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The onset of double diffusive convection is investigated in a Maxwell fluid saturated porous layer with internal heat source. The modified Darcy law for the Maxwell fluid is used to model the momentum equation of the system, and the criterion for the onset of the convection is established through the linear and nonlinear stability analyses. The linear analysis is obtained using the normal mode technique, and the nonlinear analysis of the system is studied with the help of truncated representation of Fourier series. The effects of internal Rayleigh number, stress relaxation parameter, normalized porosity, Lewis number, Vadasz number and solute Rayleigh number on the stationary, and oscillatory and weak nonlinear convection of the system are shown numerically and graphically. The effects of various parameters on transient heat and mass transfer are also discussed and presented analytically and graphically.

Double diffusive convection in porous media without heat source has been intensively studied because of its application in different branches of science and engineering, such as underground disposal of nuclear wastes, groundwater pollution, contaminant transport in fluid-saturated soils, liquid gas storage, and food processing [

The onset of thermodynamic instability in horizontal porous layer saturated with Newtonian fluid was first studied extensively on geological and engineering length scales [

Recently, viscoelastic fluid flow in porous media has attracted considerable attention, due to the large demands of such diverse fields as biorheology, geophysics, chemical industries, and petroleum industries. Wang and Tan have made the stability analysis of double diffusive convection in a Maxwell fluid saturated porous medium [

In this paper, we focus on the linear and weakly nonlinear stability analyses in a viscoelastic fluid saturated porous layer with internal heat source using the Darcy-Maxwell model. The Dufour and Soret effects are ignored. The aim of the present paper is to study how onset criteria for stationary and oscillatory double diffusive convection are affected by the viscoelastic parameter and other parameters, as well as discussing their effects on heat and mass transfer.

Assuming that an infinite horizontal porous layer saturated with Maxwell fluid mixture with internal heat source, confined between the planes,

Considering the vertically downward gravity force

The basic state of the fluid is assumed to be quiescent, and the quantities of the basic state are given by

In this section, we discuss the linear stability analysis. According to the normal mode analysis [

For the validity of principle of exchange of stabilities (i.e., steady case), we have

For oscillatory onset,

In this section, we study the nonlinear stability analysis using minimal truncated Fourier series. For simplicity, we confine ourselves to two-dimensional rolls, so that all the physical quantities are independent of

The simplified model represented by (

In the study of convection problems, the determination of heat transfer and mass transport play a very important role. Let

The linear stability analysis of double diffusive convection in a binary Maxwell fluid saturated porous layer with internal heat source has been studied analytically. In this section, we discuss the effects of the parameters in the governing equations on the onset of the double diffusive convection numerically and graphically. Figure

Variation of

The weak nonlinear analysis provides the quantification of heat and mass transport. The effects of various parameters on the rate of heat and mass transfer are shown in Figure

Variation in Nusselt number

Using the Runge-Kutta method with suitable initial conditions, we solve the autonomous system numerically given by (

Variation in Nusselt number

Variation in Nusselt number

Linear and nonlinear analysis of double diffusive convection in a Maxwell fluid saturated porous layer with internal heat source, which is heated and salted from below, is investigated analytically and numerically. The linear analysis is analyzed using the normal mode technique. On the other hand, the nonlinear analysis of the system is established through a truncated form of the Fourier series. The effects of physical parameters in governing equations, such as relaxation time, Lewis number, normalized porosity parameter, Vadasz number, solutal Rayleigh number, and internal Rayleigh number, on stationary, oscillatory convection, and heat and mass transfer are shown graphically and the following conclusions are drawn: Vadasz number

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

This work is supported by the National Natural Science Foundation of China (nos. 11002083, 51279093, and 41172268) and the National Basic Research Program of China (2013CB036000).