous medium. It is further proved that this result is uniformly valid for any combination of rigid and / or free boundaries.

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Received on 28.08.2016 Accepted on 13.09.2016

Int. J. Tech. 2016; 6(2): 223-226.

DOI: 10.5958/2231-3915.2016.00034.1