Influences of Soret and Dufour numbers on mixed convective and chemically reactive Casson fluids flow towards an inclined flat plate

The purpose of this study is to examine the magnetohydrodynamic mixed convection Casson fluid flow over an inclined flat plate along with the heat source/sink. The present flow problem is considered under the assumption of the chemical reaction and thermal radiation impacts along with heat and mass transport. The leading nonlinear partial differential equations of the flow problem were renovated into the nonlinear ordinary differential equations (ODEs) with the assistance of appropriate similarity transformations and then we solved these ODEs with the employment of the bvp4c technique using the computational software MATLAB. The consequences of numerous leading parameters such as thermophoretic parameter, local temperature Grashof number, solutal Grashof number, suction parameter, magnetic field parameter, Prandtl number, chemical reaction parameter, Dufour number, Soret number, angle of inclination, radiation parameter, heat source/sink, and Casson parameter on the fluid velocity, temperature, and concentration profiles are discoursed upon  and presented through different graphs. Some important key findings of the present investigation are that the temperature of the Casson fluid becomes lower for local temperature Grashof number and solutal Grashof number. It is initiated that the Casson fluid parameter increases the velocity of the fluid whereas the opposite effect is noticed in the temperature profile. Higher estimation of Prandtl number and magnetic parameter elevated the Casson fluid concentration. Finally, the skin friction coefficient, Nusselt number, and Sherwood number are calculated and tabulated. It is also examined that the Nusselt number is weakened for both the Dufour number and Soret number but the skin fraction coefficient is greater for both the Dufour number and Soret number.     

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.     

Influence of Soret and Dufour on mixed convection flow across a vertical cone

This contribution examines the influence of Soret and Dufour on an incompressible viscous fluid flow across a vertical cone. The flow model is framed in the form of mathematical governing equations and a nondimensionalization is performed on them for ease of the numerical computations' examination; the obtained nonsimilarity equations are solved numerically through the bivariate Chebyshev spectral collocation quasi-linearization method. Outcomes of the flow characteristics, velocity, temperature, concentration, skin friction rate, heat, and mass transfer rates are analyzed with the variations of governing parameters, Prandtl number, buoyancy parameter, Schmidt number, buoyancy ratio, Soret and Dufour parameters at various stream-wise spots of the flow. To certify the exactness of the listed computations, we performed a comparison with prior published computations, which were found with great agreement, and the residual analysis study was also portrayed to reflect the convergence and stability of the adopted numerical technique.     

MHD flow in a rotating vertical cone through a porous medium

In the context of advancements in both heat and mass transfer, the current study intends to analyze the impacts of thermal radiation, Soret, and Dufour on the magnetohydrodynamic boundary layer flow through a vertical spinning cone in porous media. The Dufour effect is the energy flux due to the mass concentration gradient with a reciprocal phenomenon, the Soret effect. Energy expression considers the physical aspects of heat generation and absorption. It is expected that the tangential, circumferential, and normal directions will all have velocity components in flow through a porous media. The governing equations are nonlinear partial differential equations that are rearranged into ordinary differential equations via similarity transformation, and then they are numerically solved using the Runge–Kutta method along with a proper shooting strategy. Graphs are used to examine the impacts of many parameters on flow characteristic velocity, temperature, and concentration, including magnetic parameters, porous parameters, Dufour and Soret parameters, chemical reaction parameters, and more. The numerical findings of the gradient of velocity, the Nusselt and Sherwood numbers, and the surface drag force are tabulated and compared with the current result and the one from the literature. The findings are found to be in good agreement. Circumferential and normal velocities are improved visually for greater values of the porosity parameter, but the tangential velocity behavior of the magnetic parameter exhibits the reverse behavior. In addition, the Dufour parameter and chemical reaction both exhibit diminishing behavior when the Soret parameter increases.     

Numerical study of chemical reaction and heat transfer of MHD slip flow with Joule heating and Soret–Dufour effect over an exponentially stretching sheet

A study of Soret–Dufour effects along with chemical reaction, viscous dissipation combining on MHD Joule heating for viscous incompressible flow is presented. It is assumed that fluid is flowing past an angled stretching sheet saturated in porous means. The slip conditions of velocity, concentration, and temperature are accounted for at the boundary. The mathematical expression of the problem contains highly nonlinear interconnected partial differential equations. To convert governing equations into ordinary differential equations, appropriate similarity transformations were utilized. These differential equations with boundary constraints are resolved by homotopy analysis method. Expression for velocity, concentration, and temperature are derived in the form of series. Effects of numerous physical parameters, for example, Schmidt number, Soret number, buoyancy ratio parameter, slip parameter, and so forth, on various flow characteristics are presented through graphs. Numerous values of velocity, concentration, and temperature gradient are tabulated against different parameters. Results show that the fluid velocity increases by enhancing the Soret number, Dufour number, or permeability parameter. The fluid's concentration rises as the Soret number increases, while it falls as the Dufour number, chemical reaction parameter, or permeability parameter increases.     

Cross–diffusion effects on mixed convection from an exponentially stretching surface in non–Darcy porous medium

The Soret and Dufour effects on mixed convection flow and heat and mass transfers from an exponentially stretching surface in a quiescent fluid–saturated non–Darcy porous medium is studied. Stretching velocity, wall temperature, and wall concentration are assumed to have specific exponential function forms. The governing partial differential equations are transformed into ordinary differential equations using similarity transformations and then solved numerically using an implicit finite difference scheme known as the Keller–box method. The present results are found to be in excellent agreement with previously published work on various special cases of the problem. The influence of buoyancy, Soret and Dufour numbers, and Darcy and non–Darcy parameters on the convective transport in the boundary layer region is analyzed. Also, the numerical values of the skin friction, heat, and mass transfer coefficients for different values of governing parameters are also tabulated. © 2013 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.21032     

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–Dufour effects on chemically reacting non-Darcian MHD flow around a plate

Here, a study of steady, magnetohydrodynamic flow of incompressible, cold fluid around a moving plate with a non-Darcian porous medium in existence of heat source and nth-order chemical reaction incorporating Soret and Dufour effects is considered. MATLAB bvp4c technique is used to solve the prevailing equations. Variations in velocity, temperature and concentration are analysed. It is observed that the applicable parameters such as non-Darcy, Soret, Dufour, chemical reaction play a significant role in controlling the flow. Chemical reaction parameter reduces skin friction, heat transfer, and mass transfer while Eckert number enhances the mass transfer and skin friction.     

Hall and ion-slip effects on MHD natural convective flow past an unbounded vertical porous channel with thermodiffusion

The consequences of Soret in addition to Dufour of natural convection heat and mass transfer for the unsteady three-dimensional boundary layer flow through a perpendicular condition of the existence of viscous dissipation, invariable suction, Hall as well as ion slip consequences into relation. The prevailing partial differential equation is dissolved digitally utilizing the implicit Crank–Nicolson finite difference method. The velocity, temperature, as well as concentration dispensations, is addressed computationally and demonstrated by the graphs. Numerical values of the Nusselt number, skin friction as well as Sherwoods numbers nearby the plate are discussed for a choice of values of substantial parameters and are displayed in a tabular manner. It is noticed that the temperature of the fluid diminishes with higher Prandtl numbers. The resulting velocity diminishes with the growing Hartmann number. Rotation, Soret, and Dufour parameters strengthen the velocity and momentum boundary layer thickness. The velocity intensifies through growing Hall and ion-slip parameters and the revoke trend is acquired with enhancement in suction parameter.     

Entropy generation analysis of free convection radiative MHD Eyring–Powell fluid flow between porous parallel plates with Soret and Dufour effects

In this present study, we have investigated the entropy generation analysis and Dufour and Soret impacts on unsteady incompressible free convective radiative MHD Eyring–Powell fluid flow between parallel plates with periodic injection and suction. The governing PDEs are converted into nondimensional coupled nonlinear ordinary differential equations by using similarity variables then numerically solved by Runge–Kutta fourth-order scheme with shooting technique. The results are discussed in detail for different flow, mass, and heat transfer profiles corresponding to various active parameters and presented in tables and graphs. Also, it is noticed that the temperature profiles are enhanced with the fluid parameter, whereas the concentration profiles are decreased with the Prandtl number. The validations of present results with the existing outcomes for the viscous case of skin friction are included and have found to be in good agreement. The present numerical study is useful for the enhancement of heat transfer in various industrial and chemical processes.     

Combined influence of cross-diffusion and variable fluid properties on free convection flow past a vertical cone

The free convective flow of an incompressible viscous fluid over an isothermal vertical cone with variable viscosity and variable thermal conductivity is examined in the presence of the Soret and Dufour effects. As thermal and solutal boundary conditions at the cone's surface, the constant temperature and concentration (WTC) and constant heat and mass flux (HMF) cases are taken into account. The successive linearization method is applied to linearize a system of nonlinear differential equations that describes the flow under investigation. The numerical solution for the resulting linear equations is attained by means of the Chebyshev spectral method. The obtained numerical results are compared and found to be in good agreement with previously published results. The impact of significant parameters on the heat and mass transfer rates is evaluated and presented graphically for the WTC and HMF situations. In both cases, the Soret number increases the skin friction coefficient and rate of heat transfer while decreasing the Sherwood number. With an increase in the Dufour parameter, the coefficient of skin friction and Sherwood numbers increase while the heat transmission rate decreases.     

Pulsating hydromagnetic flow of Casson fluid in a vertical channel filled with non-Darcian porous medium

This paper analyzes the Joule heating, Dufour number, and Soret number effects on hydromagnetic pulsatile flow of a Casson fluid in a vertical channel filled with a non-Darcian porous medium. The governing partial differential equations (PDEs) of the Casson fluid flow are transformed to ordinary differential equations (ODEs) using perturbation technique and solved by employing shooting method with Runge–Kutta (R–K) fourth-order technique using MATHEMATICA function NDSolve. The influence of Forchheimer number, Casson fluid parameter, Dufour number, radiation parameter, and Soret number on flow variables has been studied and the numerical results obtained are presented. The results reveal that the velocity rises with the rise of Darcy number, whereas it decreases for a given rise in the Forchheimer number. Furthermore, the temperature distribution enhances by increasing the Dufour number.     

Soret and Dufour effects on free convection along a vertical wavy surface in a fluid saturated Darcy porous medium

In this article, free convection of heat and mass transfer along a vertical wavy surface in a Newtonian fluid saturated Darcy porous medium is studied by considering cross diffusion (namely the Soret and the Dufour effects) in the medium. The vertical wavy wall and the flow governing equations are transformed to a plane geometry case by using a suitable transformation. Then a similarity solution to this problem is presented under the large Darcy–Rayleigh number assumption. The governing partial differential equations are reduced to a set of ordinary differential equations that are integrated using numerical methods to study the nature of the non-dimensional heat and mass transfer coefficients in the medium. The results are presented for a range of the flow governing parameters such as the diffusivity ratio parameter, the buoyancy ratio parameter, the Soret parameter, the Dufour parameter and the amplitude of the wavy surface.     

Soret and Dufour effects on free convection heat and mass transfer from an arbitrarily inclined plate in a porous medium with constant wall temperature and concentration

The free convection boundary layer flow over an arbitrarily inclined heated plate in a porous medium with Soret and Dufour effects is studied by transforming the governing equations into a universal form. The generalized equations can be used to derive the similarity solutions for limiting cases of horizontal and vertical plates and to calculate the heat and mass transfer characteristics between these two limiting cases. The heat and mass transfer characteristics are presented as functions of Soret parameter, Dufour parameter, inclination variable, Lewis number, and buoyancy ratio. Results show that an increase in the Dufour parameter tends to decrease the local heat transfer rate, and an increase in the Soret parameter tends to decrease the local mass transfer rate. As the inclination variable increases, the local Nusselt number and the local Sherwood number decrease from their respective values for horizontal plates, reach their respective minima, and then increase to their respective values for vertical plates. The minima are where the tangential and normal components of buoyancy force are comparable.     

MHD Carreau nanoliquid flow over a nonlinear stretching surface

The Dufour and Soret impacts on magnetohydrodynamic Carreau nanoliquid past a nonlinearly stretching sheet are investigated. Variations in viscosity, heat conductivity, and convective boundary conditions are considered. Suitable similarity conversions are utilized to design the governing equations nondimensional. The Optimal Homotopy Analysis Method is employed to resolve the dimensionless equations. Graphs and tables are utilized to illustrate the impacts of the relevant factors over velocity, temperature, concentration, and streamlines. For the variations of different parameters, numerical values for Nusselt number, Sherwood number, and skin friction are provided in a table. The observed results are in good agreement with the previous literature findings. Furthermore, the current research shows that when the Dufour number increases, the temperature distributions get narrower. However, with increasing Soret number, the concentration distribution has the opposite effect. One of the important outcomes of the current study is that by increasing the Weissenberg number for shear-thinning fluids, one can improve the velocity field.     

Multislip and Soret–Dufour influence on nonlinear convection flow of MHD dissipative Casson fluid over a slendering stretching sheet with generalized heat flux phenomenon

In the present investigation, Soret–Dufour and multislip's impact on magnetohydrodynamics (MHD) Casson fluid flow encompassing variable thermophysical features in the nonlinear convection process is analyzed. It is believed that to any effective heat and mass transfer enhancement, the relaxation of such fluid and material time along with the thermo-physical features, are well estimated. In this model, a magnetic field of nonuniform strength is applied perpendicular to the slendering sheet with variable thickness, and nonlinear convection flow is considered in this generalized heat flux examination. An appropriate similarity variable is implemented on the governing equations embedding the variable viscosity, thermal conductivity, and generalized Fourier's law to drive ordinary differential equations. Galerkin weighted residual approach is utilized to calculate the flow field among other flow characteristics. The novel flow features are discussed therein. Modified Fourier and multislip's parameters are seen to have downsized the velocity and temperature field greatly. Thermal and solutal buoyancy effects are more pronounced in nonlinear form compared to the linear model. Dufour number influences both velocity and energy fields positively but negates the concentration field, while the Soret number gives an opposing characterization.     

Significance of Dufour and Soret effects on the non-Darcy flow of Cross fluid by a tilted plate with radiation and chemical reaction: A Cattaneo–Christov heat flux model

The Darcy–Forchheimer flow model is substantial in the fields where a high flow rate effect is a common phenomenon, for instance, in petroleum engineering. In this paper, we aim to scrutinize the aspects of cross-diffusion effects on the non-Darcy flow of Cross fluid by a tilted plate with thermal radiation and chemical reaction. Metamorphosed equations are resolved with the combination of shooting and Runge–Kutta fourth-order procedures. The correlation coefficient is used to discuss the impact of pertinent parameters on friction factor and transfer rates (heat and mass). The main findings of this study are that the Dufour number escalates fluid temperature and the Soret number ameliorates the fluid concentration. It is observed that the fluid velocity minifies with the elevation in the Forchheimer number. And also, it is perceived that the heat transfer rate has a generous positive relationship with the thermal relaxation parameter. Furthermore, validation of current results with the earlier results under specified conditions is performed and an adequate concord is seen.     

Free convection heat and mass transfer from an isothermal sphere to a micropolar regime with Soret/Dufour effects

An analysis is presented for the steady free convection heat and mass transfer from a spherical body in a micropolar fluid with Soret/Dufour effects included. The governing boundary layer equations are transformed into a non-dimensional form and the resulting non-linear system of partial differential equations are solved using the efficient Keller-box implicit numerical finite difference method. Excellent correlation of the present numerical solutions is achieved for the case of free convection micropolar flow neglecting Sort/Dufour effects, with the earlier study by Nazar et al. (2002) [12]. With increasing Soret number, the local Nusselt number is boosted considerably, with the converse response with an increase in Dufour number. Increasing micropolar vortex viscosity parameter (K) decelerates the flow near the sphere surface but serves to accelerate the flow further from the sphere wall. With increasing K, reverse spin is induced of the micro-elements near the sphere surface. Temperature is strongly boosted throughout the boundary layer regime with increasing K with a similar response computed for the species concentration Buoyancy-assistance (N > 0) accelerates the flow whereas buoyancy-opposition (N < 0) retards flow. Reverse spin of micro-elements (i.e. negative micro-rotation) is reduced near the sphere surface with buoyancy-opposition but exacerbated with buoyancy-assistance. Buoyancy-assistance serves to depress both temperature and concentration values through the boundary layer regime with the reverse effect sustained with buoyancy-opposition. An increase in Schmidt number acts to strongly inhibit the flow and also to stifle micro-rotation of the micro-elements. Conversely both temperatures and concentration values are accentuated with increasing Schmidt number. The study has important applications in buoyancy-driven, chemical engineering flow regimes.     

Aspects of parabolic motion of MHD fluid flow past a vertical porous plate with cross-diffusion effects

In the presence of Soret and Dufour effects, a numerical analysis is performed for an unstable magnetohydrodynamics convective flow of parabolic motion with variable temperature and concentration. The finite-difference method is used to solve the set of nondimensional governing equations with boundary conditions numerically. Graphs are used to investigate the effect of various physical characteristics on flow quantities. Variations in skin friction, Nusselt number, and Sherwood number are also examined using tables for physical curiosity. This study is unique in that it takes into account changeable temperature as well as concentration with Soret and Dufour effects. The magnetic parameter, Prandtl number, heat source, radiation parameter, Schmidt number, and chemical reaction parameter show a significant increase in skin friction, whereas the Grashof number, modified Grashof number, permeability parameter, radiation absorption parameter, Dufour number, and Soret number show the opposite trend. As the Soret number rises, the concentration rises as well, whereas the opposite is true for the Schmidt number and the chemical reaction parameter. The current study is highly supported by previously published data that have been verified.