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.     

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.     

An analytical study of linear and non-linear double diffusive convection with Soret and Dufour effects in couple stress fluid

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 effect of couple stress parameter, the Soret and Dufour parameters, and the Prandtl number on the stationary and oscillatory convection are presented graphically. The Dufour parameter enhances the stability of the couple stress fluid system in case of both stationary and oscillatory mode. The effect of positive Soret parameter is to destabilize the system in case of stationary mode while it stabilizes the system in case of oscillatory mode. The negative Soret parameter enhances the stability in both stationary and oscillatory mode. The couple stress parameter enhances the stability of the system in both stationary and oscillatory modes. The Dufour parameter increases the heat transfer while the couple stress parameter has reverse effect. The Soret parameter has negligible influence on heat transfer. Both Dufour and Soret parameters increases the mass transfer while the couple stress parameter has dual effect depending on the value of the Rayleigh number.     

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.     

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 Double-Diffusive Free Convection From a Corrugated Vertical Surface in a Non-Darcy Porous Medium

Combined heat and mass transfer process by natural convection from a wavy vertical surface immersed in a fluid-saturated semi-infinite porous medium due to Soret and Dufour effects for Forchheimer extended non-Darcy model has been analyzed. A similarity transformation followed by a wavy to flat surface transformation is applied to the governing coupled non-linear partial differential equations, and they are reduced to boundary layer equations. The obtained boundary layer equations are solved by finite difference scheme based on the Keller-Box approach in conjunction with block-tridiagonal solver. Detailed simulations are carried out for a wide range of parameters like Groshof number (Gr*), Lewis number (Le), Buoyancy ratio (B), Wavy wall amplitude (a), Soret number (S _r ), and Dufour number (D _f ). Comparison tables local and average Nusselt (Nu) number, local and average Sherwood (Sh) number plots are presented.     

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.     

Stationary Fingering Instability in a Non-equilibrium Porous Medium with Coupled Molecular Diffusion

Finger type double diffusive convective instability in a fluid-saturated porous medium is studied in the presence of coupled heat-solute diffusion. A local thermal non-equilibrium (LTNE) condition is invoked to model the Darcian porous medium which takes into account the energy transfer between the fluid and solid phases. Linear stability theory is implemented to compute the critical thermal Rayleigh number and the corresponding wavenumber exactly for the onset of stationary convection. The effects of Soret and Dufour cross-diffusion parameters, inter-phase heat transfer coefficient and porosity modified conductivity ratio on the instability of the system are investigated. The analysis shows that positive Soret mass flux triggers instability and positive Dufour energy flux enhances stability whereas their combined influence depends on the product of solutal Rayleigh number and Lewis number. It also reveals that cell width at the convection threshold gets affected only in the presence of both the cross-diffusion fluxes. Besides, asymptotic solutions for both small and large values of the inter-phase heat transfer coefficient and porosity modified conductivity ratio are found. An excellent agreement is found between the exact and asymptotic solutions.     

Soret and Dufour Effects on Double-Diffusive Free Convection Over a Vertical Truncated Cone in Porous Media with Variable Wall Heat and Mass Fluxes

This work studies the Soret and Dufour effects on the double-diffusive free convection over a downward-pointing vertical truncated cone with variable wall heat and mass fluxes in fluid-saturated porous media. A coordinate transformation is used to derive the nondimensional boundary-layer governing equations, and the obtained nonsimilar equations are then solved by the cubic spline collocation method. Results for local surface temperature and the local surface concentration are presented as functions of Soret parameters, Dufour parameters, power-law exponents, buoyancy ratios, and Lewis numbers. Results show that increasing the Dufour parameter tends to increase the local surface temperature, while it tends to decrease the local surface concentration. An increase in the Soret number leads to a decrease in the local surface temperature for buoyancy assisting flows, while it leads to an increase in the local surface temperature for buoyancy opposing flows. Increasing the Soret number tends to increase the local surface concentration. Moreover, the local surface temperature and the local surface concentration of the truncated cones with higher power-law exponents are lower than those with lower exponents.     

Steady mixed convection flow of Maxwell fluid over an exponentially stretching vertical surface with magnetic field and viscous dissipation

The steady mixed convection flow and heat transfer from an exponentially stretching vertical surface in a quiescent Maxwell fluid in the presence of magnetic field, viscous dissipation and Joule heating have been studied. The stretching velocity, surface temperature and magnetic field are assumed to have specific exponential function forms for the existence of the local similarity solution. The coupled nonlinear ordinary differential equations governing the local similarity flow and heat transfer have been solved numerically by Chebyshev finite difference method. The influence of the buoyancy parameter, viscous dissipation, relaxation parameter of Maxwell fluid, magnetic field and Prandtl number on the flow and heat transfer has been considered in detail. The Nusselt number increases significantly with the Prandtl number, but the skin friction coefficient decreases. The Nusselt number slightly decreases with increasing viscous dissipation parameter, but the skin friction coefficient slightly increases. Maxwell fluid reduces both skin friction coefficient and Nusselt number, whereas buoyancy force enhances them.     

Influence of thermal radiation and Joule heating in the Eyring–Powell fluid flow with the Soret and Dufour effects

A two-dimensional magnetohydrodynamic boundary layer flow of the Eyring–Powell fluid on a stretching surface in the presence of thermal radiation and Joule heating is analyzed. The Soret and Dufour effects are taken into account. Partial differential equations are reduced to a system of ordinary differential equations, and series solutions of the resulting system are derived. Velocity, temperature, and concentration profiles are obtained. The skin friction coefficient and the local Nusselt and Sherwood numbers are computed and analyzed.     

Low-speed peristaltic transport in a vertical channel subject to the Soret and Dufour effects

Simultaneous effects of heat and mass transfer in peristaltic transport of a viscous fluid are considered. Mathematical modeling is provided in the presence of the Joule heating and the Soret and Dufour effects. The analysis is performed using the long wavelength and low Reynolds number considerations. Perturbation solutions are obtained for a small Brinkman number.     

Steady nonlinear hydromagnetic flow and heat transfer over a stretching surface of variable temperature

The steady nonlinear hydromagnetic flow of an incompressible, viscous and electrically conducting fluid with heat transfer over a surface of variable temperature stretching with a power-law velocity in the presence of variable transverse magnetic field is analysed. Utilizing similarity transformation, governing nonlinear partial differential equations are transformed to nonlinear ordinary differential equations and they are numerically solved using fourth-order Runge–Kutta shooting method. Numerical solutions are illustrated graphically by means of graphs. The effects of magnetic field, stretching parameter and Prandtl number on velocity, skin friction, temperature distribution and rate of heat transfer are discussed.     

Optimal treatment of stratified Carreau and Casson nanofluids flows in Darcy-Forchheimer porous space over porous matrix

A comparative three-dimensional (3D) analysis for Casson-nanofluid and Carreau-nanofluid flows due to a flat body in a magnetohydrodynamic (MHD) stratified environment is presented. Flow is estimated to be suspended in a Darcy-Forchheimer medium. Soret and Dufour responses are also accommodated in the flow field. A moving (rotating) coordinate system is exercised to examine the bidirectionally stretched flow fields (flow, heat transfer, and mass transfer). Nanofluid is compounded by taking ethylene glycol/sodium alginate as base fluid and ferric-oxide (Fe₃O₄) as nanoparticles. Governing equations are handled by the application of optimal homotopy asymptotic method (OHAM), where convergence parameters are optimized through the classical least square procedure. The novel mechanism (hidden physics) due to appearing parameters is explored with the assistance of tabular and graphical expositions. Outcomes reveal the double behavior state for temperature field with thermal stratification/Dufour number, and for concentration field with Soret number due to the presence of turning points.     

MHD boundary layer flow and heat transfer over a stretching sheet with induced magnetic field

In this paper, the problem of steady magnetohydrodynamic boundary layer flow and heat transfer of a viscous and electrically conducting fluid over a stretching sheet is studied. The effect of the induced magnetic field is taken into account. The transformed ordinary differential equations are solved numerically using the finite-difference scheme known as the Keller-box method. Numerical results are obtained for various values of the magnetic parameter, the reciprocal magnetic Prandtl number and the Prandtl number. The effects of these parameters on the flow and heat transfer characteristics are determined and discussed in detail. When the magnetic field is absent, the closed analytical results for the skin friction are compared with the exact numerical results. Also the numerical results for the heat flux from the stretching surface are compared with the results reported by other authors when the magnetic field is absent. It is found that very good agreement exists.     

Effect of non-uniform heat source on MHD heat transfer in a liquid film over an unsteady stretching sheet

An analysis has been carried out to study the magnetohydrodynamic boundary layer flow and heat transfer characteristics of a laminar liquid film over a flat impermeable stretching sheet in the presence of a non-uniform heat source/sink. The basic unsteady boundary layer equations governing the flow and heat transfer are in the form of partial differential equations. These equations are converted to non-linear ordinary differential equations using similarity transformation. Numerical solutions of the resulting boundary value problem are obtained by the efficient shooting technique. The effects of magnetic and the non-uniform heat source/sink parameters on the dynamics are discussed. Findings of the paper reveal that non-uniform heat sinks are better suited for effective cooling of the stretching sheet. Skin friction coefficient and the local Nusselt number are also explored for typical values of magnetic and non-uniform heat source/sink parameters. The results are in excellent agreement with the earlier published works, under some limiting cases.     

Soret and Dufour effects on unsteady MHD flow past an infinite vertical porous plate with thermal radiation

The objective of the present study is to investigate the effect of flow parameters on the free convection and mass transfer of an unsteady magnetohydrodynamic flow of an electrically conducting, viscous, and incompressible fluid past an infinite vertical porous plate under oscillatory suction velocity and thermal radiation. The Dufour （diffusion thermo） and Soret （thermal diffusion） effects are taken into account. The problem is solved numerically using the finite element method for the velocity, the temperature, and the concentration field. The expression for the skin friction, the rate of heat and mass transfer is obtained. The results are presented numerically through graphs and tables for the externally cooled plate （Gr 〉 0） and the externally heated plate （Gr 〈 0） to observe the effects of various parameters encountered in the equations.     

Nonlinear hydromagnetic flow and heat transfer over a surface stretching with a power-law velocity

 Nonlinear hydromagnetic flow and heat transfer over a surface stretching with a power-law velocity is analysed. A special form of the magnetic field is chosen to obtain similarity equations. Resulting equations are numerically solved using Runge–Kutta shooting method. Values of skin-friction and rate of heat transfer are obtained and the effect of magnetic field, stretching parameter and Prandtl number over these are discussed. Received on 2 May 2001 / Published online: 29 November 2001     

Effect of viscous dissipation and heat source on flow and heat transfer of dusty fluid over unsteady stretching sheet

This paper investigates the problem of hydrodynamic boundary layer flow and heat transfer of a dusty fluid over an unsteady stretching surface.The study considers the effects of frictional heating(viscous dissipation) and internal heat generation or absorption.The basic equations governing the flow and heat transfer are reduced to a set of non-linear ordinary differential equations by applying suitable similarity transformations.The transformed equations are numerically solved by the Runge-Kutta-Fehlberg-45 order method.An analysis is carried out for two different cases of heating processes,namely,variable wall temperature(VWT) and variable heat flux(VHF).The effects of various physical parameters such as the magnetic parameter,the fluid-particle interaction parameter,the unsteady parameter,the Prandtl number,the Eckert number,the number density of dust particles,and the heat source/sink parameter on velocity and temperature profiles are shown in several plots.The effects of the wall temperature gradient function and the wall temperature function are tabulated and discussed.     

Nonlinear convection in a non-Darcy porous medium

In this paper, the natural convection in a non-Darcy porous medium is studied using a temperature-concentration-dependent density relation. The effect of the two parameters responsible for the nonlinear convection is analyzed for different values of the inertial parameter, dispersion parameters, Rayleigh number, Lewis number, Soret number, and Dufour number. In the aiding buoyancy, the tangential velocity increases steeply with an increase in the nonlinear temperature parameter and the nonlinear concentration parameter when the inertial effect is zero. However, when the inertial effect is non-zero, the effect of the nonlinear temperature parameter and the nonlinear concentration parameter on the tangential velocity is marginal. The concentration distribution varies appreciably and spreads in different ranges for different values of the double dispersion parameters, the inertial effect parameter, and also for the parameters which control the nonlinear temperature and the nonlinear concentration. Heat and mass transfer varies extensively with an increase in the nonlinear temperature parameter and the nonlinear concentration parameter depending on Dacry and non-Darcy porous media. The variation in heat and mass transfer when all the effects, i.e., the inertial effect, double dispersion ef- fects, and Soret and Dufour effects, are simultaneously zero and non-zero. The combined effects of the nonlinear temperature parameter, the nonlinear concentration parameter and buoyancy are analyzed. The effect of the nonlinear temperature parameter and the nonlinear concentration parameter and also the cross diffusion effects on heat and mass transfer are observed to be more in Darcy porous media compared with those in non- Darcy porous media. In the opposing buoyancy, the effect of the temperature parameter is to increase the heat and mass transfer rate, whereas that of the concentration parameter is to decrease.