An analytical study of linear and non-linear double diffusive convection with Soret and Dufour effects in couple stress fluid
Introduction
If the gradients of two stratifying agencies, such as heat and salt, having different diffusivities are simultaneously present in a fluid layer, a variety of interesting convective phenomena can occur which are not possible in a single component fluid. Convection in a fluid layer with two or more stratifying agencies has been the subject of extensive theoretical and experimental investigations in the last few decades. Excellent reviews of these studies have been reported by Turner [1], [2], [3], Huppert and Turner [4] and Platten and Legros [5]. The interest in the study of two or multi-component convection has developed as a result of the marked difference between single component and multi-component systems. In contrast to single component system, convection sets is even when density decreases with height, that is, when the basic state is hydrostatically stable. The double diffusive convection is of importance in various fields such as high quality crystal production, liquid gas storage, oceanography, production of pure medication, solidification of molten alloys, and geothermally heated lakes and magmas.
Convection in a two-component fluid is characterized by well-mixed convecting layers, which are separated by relatively sharp density steps. These steps may be of the ‘finger’ or ‘diffusive’ kind and both types of interface must enable a net release of potential energy preferentially transporting the destabilizing property. Salt fingers will occur when warm salty fluid overlies cooler fresher fluid and diffusive instability will occur when warm salty fluid underlies the fresh cooler fluid. In two-component system, in the absence of cross-diffusion, instability can occur only if, at least one of the components is destabilizing. However, in the presence of cross-diffusions produced by the simultaneous interference of two transport properties e.g., Soret and Dufour effects the situations may be quite different [6], [7], [8].
Typically, the energy transport is described adequately by Fourier diffusion and the mass transport by Fickian diffusion alone. Otherwise, several investigators [9], [10], [11], [12] have shown both analytically and experimentally that both Soret and Dufour effects can be important contributions to the total mass and energy transfer, respectively. The thermal-diffusion (Soret) effect, for instance, has been utilized for isotope separation, and in mixture between gases with very light molecular weight (, He) and of medium molecular weight (, air) the diffusion-thermo (Dufour) effects was found to be of order of considerable magnitude such that it cannot be ignored [13]. In view of the importance of above-mentioned effects Atimtay and Gill [14] have shown that Soret and Dufour diffusion to be appreciable for convection on a rotating disc. Weaver and Viskanta [15] studied the influence of species interdiffusion, Soret and Dufour effects on the natural convection heat and mass transfer in a cavity due to combined temperature and concentration gradients. They have shown that contributions to the total mass flux through the cavity due to Soret diffusion can be as much as 10–15% and energy transfer due to Dufour effects can be appreciable compared to heat conduction. Kafoussias and Williams [16] studied thermal-diffusion and diffusion-thermoeffects on mixed free-forced convective and mass transfer boundary layer flow with temperature-dependent viscosity. Thermal convection in a binary fluid driven by the Soret and Dufour effects has been investigated by Knobloch [17]. He has shown that equations are identical to the thermosolutal problem except for a relation between the thermal and solute Rayleigh numbers. Mc Dougall [6] has made an in-depth study of double diffusive convection caused by molecular diffusion in a solute–solute pair for which both Soret and Dufour effects are important. Of particular interest, crystal growth from the vapour is sometimes carried out under conditions conducive to Soret and Dufour effects. As greater demands are made for tighter control of industrial process, second order effects such as Soret and Dufour diffusion must be considered. Because of the limited number of studies available, the knowledge concerning the influence of these effects on the heat and mass transfer and fluid flow is incomplete.
With the growing importance of non-Newtonian fluids in modern technology and industries, the investigations on such fluids are desirable. During recent years the theory of polar fluids has received much attention and this is because the traditional Newtonian fluids cannot precisely describe the characteristics of the fluid flow with suspended particles. The study of such fluids have applications in a number of process that occur in industry such as the extrusion of polymer fluids, solidification of liquid crystals, cooling of metallic plate in a bath, exotic lubricants and colloidal and suspension solutions. In the category of non-Newtonian fluids couple stress fluid has distinct features, such as polar effects. The theory of polar fluids and related theories are models for fluids whose microstructure is mechanically significant. The constitutive equations for couple stress fluids were given by Stokes [18]. The theory proposed by Stokes [18] is the simplest one for microfluids, which allows polar effect such as the presence of couple stress, body couples and non-symmetric tensors. Couple stresses are found to appear in noticeable magnitude in fluids with very large molecules. Rayleigh–Benard convection in fluids with stress non-linearly proportional to velocity gradient is studied by few researchers (see e.g., [19], [20], [21]). However, the study on double diffusive convection in couple stress fluids is not available to the authors’ knowledge. Therefore, in the present paper we investigate the effects of Soret coefficient and Dufour coefficient on the onset of double diffusive convection using the Stokes’ [18] couple stress model. The important objective of the study is to perform a non-linear stability analysis of the problem using the minimal representation of Fourier series to compute heat and mass transports.
Section snippets
Basic equations
The constitutive equations for couple stress fluids are listed below [18]. The stress tensor can be decomposed into symmetric and antisymmetric parts, which are given byand the coupled stress tensorwhere the deformation tensorthe velocity tensorandHere is the velocity field, and are material constants with the dimension of viscosity, and and are
Linear stability theory
In this section, we discuss the linear stability analysis, which is very useful in the local non-linear stability analysis discussed in the next section. To study this we neglect the Jacobians in Eqs. (2.23)–(2.25) and assume the solutions to be periodic waves of the formwhere is the growth rate and in general a complex quantity (, is horizontal wavenumber. Substituting Eq. (3.1) in the linearized version of Eqs. (2.23)–(2.25), we get
Non-linear theory
The finite amplitude analysis is carried out here via truncated Fourier series representation for the stream function , temperature T and concentration S in the formwhere the amplitudes , and are to be determined from the dynamics of the system.
Substituting Eqs. (4.1)–(4.3) into Eqs. (2.23)–(2.25) and equating the coefficients of like terms we obtain the following non-linear
Results and discussions
The onset of double diffusive convection in a couple stress fluid layer in the presence of Soret and Dufour effects is investigated using linear theory. The linear theory provides the condition for the onset of stationary and oscillatory convection. The non-linear theory, which is based on the truncated representation of Fourier series, gives information about quantity of heat and mass transfer. In the present case the behavior of the system as a function of depends upon the diffusivity
Conclusions
The onset of double diffusive convection in a two component couple stress fluid layer with Soret and Dufour effects has been studied using both linear and non-linear stability analysis. The linear theory depends on normal mode technique and non-linear analysis depends on a minimal representation of double Fourier series. The following conclusions are drawn:
- 1.
The Dufour parameter enhances the stability of the couple stress fluid system in case of both stationary and oscillatory mode.
- 2.
The effect of
Acknowledgment
This work is supported by UGC New Delhi, under the Special Assistance Programme DRS. The authors thank the reviewer for his critical comments and useful suggestions which have led to the improvements in the paper.
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