Influence of chemical reaction on heat and mass transfer by natural convection from vertical surfaces in porous media considering Soret and Dufour effects

The heat and mass transfer characteristics of natural convection about a vertical surface embedded in a saturated porous medium subjected to a chemical reaction is numerically analyzed, by taking into account the diffusion-thermo (Dufour) and thermal-diffusion (Soret) effects. The transformed governing equations are solved by a very efficient numerical method, namely, a modified version of the Keller-box method for ordinary differential equations. The parameters of the problem are Lewis, Dufour and Soret numbers, sustentation parameter, the order of the chemical reaction n and the chemical reaction parameter γ. Local Nusselt number and local Sherwood number variations and dimensionless concentration profiles in the boundary layer are presented graphically and in tables for various values of problem parameters and it is concluded that γ and n play a crucial role in the solution.     

Soret and Dufour effects on heat and mass transfer by mixed convection over a vertical surface saturated porous medium with temperature dependent viscosity

The diffusion‐thermo and thermal‐diffusion effects on heat and mass transfer by mixed convection boundary layer flow over a vertical isothermal permeable surface embedded in a porous medium were studied numerically in the presence of chemical reaction with temperature‐dependent viscosity. The governing nonlinear partial differential equations are transformed into a set of coupled ordinary differential equations, which are solved numerically by using Runge–Kutta method with shooting technique. Numerical results are obtained for the velocity, temperature and concentration distributions, and the local skin friction coefficient, local Nusselt number and local Sherwood number for several values of the parameters, namely, the variable viscosity parameter, suction/injection parameter, Darcy number, chemical reaction parameter, and Dufour and Soret numbers. The obtained results are presented graphically and in tabulated form, and the physical aspects of the problem are discussed. Copyright © 2011 John Wiley & Sons, Ltd.     

MHD free convective flow and mass transfer over a stretching sheet with chemical reaction

The effect of chemical reaction on free convective flow and mass transfer of a viscous, incompressible and electrically conducting fluid over a stretching surface is investigated in the presence of a constant transverse magnetic field. The non-linear boundary layer equations with the boundary conditions are transferred by a similarity transformation into a system of non-linear ordinary differential equations with the appropriate boundary conditions. Furthermore, the similarity equations are solved numerically by using a fourth order Runge-Kutta scheme with the shooting method. Numerical results of the skin friction coefficient, the local Nusselt number Nu, the local Sherwood number Sh, as will as the velocity, temperature and concentration profiles are presented for gases with a Prandtl number of 0.71 for various values of chemical reaction parameter, order of reaction, magnetic parameter and Schmidt number.     

Soret and dufour effects on MHD free convective heat and mass transfer with thermophoresis and chemical reaction over a porous stretching surface: Group theory transformation

The group theoretic method is applied for solving the problem of the combined influence of the thermal diffusion and diffusion thermoeffect on magnetohydrodynamic free convective heat and mass transfer over a porous stretching surface in the presence of thermophoresis particle deposition with variable stream conditions. The application of one-parameter groups reduces the number of independent variables by one; consequently, the system of governing partial differential equations with boundary conditions reduces to a system of ordinary differential equations with appropriate boundary conditions. The equations along with the boundary conditions are solved numerically by using the Runge-Kutta-Gill integration scheme with the shooting technique. The impact of the Soret and Dufour effects in the presence of thermophoresis particle deposition with a chemical reaction plays an important role on the flow field.     

Linear Stability of the Double-Diffusive Convection in a Horizontal Porous Layer with Open Top: Soret and Viscous Dissipation Effects

The linear stability of the double-diffusive convection in a horizontal porous layer is studied considering the upper boundary to be open. A horizontal temperature gradient is applied along the upper boundary. It is assumed that the viscous dissipation and Soret effect are significant in the medium. The governing parameters are horizontal Rayleigh number (\(Ra_\mathrm{H}\)), solutal Rayleigh number (\(Ra_\mathrm{S}\)), Lewis number (Le), Gebhart number (Ge) and Soret parameter (Sr). The Rayleigh number (Ra) corresponding to the applied heat flux at the bottom boundary is considered as the eigenvalue. The influence of the solutal gradient caused due to the thermal diffusion on the double-diffusive instability is investigated by varying the Soret parameter. A horizontal basic flow is induced by the applied horizontal temperature gradient. The stability of this basic flow is analyzed by calculating the critical Rayleigh number (\(Ra_\mathrm{cr}\)) using the Runge–Kutta scheme accompanied by the Shooting method. The longitudinal rolls are more unstable except for some special cases. The Soret parameter has a significant effect on the stability of the flow when the upper boundary is at constant pressure. The critical Rayleigh number is decreasing in the presence of viscous dissipation except for some positive values of the Soret parameter. How a change in Soret parameter is attributing to the convective rolls is presented.     

Double-Diffusive Free Convection Flow Past an Inclined Plate Embedded in a Non-Darcy Porous Medium Saturated with a Nanofluid

In this investigation, we intend to present the influence of the prominent Soret effect on double-diffusive free convection heat and mass transfer in the boundary layer region of a semi-infinite inclined flat plate in a nanofluid saturated non-Darcy porous medium. The transformed boundary layer ordinary differential equations are solved numerically using the shooting and matching technique. Consideration of the nanofluid and the coupled convective process enhanced the number of non-dimensional parameters considerably thereby increasing the complexity of the present problem. A wide range of parameter values are chosen to bring out the effect of Soret parameter on the free convection process with varying angle of inclinations making the wall geometry from vertical to horizontal plate. The effects of angle of inclination and Soret parameter on the flow, heat and mass transfer coefficients are analyzed. The numerical results obtained for the velocity, temperature, volume fraction, and concentration profiles, local wall temperature, local nanoparticle concentration, and local wall concentration reveal interesting phenomenon, and some of these qualitative results are presented through the plots.     

MHD flow and mass transfer of chemically reactive upper convected Maxwell fluid past porous surface

The magnetohydrodynamic (MHD) flow and mass transfer of an electrically conducting upper convected Maxwell (UCM) fluid at a porous surface are studied in the presence of a chemically reactive species. The governing nonlinear partial differential equations along with the appropriate boundary conditions are transformed into nonlinear ordinary differential equations and numerically solved by the Keller-box method. The effects of various physical parameters on the flow and mass transfer characteristics are graphically presented and discussed. It is observed that the order of the chemical reaction is to increase the thickness of the diffusion boundary layer. Also, the mass transfer rate strongly depends on the Schmidt number and the reaction rate parameter. Furthermore, available results in the literature are obtained as a special case.     

Similarity solution of MHD boundary layer flow with diffusion and chemical reaction over a porous flat plate with suction/blowing

Similarity analysis of diffusion of chemically reactive solute distribution in MHD boundary layer flow of an electrically conducting incompressible fluid over a porous flat plate is presented. The reaction rate of the solute is considered inversely proportional along the plate. Adopting the similarity transformation technique the governing equations are converted into the self-similar ordinary differential equations which are solved by shooting procedure using Runge-Kutta method. For increase of the Schmidt number the solute boundary layer thickness is reduced. Most importantly, the effects of reaction rate and order of reaction on concentration field are of conflicting natures, due to increasing reaction rate parameter the concentration decreases, but for the increase in order of reaction it increases. In presence of chemical reaction, the concentration profiles attain negative value when Schmidt number is large.     

On the diffusion of a chemically reactive species in a convective flow past a vertical plate

A numerical solution of the first-order homogeneous chemical reaction in an unsteady free convective flow past a semi-infinite vertical plate is studied. The dimensionless governing equations are solved by an efficient, more accurate, unconditionally stable, and rapidly converging implicit finite-difference scheme. The effect of various parameters, such as the Prandtl number, Schmidt number, buoyancy ratio parameter, and chemical reaction parameter on flow velocity and temperature is determined. The velocity profiles are in excellent agreement with available results in the literature. The local and average values of skin friction and Nusselt and Sherwood numbers are calculated. The effects of the chemical reaction parameters on these values are discussed for both generative and destructive reactions. Owing to the presence of the first-order chemical reaction, the velocity is found to increase in the generative reaction and to decrease in the destructive reaction.     

Numerical investigation of Dufour and Soret effects on unsteady MHD natural convection flow past vertical plate embedded in non-Darcy porous medium

The Dufour and Soret effects on the unsteady two-dimensional magnetohydro-dynamics(MHD) double-diffusive free convective flow of an electrically conducting fluidpast a vertical plate embedded in a non-Darcy porous medium are investigated numeri-cally.The governing non-linear dimensionless equations are solved by an implicit finitedifference scheme of the Crank-Nicolson type with a tridiagonal matrix manipulation.The effects of various parameters entering into the problem on the unsteady dimension-less velocity,temperature,and concentration profiles are studied in detail.Furthermore,the time variation of the skin friction coefficient,the Nusselt number,and the Sherwoodnumber is presented and analyzed.The results show that the unsteady velocity,tem-perature,and concentration profiles are substantially influenced by the Dufour and Soreteffects.When the Dufour number increases or the Soret number decreases,both the skinfriction and the Sherwood number decrease,while the Nusselt number increases.It isfound that,when the magnetic parameter increases,the velocity and the temperaturedecrease in the boundary layer.     

MHD mixed convection–radiation interaction along a permeable surface immersed in a porous medium in the presence of Soret and Dufour’s Effects

This work is focused on the numerical modeling of steady, laminar, heat and mass transfer by MHD mixed convection from a semi-infinite, isothermal, vertical and permeable surface immersed in a uniform porous medium in the presence of thermal radiation and Dufour and Soret effects. A mixed convection parameter for the entire range of free-forced-mixed convection is employed and the governing equations are transformed into non-similar equations. These equations are solved numerically by an efficient, implicit, iterative, finite-difference scheme. The obtained results are checked against previously published work on special cases of the problem and are found to be in excellent agreement. A parametric study illustrating the influence of the thermal radiation coefficient, magnetic field, porous medium inertia parameter, concentration to thermal buoyancy ratio, and the Dufour and Soret numbers on the fluid velocity, temperature and concentration as well as the local Nusselt and the Sherwood numbers is conducted. The obtained results are shown graphically and the physical aspects of the problem are discussed.     

Dufour and Soret effects on MHD flow of viscous fluid between radially stretching sheets in porous medium

The aim of this paper is to examine the Dufour and Soret effects on the two-dimensional magnetohydrodynamic (MHD) steady flow of an electrically conducting viscous fluid bounded by infinite sheets. An incompressible viscous fluid fills the porous space. The mathematical analysis is performed in the presence of viscous dissipation, Joule heating, and a first-order chemical reaction. With suitable transformations, the governing partial differential equations through momentum, energy, and concentration laws are transformed into ordinary differential equations. The resulting equations are solved by the homotopy analysis method (HAM). The convergence of the series solutions is ensured. The effects of the emerging parameters, the skin friction coefficient, the Nusselt number, and the Sherwood number are analyzed on the dimensionless velocities, temperature, and concentration fields.     

Similarity solutions for MHD thermosolutal Marangoni convection over a flat surface in the presence of heat generation or absorption effects

The problem of steady, laminar, thermosolutal Marangoni convection flow of an electrically-conducting fluid along a vertical permeable surface in the presence of a magnetic field, heat generation or absorption and a first-order chemical reaction effects is studied numerically. The general governing partial differential equations are converted into a set of self-similar equations using unique similarity transformations. Numerical solution of the similarity equations is performed using an implicit, iterative, tri-diagonal finite-difference method. Comparisons with previously published work is performed and the results are found to be in excellent agreement. Approximate analytical results for the temperature and concentration profiles as well as the local Nusselt and sherwood numbers are obtained for the conditions of small and large Prandtl and Schmidt numbers are obtained and favorably compared with the numerical solutions. The effects of Hartmann number, heat generation or absorption coefficient, the suction or injection parameter, the thermo-solutal surface tension ratio and the chemical reaction coefficient on the velocity, temperature and concentration profiles as well as quantitites related to the wall velocity, boundary-layer mass flow rate and the Nusselt and Sherwood numbers are presented in graphical and tabular form and discussed. It is found that a first-order chemical reaction increases all of the wall velocity, Nusselt and Sherwood numbers while it decreases the mass flow rate in the boundary layer. Also, as the thermo-solutal surface tension ratio is increased, all of the wall velocity, boundary-layer mass flow rate and the Nusselt and Sherwood numbers are predicted to increase. However, the exact opposite behavior is predicted as the magnetic field strength is increased.     

Effects of surface mass transfer on unsteady mixed convection flow over a vertical cone with chemical reaction

The unsteady mixed convection boundary layer flow over a vertical cone is considered to investigate the combined effects of the buoyancy force, thermal and mass diffusion in the presence of the first order chemical reaction and surface mass transfer. The unsteadiness is caused by the time dependent free stream velocity varying arbitrarily with time. The governing boundary layer equations are transformed into a non-dimensional form by a group of non-similar transformations. The resulting system of coupled non-linear partial differential equations is solved numerically by the combination of quasi-linearization technique and an implicit finite difference scheme. Numerical computations are performed for different values of the parameters to display the velocity, temperature and concentration profiles graphically. Both accelerating and decelerating free stream velocities are considered. Numerical results are presented for the velocity, temperature and concentration profiles as well as for the skin-friction coefficient, local Nusselt number and local Sherwood number. The obtained results are compared with previously reported ones and are found to be in excellent agreement.     

Melting heat transfer effects on stagnation point flow of micropolar fluid saturated in porous medium with internal heat generation （absorption）

The effect of melting heat transfer on the two dimensional boundary layer flow of a micropolar fluid near a stagnation point embedded in a porous medium in the presence of internal heat generation/absorption is investigated. The governing non-linear partial differential equations describing the problem are reduced to a system of non-linear ordinary differential equations using similarity transformations solved numerically using the Chebyshev spectral method. Numerical results for velocity, angular velocity and temperature profiles are shown graphically and discussed for different values of the inverse Darcy number, the heat generation/absorption parameter, and the melting parameter. The effects of the pertinent parameters on the local skin-friction coefficient, the wall couple stress, and the local Nusselt number are tabulated and discussed. The results show that the inverse Darcy number has the effect of enhancing both velocity and temperature and suppressing angular velocity. It is also found that the local skin-friction coefficient decreases, while the local Nusselt number increases as the melting parameter increases.     

Computational modeling of heat transfer over an unsteady stretching surface embedded in a porous medium

The present paper deals with the study of heat transfer characteristics in the laminar boundary layer flow of an incompressible viscous fluid over an unsteady stretching sheet which is placed in a porous medium in the presence of viscous dissipation and internal absorption or generation. Similarity transformations are used to convert the governing time dependent nonlinear boundary layer equations into a system of non-linear ordinary differential equations containing Prandtl number, Eckert number, heat source/sink parameter, porous parameter and unsteadiness parameter with appropriate boundary conditions. These equations are solved numerically by applying shooting method using Runge-Kutta-Fehlberg method. Comparison of numerical results is made with the earlier published results under limiting cases. The effects of the parameters which determine the velocity and temperature fields are discussed in detail.     

Local non-similarity solution to impact of chemical reaction on MHD mixed convection heat and mass transfer flow over porous wedge in the presence of suction / injection

Combined heat and mass transfer on free, forced, and mixed convection flow along a porous wedge with magnetic effect in the presence of chemical reaction is investigated. The flow field characteristics are analyzed by the Runge-Kutta-Gill scheme with the shooting method as well as the local non-similarity method up to the third level of truncation, which are used to reduce the governing partial differential equations into nine ordinary differential equations. The governing boundary layer equations are converted to a dimensionless form by Falkner-Skan transformations. Because of the effect of suction/injection on the wall of the wedge with buoyancy force and variable wall temperature, the flow field is locally non-similar. Numerical calculations up to the third order level of truncation are carried out as a special case for different values of dimensionless parameters. Effects of the magnetic field strength in the presence of chemical reaction with variable wall temperature and concentration on the dimensionless velocity, temperature and concentration profiles are shown graphically.     

Stability analysis of soret-driven double-diffusive convection of Maxwell fluid in a porous medium

Stability analysis of double-diffusive convection for viscoelastic fluid with Soret effect in a porous medium is investigated using a modified-Maxwell-Darcy model. We use the linear stability analysis to investigate how the Soret parameter and the relaxation time of viscoelastic fluid effect the onset of convection and the selection of an unstable wavenumber. It is found that the Soret effect is to destabilize the system for oscillatory convection. The relaxation time also enhances the instability of the system. The effects of Soret coefficient and relaxation time on the heat transfer rate in a porous medium are studied using the nonlinear stability analysis, the variation of the Nusselt number with respect to the Rayleigh number is derived for stationary and oscillatory convection modes. Some previous results can be reduced as the special cases of the present paper.     

Locally similar solutions for hydromagnetic and thermal slip flow boundary layers over a flat plate with variable fluid properties and convective surface boundary condition

This paper presents heat transfer process in a two-dimensional steady hydromagnetic convective flow of an electrically conducting fluid over a flat plate with partial slip at the surface of the boundary subjected to the convective surface heat flux at the boundary. The analysis accounts for both temperature-dependent viscosity and temperature dependent thermal conductivity. The local similarity equations are derived and solved numerically using the Nachtsheim-Swigert iteration procedure. Results for the dimensionless velocity, temperature and ambient Prandtl number within the boundary layer are displayed graphically delineating the effect of various parameters characterizing the flow. The results show that momentum boundary layer thickness significantly depends on the surface convection parameter, Hartmann number and on the sign of the variable viscosity parameter. The results also show that plate surface temperature is higher when there is no slip at the plate compared to its presence. For both slip and no-slip cases surface temperature of the plate can be controlled by controlling the strength of the applied magnetic field. In modelling the thermal boundary layer flow with variable viscosity and variable thermal conductivity, the Prandtl number must be treated as a variable irrespective of flow conditions whether there is slip or no-slip at the boundary to obtain realistic results.     

Impact of thermophoresis particle deposition and chemical reaction on unsteady non-Darcy mixed convective flow over a porous wedge in the presence of temperature-dependent viscosity

An analysis is presented to investigate the effect of thermophoresis particle deposition and temperature dependent viscosity on unsteady non-Darcy mixed convective heat and mass transfer of a viscous and incompressible fluid past a porous wedge in the presence of chemical reaction. The wall of the wedge is embedded in a uniform non-Darcian porous medium in order to allow for possible fluid wall suction or injection. The governing partial differential equations of the problem, subjected to their boundary conditions, are solved numerically by applying an efficient solution scheme for local nonsimilarity boundary layer analysis. Numerical calculations are carried out for different values of dimensionless parameters arising in the problem. The results are compared with available ones in the literature and excellent agreement is obtained. An analysis of the obtained results shows that the flow field is influenced appreciably by the chemical reaction and thermophoresis particle deposition.