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Double diffusive convection in a horizontal layer of Maxwell viscoelastic fluid in a porous medium in the presence of temperature gradient (Soret effects) and concentration gradient (Dufour effects) is investigated. For the porous medium Darcy model is considered. A linear stability analysis based upon normal mode technique is used to study the onset of instabilities of the Maxwell viscolastic fluid layer confined between two free-free boundaries. Rayleigh number on the onset of stationary and oscillatory convection has been derived and graphs have been plotted to study the effects of the Dufour parameter, Soret parameter, Lewis number, and solutal Rayleigh number on stationary convection.

Bénard convection originated from the experimental works of Bénard [

Double-diffusive convection is referred to buoyancy-driven flows induced by combined temperature and concentration gradients. The onset of double diffusive convection in a fluid saturated in porous medium is regarded as a classical problem due to its wide range of applications in many engineering fields such as evaporative cooling of high temperature systems, agricultural product storage, soil sciences, enhanced oil recovery, packed-bed catalytic reactors, and the pollutant transport in underground. A detailed review of the literature concerning double diffusive convection in binary fluid in a porous medium was given by Nield and Bejan [

The study of natural convection of non-Newtonian fluids in a porous medium had gained much attention because of its engineering and industrial applications. These applications included design of chemical processing equipment, formation and dispersion of fog, distributions of temperature and moisture over agricultural fields and groves of fruit trees, and damage of crops due to freezing and pollution of the environment.

The fluids that show distinct deviation from “Newtonian hypothesis” (stress on fluid is linearly proportional to strain rate of fluid) are called non-Newtonian fluids. Different models had been proposed to explain the behavior of non-Newtonian fluids. Maxwell model is one of them. These fluids help us to understand the wide variety of fluids that exist in the physical world and characterized by power-law model. The work on viscoelastic fluid appears to be that of Herbert on plane coquette flow heated from below. He found a finite elastic stress in the undistributed state to be required for the elasticity to affect the stability. Using a three constants rheological model due to Oldroyd [

The importance of the study of viscoelastic fluids in a porous medium has been increasing for the last few years. This is mainly due to their applications in petroleum drilling, manufacturing of foods and paper, and many others. The problem of convective instability of viscoelastic fluid heated from below was first studied by Green [

In this paper an attempt has been made to study the Dufour and Soret effects on the onset of instability in a horizontal layer of Maxwell viscoelastic fluid in a porous medium.

Consider an infinite horizontal layer of Maxwell viscoelastic fluid of thickness “

Let

The mathematical equations describing the physical model are based upon the following assumptions.

Thermophysical properties expect for density in the buoyancy force (Boussinesq hypothesis) are constant.

Darcy’s model with time derivative is employed for the momentum equation.

The porous medium is assumed to be isotropic and homogeneous.

No chemical reaction takes place in a layer of fluid.

The fluid and solid matrix are in thermal equilibrium state.

Radiation heat transfer between the sides of the wall is negligible when compared with other modes of the heat transfer.

According to the works of Bhatia and Steiner [

We assume that temperature and concentration are constant at the boundaries of the fluid layer. Therefore, boundary conditions are

The steady state is given by

To study the stability of the system, we superimposed infinitesimal perturbations on the basic state, which are of the forms

Equation (

The nondimensional boundary conditions are

Analyze the disturbances into the normal modes and assume that the perturbed quantities are of the form

Using (

The boundary conditions are

Substituting solution (

The nontrivial solution of the above matrix requires that

The critical cell size at the onset of instability is obtained from the condition

This result is the same as obtained by Lapwood [

The corresponding critical Rayleigh number

If

The onset of double diffusive convection in a horizontal layer of Maxwell viscoelastic fluid in the presence of Soret and Dufour in a porous medium is investigated analytically and graphically. The expressions for both the stationary and oscillatory Rayleigh numbers, which characterize the stability of the system, are obtained analytically. The stationary critical Rayleigh number is found to be independent of the viscoelastic parameter

In order to investigate effects of the Dufour parameter

From (

Thus for the stationary convection Dufour parameter

Thus for stationary convection Soret parameter

Thus for stationary convection Lewis number Le has a stabilizing effect if

Now we discussed the effects of various parameters on the onset of double diffusive convection of Maxwell viscoelastic fluid in a porous medium for stationary convection graphically. The convection curves for solutal Rayleigh number Rs, Soret parameter

Variation of Rayleigh number Ra with wave number

Variation of Rayleigh number

Variation of Rayleigh number Ra with wave number

Variation of Rayleigh number Ra with wave number

Figure

Figure

Figure

Figure

Curves in Figures

A linear stability analysis of double diffusive convection in a horizontal layer of Maxwell viscoelastic fluid in the presence of Soret and Dufour in a porous medium is investigated analytically and graphically. The expressions for both the stationary and oscillatory Rayleigh numbers, which characterize the stability of the system, are obtained.

The main conclusions are as follows.

In stationary convection Maxwell viscoelastic fluid behaves like ordinary Newtonian fluid.

Dufour parameter, Soret parameter, and Lewis parameter have both stabilizing and destabilizing effects on the stationary convection.

Solutal Rayleigh number destabilizes the stationary convection.

In limiting case when Rs =

Solute concentration

Heat capacity

Dufour parameter

Dufour coefficient

Soret coefficient

Thickness of fluid layer

Stress relaxation parameter

Acceleration due to gravity

Medium permeability

Lewis number

Growth rate of disturbances

Pressure

Darcy fluid velocity

Thermal Rayleigh number

Critical Rayleigh number

Solutal Rayleigh number

Soret parameter

Time

Temperature

Components of fluid velocity

Space coordinates.

Density of fluid

Coefficient of thermal expansion

Analogous solvent coefficient of expansion

Viscosity

Thermal diffusivity

Solute diffusivity

Relaxation time

Curly operator

Porosity

Dimensionless frequency of oscillation

Thermal capacity ratio.

Nondimensional variables

Perturbed quantity.

Value of variables at lower boundary

Value of variables at upper boundary

Steady state

Fluid

Porous medium.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors are grateful to the reviewers for their lucid comments and suggestions which have served to improve the research paper.