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.     

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.     

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.     

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.     

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.     

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.     

Numerical solution of MHD,Soret, Dufour,and thermal radiation contributions on unsteady free convection motion of Casson liquid past a semi-infinite vertical porous plate

In this paper, the unsteady motion of Casson liquid over a half-infinite penetrable vertical plate with MHD, thermal radiation, Soret, and Dufour contributions have been explored numerically. In the physical geometry, the Casson liquid flows to the layer from the penetrable vertical plate. At the layer, Casson liquid is set into motion and the flow equations are illustrated using coupled partial differential equations (PDEs). This set of PDEs is simplified to form dimensionless PDEs with the use of normal nondimensional transformation. The controlling parameters' effects on the working fluid are extensively discussed on velocity, concentration, and temperature and presented graphically. Computational values of Nusselt plus Sherwood number and skin friction for controlling parameters are depicted in a tabular form. Our outcomes show that a raise in the Casson term depreciates the velocity because of the magnetic parameter influence on the fluid flow. The Soret parameter was found to accelerate the skin friction along with the Sherwood number coefficients. An incremental value of the Dufour parameter was detected to hike the skin friction alongside the Nusselt number. Results of this study were found to be in conformity with previously published work.     

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.     

Influence of thermophoretic deposition and viscous dissipation on magnetohydrodynamic flow with variable viscosity and thermal conductivity

In this study, we numerically explore the impact of varying viscosity and thermal conductivity on a magnetohydrodynamic flow problem over a moving nonisothermal vertical plate with thermophoretic effect and viscous dissipation. The boundary conditions and flow-regulating equations are converted into ordinary differential equations with the aid of similarity substitution. The MATLAB bvp4c solver is used to evaluate the numerical solution of the problem and it is validated by executing the numerical solution with previously published studies. The impacts of several factors, including the magnetic parameter, Eckert number, heat source parameter, thermal conductivity parameter, stratification parameter, Soret, Dufour, Prandtl number, and Schmidt number are calculated and shown graphically. Also, the skin friction coefficient, Nusselt number, and Sherwood number are calculated. Fluid velocity, temperature, and concentration significantly drop as the thermophoretic parameter and thermal stratification parameter increases. As thermal conductivity rises, it is seen that the velocity of the fluid and temperature inside the boundary layer rise as well. Also, the Soret effect drops temperature and concentration profile. The applications of this type of problem are found in the processes of nuclear reactors, corrosion of heat exchangers, lubrication theory, and so forth.     

MHD,thermal radiation,thermophoresis and Soret–Dufour mechanism contributions on unsteady motion of Casson fluid over a semi-infinite porous vertical plate

This paper examined the contribution of MHD, thermal radiation, thermophoresis and Soret–Dufour mechanism on unsteady motion of Casson non-Newtonian fluid. The flow model which resulted in partial differential equations (PDES) was transformed into a dimensionless form of PDES using suitable quantities. The contributions of controlling flow parameters were graphically presented for velocity, temperature, and concentration. The quantities of engineering interest were calculated for flow parameters and presented using table. A considerable value of Soret parameter was noticed to elevate the skin friction and Sherwood number. Impact of the magnetic parameter has great impact on the motion of Casson fluid by lowering its motion. This is because as the value of magnetic parameter increases the Lorentz force added more strength and hereby slow down the motion of an electrically conducting fluids. The present outcomes were examined with previously published work and was in conformity with each other.     

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.     

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.     

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.     

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.     

Soret and Dufour effects between two rectangular plane walls with heat source/sink

This study addresses the thermo‐diffusion and the diffusion‐thermo phenomena in a semi‐infinite absorbent channel whose walls are contracting/expanding, with heat source/sink effects. The governing partial differential equations with suitable boundary conditions are transformed to a system of dimensionless ordinary differential equations. An analytic solution of the problem has been found using a technique called homotopy analysis method (HAM). HAM gives consistently valid answers to the problem over an extensive variety of parameters and also provides better accuracy. To validate the analytical results, a comparison has been presented with a numerical solution calculated by using the parallel shooting method. The effects of dimensionless parameters, that is, deformation parameter, Reynolds number, Soret and Dufour numbers, and heat source/sink parameter on the expressions of velocity, temperature, and concentration profiles are analyzed graphically to understand the physics of the deformable channel. It has been noted that the velocity across the channel is higher for the expanding channel, as compared to that for the contracting channel. Also the Soret and Dufour number increases the temperature of the fluid, and decreases the concentration. The temperature profile has an increasing behavior in the case of heat source, and a decreasing behavior in the case of heat sink.     

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.     

Mechanism of Soret‐Dufour,magnetohydrodynamics, heat and mass transfer flow with buoyancy force,and viscous dissipation effects

This paper examined the mechanism of both positive and negative effects of Soret‐Dufour with heat and mass transfer processes over an accelerating permeable surface. The partial differential flow equations were simplified using similarity variables, and the resulting equations were solved numerically using the spectral homotopy analysis method (SHAM). The SHAM is used in separating nonlinear equations into linear and nonlinear. The physics of each pertinent flow parameters was used to examine their influence on velocity, temperature, and concentration fields. The effect of Soret‐Dufour was examined separately, and its negative effect was used to determine its influence on velocity, temperature, and concentration fields. The result revealed that positive Soret‐Dufour enhances the boundary layer, whereas negative Soret‐Dufour parameter decreases the boundary layer. The result presented in this paper is in good agreement with existing works in literature.     

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 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.     

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.