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 nonlinear natural convection from a uniformly moving porous vertical plate with constant heat and mass flux in the presence of thermal radiation,diffusion-thermo,heat sink,and chemical reaction

An attempt has been made to investigate the problem of nonlinear free convection heat and mass transfer flow past an infinite vertical porous plate embedded in a porous medium by taking into account thermal radiation and heat sink with constant heat and mass flux. Transversely oriented and of uniform strength $B_0$, a magnetic field has been introduced to the fluid area. The nonlinear density variation with temperature as well as concentration are the basis for the current physical situation, which is explained by this mathematical model. Exact solutions are derived for momentum equation, energy equation, and species continuity equation under the relevant boundary conditions. The dimensionless governing equations are analytically solved. The influence of various physical parameters, such as Dufour number, Schmidt number, thermal Grashof number, magnetic parameter, mass Grashof number, heat sink, thermal radiation, Prandtl number, chemical reaction parameter on the flow, and transport characteristic, has been presented graphically and in tabular form. The novelty of the present investigation is that here both constant heat and mass flux at the plate are taken into account in addition to thermal radiation and heat sink. The findings of the mathematical study demonstrate that velocity, temperature, and skin friction intensify with a rise in the Dufour number this is due to the fact that the convection current becomes stronger as the Dufour number rises. Fluid's concentration declines as the Schmidt number grows, or the concentration rises as the mass diffusivity rises. Fluid temperature is enhanced with high thermal diffusivity. Frictional resistance on the plate hikes due to thermal buoyancy force.     

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–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.     

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.     

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.     

Impact of viscous dissipation and Dufour on MHD natural convective flow past an accelerated vertical plate with Hall current

The present research work concentrates on viscous dissipation, Dufour, and heat source on an unsteady magnetohydrodynamics natural convective flow of a viscous, incompressible, and electrically conducting fluid past an exponentially accelerated infinite vertical plate in the existence of a strong magnetic field. The presence of the Hall current induces a secondary flow in the problem. The distinguishing features of viscous dissipation and heat flux produced due to gradient of concentration included in the model along with heat source as they are known to arise in thermal-magnetic polymeric processing. The flow equations are discretized implicitly using the finite difference method and solved using MATLAB fsolve routine. Numerical values of the primary and secondary velocities, temperature, concentration, skin friction, Nusselt number, and Sherwood number are illustrated and presented via graphs and tables for various pertinent parametric values. The Dufour effect was observed to strengthen the velocity and temperature profile in the flow domain. In contrast, due to the impact of viscous dissipation, the local Nusselt number reduces. The study also reveals that the inclusion of the chemical reaction term augments the mass transfer rate and diminishes the heat transfer rate at the plate.     

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.     

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.     

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.     

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.     

Heat and Mass Transfer by Mixed Convection from a Vertical Slender Cylinder with Chemical Reaction and Soret and Dufour Effects

This work is focused on the study of heat and mass transfer by mixed convection over a vertical slender cylinder in the presence of chemical reaction and thermal‐diffusion and diffusion‐thermo effects. The resulting equations have the property whereby they reduce to various special cases previously considered in the literature. An adequate 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. Representative results for the local skin‐friction coefficient, local Nusselt number, and the local Sherwood number illustrating the influence of the surface transverse curvature parameter, Richardson number, concentration to thermal buoyancy ratio, Schmidt number, chemical reaction, and the Dufour and Soret numbers are presented and discussed. © 2013 Wiley Periodicals, Inc. Heat Trans Asian Res, 42(7): 618–629, 2013; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.21045     

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.     

The skin friction in the MHD mixed convection stagnation point with mass transfer

Numerical results are presented for the effects of mass transfer on steady two-dimensional laminar MHD mixed convection owing to the stagnation flow against a heated vertical semi-infinite permeable surface. These results are obtained by solving the coupled nonlinear partial differential equations describing the conservation of mass, momentum and energy by a perturbation technique. These results are presented to illustrate the influence of the Hartmann number, wall mass transfer coefficient, heat absorption coefficient, Prandtl number and the mixed convection or buoyancy parameter. Numerical results for the dimensionless velocity profiles, the temperature profiles, the local friction coefficient and the local Nusselt number are presented for various parameters. These effects of the different parameters on the velocity and temperature as well as the skin friction and wall heat transfer are presented graphically.     

Viscous dissipation and radiation absorption effects on unsteady MHD free convective fluid flow past an inclined porous plate with chemical reaction

The current scrutinization concentrated on the consequences of viscous dissipation and chemical reaction on unsteady MHD two-dimensional free convective fluid flow past a semi-infinite inclined permeable plate with radiation absorption and heat generation. The governing equations are determined analytically by employing the perturbation technique. The impact of various physical estimators on velocity, temperature, concentration, skin friction, and Nusselt number along with Sherwood number were exemplified quantitatively through graphs. It was concluded that velocity declined with the incremental values of Eckert number, but contradictory impact occurred in the case of skin friction. In addition temperature, Nusselt number, as well as velocity, declined with the progressive values of radiation absorption. However, skin friction was accelerated with the augmented values of radiation absorption. Velocity accelerated, with the progressive values of angle of inclination. Concentration declined with the various augmentation values of chemical reaction as well as Schmidt number.     

The eventuality of diffusiophoresis and chemical reaction in a flow with variable fluid properties through an upright channel

The quintessence of this article encompasses the effect of diffusiophoresis, chemical reaction, and varying viscosity as well as thermal conductivity on a fully developed dissipative flow through an upright channel. The fluid is electrically conducting undergoing mixed convection. The governing equations, after transfiguring into dimensionless formation, are solved through a numerical procedure for boundary value problems incorporating MATLAB solver. Imperative scrutiny is made to visualize associative impacts of flow parameters, namely, magnetic parameter, the Brinkmann number, the Schmidt number, variable viscosity parameter, variable thermal conductivity parameter, chemical reaction parameter, diffusiophoretic parameter, and particle diffusion parameter, on the flow. The velocity field, the temperature field, the solute concentration field, the particle concentration field, the skin friction, the Nusselt number, the Sherwood number, and the particle concentration gradient are assessed in view of alteration of the aforesaid parameters with the help of visual illustrations in graphical form and tabular form. Solute mass transposition and colloidal particle locomotion are the fresh inclusions to the scrutiny of upright channel flow in light of solving scheme of bvp4c. Chemical reaction engulfs both solute and particle concentration. Growing viscosity hinders the fluid velocity and heats up the flow encouraging interlayer friction.     

Natural convection laminar flow between two vertical walls with a nonlinear variation of density and viscosity with temperature

Under the combined influence of buoyancy force and constant pressure gradient, the free convection flow and heat transmission inside a channel placed vertically and consisting of two parallel walls are examined. Through the application of the Runge–Kutta fourth-order approach and the shooting procedure, the modeled equations of the problem including nonlinear density–temperature and quadratic viscosity–temperature change at constant but different wall temperatures are numerically solved. For distinct values of the embedded parameters, the fluctuations in fluid's velocity and temperature are examined. For various situations of linear, quadratic, and nonlinear dependence of density on temperature, numerical values for heat transmission rate, volume flow rate and skin friction are determined and tabulated. The fluctuations of the Nusselt number with the Grashof number corresponding to thermal expansion coefficients are depicted through graphical interpretations. It was found that the combined effect of thermal expansion coefficients raises the values of the physical entities, Nusselt number, volume flow rate, and heat transmission rate. For the lower Reynolds number values, the Nusselt number value stays around one, indicating the same impact of conduction and convection on heat transmission.     

Effect of Dufour and chemical reaction on an unsteady magneto hydrodynamics flow past an exponentially moving plate

The aim of the study is to measure the Dufour number effects on the flow patterns and heat transfer in an exponentially accelerated infinite vertical plate embedded in a porous medium in the presence of heat source and chemical reaction. Time-dependent variations in temperature, velocity, and other factors should be taken into consideration due to the flow's unsteadiness. The fluid considered is a gray, absorbing/emitting radiation but nonscattering medium. Using the finite element method, a set of nondimensionless equations is solved analytically. Results are discussed graphically for concentration, temperature, and velocity profiles. Skin friction, Sherwood number, and Nusselt number are also explained for flow parameters through graphs.     

Natural MHD convection for an impulsively started infinite vertical plate with diffusion-thermo effect,induced magnetic field,and ramped wall temperature and concentration

The major motive of the current investigation is to analyze an exact solution for an unsteady magnetohydrodynamics natural convective flow of a viscous incompressible electrically conducting non-gray optically thick fluid past an impulsively started infinite vertical plate with ramped wall temperature and concentration in the presence of radiation, diffusion-thermo effect, and induced magnetic field. Laplace transform method is used to acquire the specific solutions of the dimensionless domain equations. The impact of various physical parameters on fluid velocities, temperature, concentration, and moreover on the rate of heat transfer, mass transfer, and skin friction at the wall are shown graphically. It is also found that the Dufour effect causes an increase in fluid velocity as well as temperature. The ramped condition influences the rise in Nusselt number.     

Variable viscosity and thermophoresis for Soret and Dufour's effects on MHD generation in a non‐Daracian porosity fluid

The effect of thermophoresis on a magnetic field generation in a non‐Daracian porous medium flow with the variation of the viscosity fluid in the presence of Soret and Dufour, thermal reaction and diffusion effects over a stretching surface is investigated in the present analysis. The governing equations of continuity, momentum, energy, and concentration are transformed into nonlinear ordinary differential equations, using similarity transformations and then solved numerically. The influence of various physical parameters on velocity, temperature, and concentration profiles are illustrated graphically, and the physical aspects are discussed in detail. Finally, the effects of related physical parameters on the skin friction, the rate of heat and mass transfer are also studied.