Soret and Dufour effects on free convection boundary layers over an inclined wavy surface in a porous medium

This work studies the heat and mass transfer characteristics of natural convection near a vertical wavy cone in a fluid saturated porous medium with Soret and Dufour effects. The surface of the wavy cone is kept at constant temperature and concentration. The governing equations are transformed into a set of coupled differential equations, and the obtained boundary layer equations are solved by the cubic spline collocation method. The heat and mass transfer characteristics are presented as functions of Soret parameter, Dufour parameter, half angle of the cone, Lewis number, buoyancy ratio, and dimensionless amplitude. Results show that an increase in the Dufour parameter tends to decrease the local Nusselt number, and an increase in the Soret parameter tends to decrease the local Sherwood number. Moreover, a greater half angle of the cone leads to a greater fluctuation of the local Nusselt and Sherwood numbers with the streamwise coordinates.     

Soret and Dufour effects on free convection boundary layers of non-Newtonian power law fluids with yield stress in porous media over a vertical plate with variable wall heat and mass fluxes

This work studies the Soret and Dufour effects on the free convection boundary layers over a vertical plate with variable wall heat and mass fluxes in a porous medium saturated with a non-Newtonian power law fluid with yield stress. The governing equations are transformed into a dimensionless form by the similarity transformation and then solved by a cubic spline collocation method. Results are presented for the local surface temperature and concentration for various parameters of the power law fluid with yield stress in porous media. An increase in the power law exponent decreases the local surface temperature and concentration, thus increasing the local Nusselt and Sherwood numbers. An increase in the Soret parameter tends to increase the local surface concentration, thus decreasing the local Sherwood number. Moreover, increasing the Dufour number increases the surface temperature and thus decreases the local Nusselt number.     

Soret and Dufour effects on heat and mass transfer by natural convection from a vertical truncated cone in a fluid-saturated porous medium with variable wall temperature and concentration

This work studies the Soret and Dufour effects on the natural convection heat and mass transfer near a vertical truncated cone with variable wall temperature and concentration in a fluid-saturated porous medium. A coordinate transform is used to obtain the nonsimilar governing equations, and the transformed boundary layer equations are solved by the cubic spline collocation method. Results for local Nusselt number and the local Sherwood number are presented as functions of Soret parameters, Dufour parameters, surface temperature and concentration exponents, buoyancy ratios, and Lewis numbers. Results show that increasing the Dufour parameter tends to decrease the local Nusselt number, while it tends to increase the local Sherwood number. An increase in the Soret number leads to an increase in the Nusselt number and a decrease in the Sherwood number from a vertical truncated cone in a fluid-saturated porous medium. The local Nusselt number and the local Sherwood number of the truncated cones with higher surface temperature and concentration exponents are higher than those with lower exponents.     

Soret and Dufour effects on natural convection heat and mass transfer from a vertical cone in a porous medium

This work studies the Soret and Dufour effects on the boundary layer flow due to natural convection heat and mass transfer over a downward-pointing vertical cone in a porous medium saturated with Newtonian fluids with constant wall temperature and concentration. A similarity analysis is performed, and the obtained similar equations are solved by cubic spline collocation method. The effects of the Dufour parameter, Soret parameter, Lewis number, and buoyancy ratio on the heat and mass transfer characteristics have been studied. The local Nusselt number tends to decrease as the Dufour parameter is increased. The effect of the Dufour parameter on the local Nusselt number becomes more significant as the Lewis number is increased. Moreover, an increase in the Soret number leads to a decrease in the local Sherwood number and an increase in the local Nusselt number.     

Soret and Dufour effects on natural convection boundary layer flow over a vertical cone in a porous medium with constant wall heat and mass fluxes

This work studies the Soret and Dufour effects on the boundary layer flow due to natural convection heat and mass transfer over a vertical cone in a fluid-saturated porous medium with constant wall heat and mass fluxes. A similarity analysis is performed, and the obtained similar equations are solved by the cubic spline collocation method. The effects of the Dufour parameter, Soret parameter, Lewis number, and buoyancy ratio on the heat and mass transfer characteristics have been studied. The local surface temperature tends to increase as the Dufour parameter is increased. The effect of the Dufour parameter on the local surface temperature becomes more significant as the Lewis number is increased. Moreover, an increase in the Soret parameter leads to an increase in the local surface concentration and a decrease in the local surface temperature.     

Soret and Dufour effects on free convection boundary layer over a vertical cylinder in a saturated porous medium

This paper deals with an analysis of the Soret and Dufour effects on the boundary layer flow due to free convection heat and mass transfer over a vertical cylinder in a porous medium saturated with Newtonian fluids with constant wall temperature and concentration. A suitable coordination transformation is used to derive the similar governing boundary-layer equations, and the cubic spline collocation method is then employed to solve the similar governing boundary-layer equations. The variation of the Nusselt number and the Sherwood number with the Dufour parameter and the Soret parameter for various Lewis numbers and buoyancy ratios have been presented in this work. Results show that an increase in the Soret number leads to a decrease in the local Sherwood number and an increase in the local Nusselt number. The local Nusselt number tends to decrease as the Dufour parameter is increased. Moreover, an increase in the Lewis number enhances the effect of the Dufour parameter on the local Nusselt number.     

Finite element Soret Dufour effects on an unsteady MHD heat and mass transfer flow past an accelerated inclined vertical plate

The present study is focused on the Soret and Dufour effects on magnetohydrodynamics unsteady fluid flow over an accelerated inclined vertical plate with thermal radiation and heat source. Solution of the nondimensional governing differential equations are worked out by the efficient Galerkin finite element method. The influence of several relevant flow parameters on velocity, temperature, and concentration distributions, as well as the numerical results, are studied and graphically displayed. The nondimensional skin friction and the rate of heat and mass transfer parameters are presented in the Tables 1-3 below. Raising the Soret number results in growing concentrations, but the converse is true for the Schmidt number. Skin friction reduces when Soret and Dufour numbers increase. The present simulations apply to the processing of magnetic materials in the chemical and metallurgical industries.     

Soret and Dufour effects on free convection flow of non-Newtonian fluids along a vertical plate embedded in a porous medium with thermal radiation

This article numerically studies the combined laminar free convection flow with thermal radiation and mass transfer of non-Newtonian power-law fluids along a vertical plate within a porous medium. The solution takes the diffusion-thermo (Dufour), thermal-diffusion (Soret), thermal radiation and power-law fluid index effects into consideration. The governing boundary layer equations along with the boundary conditions are first cast into a dimensionless form by a similarity transformation and the resulting coupled differential equations are then solved by the differential quadrature method (DQM). The effects of the radiation parameter R, the power-law index n, the Dufour number D_f, and the Soret number S_r on the fluid flow, thermal and concentration fields are discussed in detail. The results indicate that when the buoyancy ratio of concentration to temperature is positive, N > 0, the local Nusselt number increases with an increase in the power-law index and the Soret number or a decrease in the radiation parameter and the Dufour number. In addition, the local Sherwood number for different values of the controlling parameters is also obtained.     

Chemical reaction and Dufour effects on nonlinear free convection heat and mass transfer flow near a vertical moving porous plate

A theoretical analysis is made for steady fully developed free convection and mass transfer flow near an infinite vertical moving porous plate by taking into consideration the first‐order chemical reaction and Dufour effects. The mathematical model responsible for the present physical situation is based on the nonlinear density variation with temperature as well as nonlinear density variation with concentration. Exact solutions are derived for heat mass and momentum equations under relevant boundary conditions. The dimensionless velocity, temperature, and concentration are presented in terms of exponential functions. The impact of controlling parameters such as Dufour number (diffusion thermo effect), chemical reaction parameter, Prandlt number, Schmidt number, on velocity, temperature, Nusselt number, and skin friction are discussed with the aid of line graphs, contours, and tables. The analysis of the result shows that Nusselt number, skin friction, and velocity increases with increase in Dufour number. Furthermore, velocity and skin friction are higher in case of nonlinear convection in comparison to linear convection.     

MHD non-Darcian mixed convection heat and mass transfer over a non-linear stretching sheet with Soret-Dufour effects and chemical reaction

A study has been carried out to analyze the effects of variable thermal conductivity, Soret (thermal-diffusion) and Dufour (diffusion-thermo) on MHD non-Darcy mixed convection heat and mass transfer over a non-linear stretching sheet embedded in a saturated porous medium in the presence of thermal radiation, viscous dissipation, non-uniform heat source/sink and first-order chemical reaction. The governing differential equations transform into a set of non-linear coupled ordinary differential equations using similarity analysis. Similarity equations are then solved numerically using shooting algorithm with Runge-Kutta Fehlberg integration scheme over the entire range of physical parameters. A comparison with previously published work has been carried out and the results are found to be in good agreement. Graphical presentation of the local skin-friction coefficient, the local Nusselt number and the local Sherwood number as well as the temperature profiles show interesting features of the physical parameters.     

Coupled heat and mass transfer by MHD free convection flow along a vertical plate with streamwise temperature and species concentration variations

The problem of steady, laminar, coupled heat and mass transfer by MHD free convective boundary‐layer flow along a vertical flat plate with the combined effects of streamwise sinusoidal variations of both the surface temperature and the species concentration in the presence of Soret and Dufour effects is considered. A suitable set of dimensionless variables is used to transform the governing equations of the problem into a non‐similar form. The resulting non‐similar equations have the property that they reduce to various special cases previously considered in the literature. An adequate and efficient implicit, tri‐diagonal finite difference scheme is employed for the numerical solution of the obtained equations. Various comparisons with previously published work are performed and the results are found to be in excellent agreement. A representative set of numerical results for the velocity, temperature, and concentration profiles as well as the surface shear stress, rate of heat transfer, and the rate of mass transfer is presented graphically for various parametric conditions and is discussed. © 2012 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.21033     

MHD non-Darcian free convection from a vertical wavy surface embedded in porous media in the presence of Soret and Dufour effect

The objective of this paper is to examine the combined effect of spatially stationary surface waves and the presence of fluid inertia on the free convection along a heated vertical wavy surface embedded in an electrically conducting fluid saturated porous medium, subject to the diffusion-thermo (Dufour), thermo-diffusion (Soret) and magnetic field effects. Diffusion-thermo implies that the heat transfer is induced by concentration gradient, and thermo-diffusion implies that the mass diffusion is induced by thermal gradient. The boundary-layer regime is considered where the Darcy–Rayleigh number is very large. A suitable coordinate transformation was considered to reduce the governing boundary-layer equations into non-similar form. The resulting nonlinear, coupled differential equations were solved numerically employing the Runge–Kutta algorithm with shooting iteration technique. Dimensionless velocity, temperature, concentration distributions, as well as local Nusselt number and Sherwood number are presented graphically for various values of Dufour number D_u, Soret number S_r, Buoyancy ratio N, amplitude of the wavy surface a, Lewis number Le, Grashof number Gr, and magnetic field effect M_g.     

Natural convection heat and mass transfer from a vertical truncated cone in a porous medium saturated with a non-Newtonian fluid with variable wall temperature and concentration

This work presents a boundary-layer analysis about the natural convection heat and mass transfer near a vertical truncated cone with variable wall temperature and concentration in a porous medium saturated with non-Newtonian power-law fluids. A coordinate transform is used to obtain the nonsimilar governing equations, and the transformed boundary-layer equations are solved by the cubic spline collocation method. Results for local Nusselt numbers are presented as functions of power-law indexes, surface temperature and concentration exponents, buoyancy ratios, and Lewis numbers. The heat and mass transfer rates of the truncated cones with higher surface temperature and concentration exponents are higher than those with lower exponents. Moreover, an increase in the power-law index of fluids tends to decrease the heat and mass transfer from a vertical truncated cone in a porous medium saturated with non-Newtonian power-law fluids.     

Mixed Convection Effect on Melting from a Vertical Plate Embedded in Porous Medium with Soret and Dufour Effects

The present work analyzed the impact of mixed convection on melting from a vertical flat plate embedded in porous medium in the presence of Dufour and Soret effects. The partial differential equations governing the problem under consideration have been transformed by a similarity transformation into a system of ordinary differential equation which is solved numerically by Runge–Kutta–Gill methods. Dimensionless velocity, temperature, and concentration profiles are presented graphically for various values of the Dufour number (Df), Soret number (Sr), melting parameter (M), and buoyancy parameter (Gr/Re). During the investigation, it was found that the melting phenomenon decreases the local Nusselt number and local Sherwood number at the solid–liquid interface. Also, it is interesting to note that the velocity as well as temperature increases while the concentration decreases with an increase in the Dufour number Df (or simultaneous decrease in the Soret number Sr). © 2013 Wiley Periodicals, Inc. Heat Trans Asian Res, 43(7): 667–676, 2014; Published online 3 October 2013 in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.21113     

Heat and mass transfer for Soret and Dufour's consequences on unsteady MHD free convection flow over a porous media with heat absorption

The significance of this article lies in explaining the influence of Soret and Dufour numbers on an unsteady MHD free convection of flow of heat and mass transfer through porous media. The substances and radiation along the viscous, incompressible, and conductive compounds respond to the unstable convection of the liquid. Using physical quantities, the dimensional governing equations are converted to non-dimensional equations. Finite element Galerkin method is applied to numerically solve the resulting partial differential equations. Flow parameters on velocity, temperature, and concentration are studied and explained graphically to reflect their effects. Similarly, the skin friction number and Nusselt number are also observed and recorded in tables.     

Unsteady hydromagnetic free convection flow with hall current mass transfer and variable suction through a porous medium near an infinite vertical porous plate with constant heat flux

The effects of Hall current on unsteady free convection flow of an MHD viscous incompressible fluid along an infinite vertical porous plate are investigated in the presence of a uniformly applied magnetic field acting in a plane which makes an angle α with the plane transverse to the plate. A similarity parameter, taken to be a function of time, is introduced, and the suction velocity is considered to be inversely proportional to this parameter. Similarity equations are then derived and solved numerically. The effects of the Hall parameter, the magnetic parameter and the permeability are discussed and shown graphically.     

Numerical simulation of Dufour and Soret impacts on MHD Williamson nanofluid flow through an inclined surface

The carry-outs of Dufour and Soret, as well as radiation, and chemical response on a non-Newtonian MHD Williamson nanofluid flow through an inclined extended plane are discussed in this article. Keller-box analysis is being used to explore the influence of the Williamson factor here on the fluid domain quantitatively. Ordinary differential equations (ODEs) are recovered from boundary flow equations using appropriate similarity transformations. These ODEs are numerically addressed. Graphs and comparisons are used to simulate and study the features of flow characteristics such as velocity, temperature, and concentration of Williamson nanofluids distributions in response to various emerging parameters. The numerical computations show that our results are in reasonable harmony with previous studies. The numerical computations revealed that for the time being, the density of the momentum fluid layers is diminishing for the values of $ᴦ$, Le, $\Omega$, M, and increasing for Gc, Gr. The thickness of the thermal boundary layer is decreasing for Sr, Df, Pr, Gc, and Gr. M, $ᴦ$, $\Omega$, R, N, and Le are all on the rise. The concentration profile for R, Le, Nb, Nt, Gr, Gc, and N is decreasing, while Pr, Df, Sr, M, $ᴦ$, and $\Omega$ are increasing.     

Impact of Soret and Dufour on bioconvective flow of nanofluid in porous square cavity

This article addresses the bioconvection in a porous cavity associated with Soret and Dufour effects. The bioconvective flow in a porous medium is based on Hillesdon and Pedley's model and is governed by nonlinear partial differential equations. These equations are transformed into a dimensionless form with suitable nondimensional parameters. The finite element method is employed to solve the dimensionless equations. The outcomes of the study are presented by streamlines, temperature distributions, isoconcentrations of solute, nanoparticles, and microorganisms. Furthermore, the tendency of average Nusselt number and average Sherwood number and the influence of Soret parameter, Dufour parameter, Peclet number, and bioconvective Rayleigh number is interpreted. Thermophoresis and Soret number show a strong effect on the concentration of nanoparticles. Brownian motion and thermophoresis exhibit a significant effect on the density distributions of microorganisms. The novelty of the paper is to combine the effects of Soret–Dufour and oxytactic bioconvection. The present study can be useful in the following applications: microbial-enhanced oil recovery, toxin removal, antibiotics, and modeling of microfluidic devices.     

Chemical reaction and magnetohydrodynamics effects on heat absorbing fluid past an inclined porous plate

The study of a heat-absorbing, chemically bonding fluid over a porous channel in a conducting field with ramped wall temperature is considered. The Dufour effect presence is also considered with thermal radiation. The novelty is the consideration of radiation absorption and the angle of inclination. In this approach, the dimensional governing equations and boundary forms are transformed into a dimensionless form using standard nondimensional parameters and variables. The simplified governing equations and boundary forms are then calculated using the Laplace transform method. We get accurate answers in the speed, temperature, and concentration spaces. Calculations of surface friction, the Nusselt number, and the Sherwood number are also performed. Several physical parameters' influences on the quantified flows are analysed using graphics. A comparison is also made with the results available in the literature and found a good agreement in the absence of radiation absorption. When a chemical is added to a fluid to dilute it, the velocity area and concentration area both decrease, but the temperature area increases as a result of an increase in the Schmidt Number, the Nusselt Number, and the skin friction. Our research revealed that the Dufour effect and arbitrarily ramped temperatures had a similar effect on fluid velocity.     

An explicit solution for the combined heat and mass transfer by natural convection from a vertical wall in a non-Darcy porous medium

An analytic technique, namely the homotopy analysis method, is applied to solve the combined heat and mass transfer by natural convection adjacent to a vertical wall in a non-Darcy porous medium governed by a set of three fully coupled, highly nonlinear similarity equations. An explicit, totally analytic and uniformly valid solution is derived, which agrees well with numerical results.