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.     

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.     

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 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.     

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.     

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     

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.     

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 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.     

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.     

Mixed convection flow over a vertical cone with double dispersion and chemical reaction effects

This contribution brings mixed convection flow across a vertical cone in the presence of double dispersion and chemical reaction effects. The model of the problem is designed mathematically in the forms of governing equations; it is nondimensionalized for ease of numerical computations and the gained nonsimilarity equations are solved numerically throughout the detailed numerical technique. The outcomes are summarized in graphical and numerical forms to illustrate the impacts of governing parameters Prandtl number, Schmidt number, buoyancy ratio, thermal dispersion, chemical reaction, solutal dispersion, and buoyancy parameters at various streamwise spots of velocity, temperature, and concentration profiles. Moreover, skin friction, heat, and mass transfer rates are tabulated. To establish the exactness of the adopted numerical technique, residual analysis study also portrayed; we made a comparison with prior published outcomes and found them to be in great consent.     

An application of paired quasilinearization on double‐diffusive convection flow over a cone embedded in a porous medium in the presence of nanoparticles

Mixed convection flow of a nanofluid near a vertical cone embedded in a a porous medium with Soret and Dufour effects is exercised. The bearing of a porous medium is recounted by the Darcy model. The partial differential equations, modeling the concerned problem, is nondimensionalised by implementing compatible transformations, which results in a similar form. A new paired spectral quasilinearization method is adopted to get the accurate numerical solution. Convergence and accuracy of the solution is elaborated by analyzing the norm of residual and solution errors. Alteration of velocity, temperature, nanoparticle and solute concentration profiles due to flow controlling parameters, namely, Brownian motion, thermophoresis, Soret, Dufour, Lewis number, and buoyancy ratio is outlined by reproducing the obtained numerical solution in graphs and tables. Analysis reveals that the flow profiles are greatly influenced by the physical parameters under investigation.     

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.     

Buoyancy-motivated dissipative free convection flow of Walters-B fluid along a stretching sheet under the Soret effect and Lorentz force influence

A two-dimensional numerical model has been framed to investigate the effect of buoyancy forces on magnetized free convective Walters-B fluid flow over a stretching sheet with Soret effect, heat radiation, thermal source/sink, and viscous dissipation. The current physical model is developed based on the stretching sheet geometry. The impact of Lorentz force on the nonlinear system is investigated and considered in the velocity equation. The influence of thermal radiation, heat source/sink, viscous dissipation, and Joule heating is considered in the energy equation. The effect of Soret parameter and chemical reaction on mass transfer is accounted in the concentration equation. The current physical model is governed by the highly coupled nonlinear system of partial differential equations. Owing to the inadequacy in the analytical techniques, the obtained governing equations are solved by using the bvp4c Matlab function via similarity transformations approach. Numerical computations are performed for the varying values of physical parameters, which are expressed in terms of tables and graphs. Magnifying viscoelastic parameter decays the velocity profile and enhances the thermal and concentration fields. Enhancing free convection parameters diminishe the velocity fields and magnifies the thermal profile. Thermal field magnifies with enhancing thermal radiation parameter and Eckert number. Enhancing the Soret number raises the concentration field. Also, the bvp4c Matlab function adequately simplifies the highly nonlinear coupled system of equations occurring in nature. The present similarity solutions presented in this paper coincides with previously published results in the literature.     

Magnetoconvection in an enclosure of water near its density maximum with Soret and Dufour effects

In this paper, unsteady double-diffusive magnetoconvection of water in an enclosure with Soret and Dufour effects around the density maximum has been numerically investigated. The right vertical wall has constant temperature, θ_c, while left vertical wall is θ_h, with θ_h > θ_c. The concentration in right wall is maintained lower than left wall (c_h > c_c). The remaining horizontal walls are adiabatic. The governing equations are solved by control volume method using SIMPLE algorithm with QUICK scheme. Representative results illustrating the effects of the thermal Rayleigh number, Hartmann number, the direction of magnetic field, density inversion parameter, buoyancy ratio, Schmidt number, and Soret and Dufour parameters on the contour maps of the fluid flow, temperature and concentration as well as the profile of velocity at mid-section of the enclosure are reported. In addition, numerical results for the average Nusselt and Sherwood numbers are presented for various parametric conditions and discussed.     

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.     

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.     

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.     

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.     

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.