Dufour-Soret Driven Double-Diffusive Convection in Nonreactive Binary Fluids with Temperature Dependent Viscosity

Joginder S. Dhiman¹*, Praveen K. Sharma², Poonam Sharma¹

¹Department of Mathematics, Himachal Pradesh University, Summerhill, Shimla-171005, INDIA.

²UIIT, Himachal Pradesh University, Summerhill, Shimla-171005, INDIA.

*Corresponding Author Email: dhiman_jp@yahoo.com

ABSTRACT:

The aim of the present paper is to investigate the effects of temperature dependent viscosity on the stability of Dufour-Soret driven double-diffusive convection in nonreactive binary fluids, for general boundary conditions. Some analytical results concerning the stability of oscillatory motions and otherwise the complex growth rate of oscillatory motions are derived for this general problem. The expressions for Rayleigh numbers, when instability sets in as stationary modes, for each combination of rigid(slip free) and dynamically free(stress free) boundary conditions are derived numerically using Galerkin’s method. The effects of linear and exponential temperature dependent viscosity on the onset stationary convection are studied and computed numerically. Various consequences of the obtained results for different convective problems have been worked out, as special cases.

KEYWORDS: Binary fluids; Dufour-Soret effects; Double-diffusive convection; Principle of exchange of stabilities; Galerkin's technique; Rayleigh number.

1. INTRODUCTION:

Double-diffusive convection becomes complicated when the diffusivity of one of the diffusing component is much greater than the other. When two transport processes take place simultaneously, they interfere with each other, producing cross-diffusion effects and these cross-diffusion effects are significant in the stability of the system. The flux of concentration caused by temperature gradient and the flux of heat caused by concentration gradient are known as Soret and Dufour effects, respectively. Mojtabi and Charrier-Mojtabi [1] reported that the phenomenon of cross-diffusion is complicated as accurate values of the cross-diffusion coefficients are not available. However, they have a large influence on hydrodynamic stability relative to their contributions to the buoyancy of the fluid. Comprehensive reviews on this area have been made by Takashima [2], Knobloch [3] and Nield and Bejan [4].

It is well known fact, that the viscosity of a fluid is one of the properties which are most sensitive to temperature (cf. Straughan [5]). In the majority of the cases, viscosity becomes the only property which may have considerable effect on the heat transfer and the temperature variation and the dependence of other thermo-physical properties to temperature is often negligible. Torrance & Turcotte [6] observed that the viscosity of the liquids decreases with increasing temperature, while the reverse trend is observed in gases. Heat transfer and pressure drop characteristics are also affected significantly with the variations in the fluid viscosity. Therefore, the type of fluid and the range of operating temperatures are very important and crucial parameters in the study of fluid dynamics, and in particular in the fields of oceanography, astrophysics etc. In most of the studies pertaining to single or double-diffusive convection, authors have considered the viscosity of the fluid to be constant. Because, when the viscosity of the fluid is varying with temperature in convective instability problems, then the eigenvalue problems results in the forms of differential equations with variable coefficients, which introduces additional mathematical complexities. Some authors, including Trompert and Hansen [7], Stengel et al. [8] and Dhiman and Kumar [9]) have considered the effect of variable viscosity on the thermal convection problems. Gupta and Kaushal [10] have also analyzed the stability of double-diffusive convection problems of Veronis and Stern types, by taking into account the variations in viscosity. However, these results are of limited utility, because these do not involves the impersonation of the viscosity variation. Dhiman and Kumar [11] studied the effect of temperature dependent viscosity on the onset of instability in thermohaline convection problems of Veronis and Stern type configurations, using linear stability theory and derived some general qualitative and quantitative results and discussed the effect of temperature dependent viscosity on the onset of stationary convection for each combination of rigid(slip free) and dynamically free(stress free) boundary conditions.

The aim of the present paper is to investigate the effects of temperature dependent viscosity on the stability of Dufour-Soret driven double-diffusive convection in nonreactive binary fluids, for general boundary conditions. The disparate behavior of the eigenvalue problem arising due to the presence of coupling due to cross diffusions has been arrested by the utilizing some linear transformations. Some analytical results concerning the stability of oscillatory motions and otherwise the complex growth rate of oscillatory motions are derived for this general problem. Further, the expressions for Rayleigh numbers, when instability sets in as stationary modes, for each combination of rigid and dynamically free boundary conditions are derived numerically using Galerkin’s method. The effects of linear and exponential temperature dependent viscosity on the onset stationary convection are studied and computed numerically. Various consequences of the obtained results for different convective problems have been worked out, as special cases.

2. CONCLUSIONS:

The present analysis establishes that the positive/negative values of the temperature dependent viscosity parameter has stabilizing/destabilizing effect on the onset stationary convection of Dufour-Soret driven double-diffusive convection in nonreactive binary fluids. It is observed from the values presented in Tables 1-4 that the values of the critical Rayleigh number increases with the increasing temperature dependent viscosity factor It is also observed that the exponential variation of temperature dependent viscosity has more stabilizing effect than the linear variation of viscosity in each case of boundary conditions.

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Received on 24.08.2016 Accepted on 09.09.2016

Int. J. Tech. 2016; 6(2): 161-169.

DOI: 10.5958/2231-3915.2016.00025.0