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.     

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.     

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.     

Finite difference analysis for entropy optimized flow of Casson fluid with thermo diffusion and diffusion-thermo effects

Heat transportation is a novel prospective in many thermal processes and presents dynamic applications in industrial and thermal polymer processing optimization. The importance of heat transportation is noted in heat exchangers, production of crude oils, combustion, petroleum reservoirs turbine systems, thermal systems, porous media, modeling of resin transfer nuclear reactions etc. In view of such thermal applications the main objective here is to examine entropy in unsteady magnetohydrodynamic of Casson fluid flow. Radiation in addition to dissipation and ohmic heating are analyzed. Entropy is scrutinized employing thermodynamic second law. Characteristics of Soret and Dufour are also examined. Main objective here is to examine irreversibility. Dimensionless version of differential system is obtained through suitable variables. The obtained partial differential system is solved through numerical scheme (Finite difference method). Physical features of fluid flow, temperature, entropy optimization and concentration have been explained. Variations of parameters on drag force, Nusselt number and solutal transfer rate are graphically discussed. Higher fluid parameter leads to improve in velocity and entropy rate. Larger values of radiation parameter boost up thermal field. Entropy rate and velocity have reverse trend for magnetic field. An intensification for concentration is found through Soret number. Higher approximation of Reynold number enhances skin friction and velocity. Thermal transfer rate is augmented versus radiation and magnetic variables.     

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.     

Heat and mass transfer in MHD Casson nanofluid flow past a stretching sheet with thermophoresis and Brownian motion

The foremost objective of the current article is to explore the impact of Brownian motion on magnetohydrodynamic Casson nanofluid flow toward a stretching sheet in the attendance of nonlinear thermal radiation. The combined heat and mass transfer characteristics are investigated. The influence of chemical reaction, nonuniform heat source/sink, Soret, and Dufour is deemed. The convective boundary condition is taken. The appropriate transformations are utilized to transform the flow regulating partial differential equations into dimensionless ordinary differential equations (coupled). The numerical outcomes of the converted nonlinear system are solved by the Runge-Kutta based Shooting procedure. Results indicate that the temperature is an increasing function of both thermophoresis and Brownian motion parameters. The concentration of the fluid and the corresponding boundary layer thickness reduces with an enhancement in Lewis number.     

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.     

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.     

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.     

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.     

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.     

Hydromagnetic mixed convection unsteady radiative Casson fluid flow towards a stagnation-point with chemical reaction,induced magnetic field,Soret effect,and convective boundary conditions

This article presents the two-dimensional mixed convective MHD unsteady stagnation-point flow with heat and mass transfer on chemically reactive Casson fluid towards a vertical stretching surface. This fluid flow model is influenced by the induced magnetic field, thermal radiation, viscous dissipation, heat absorption, and Soret effect with convective boundary conditions and solved numerically by shooting technique. The calculations are accomplished by MATLAB bvp4c. The velocity, induced magnetic field, temperature, and concentration distributions are displayed by graphs for pertinent influential parameters. The numerical results for skin friction coefficient, rate of heat, and mass transfer are analyzed via tables for different influential parameters for both assisting and opposing flows. The results reveal that the enhancement of the unsteadiness parameter diminishes velocity and induced magnetic field but it rises temperature and concentration distributions. Moreover, higher values of magnetic Prandtl number enhance Nusselt number and skin friction coefficient, but it has the opposite impact on Sherwood number. We observe that the amplitude is higher in assisting flow compared to opposing flow for skin friction coefficient and Nusselt number whereas opposite trends are noticed for Sherwood number. Our model will be applicable to various magnetohydrodynamic devices and medical sciences.     

Numerical solution for natural convection flow near a vertical porous plate having convective boundary condition with nonlinear thermal radiation

This article presents the theoretical study of the effects of suction/injection and nonlinear thermal radiation on boundary layer flow near a vertical porous plate. The importance of the convective boundary condition as regards the heat transfer rate is taken into account. The coupled nonlinear boundary layer equations are translated into a set of ordinary differential equations via a similarity transformation. The consequences of the active parameters like the suction parameter, injection parameter, convective heat transfer parameter, nonlinear thermal radiation parameters, and Grashof number dictating the flow transport are examined. The numerical result obtained shows that with suction/injection, the heat transfer rate could be increased with nonlinear thermal radiation parameter augment whereas decays with the convective heat transfer parameter and Grashof number. In the presence of suction/injection, the wall shear stress generally increases with nonlinear thermal radiation parameter, convective heat transfer parameter, and Grashof number. The suction has an increasing effect on Nusselt number and shear stress whereas a decreasing effect on Nusselt number and skin friction is seen with injection augment. The nonlinear thermal radiation is an increasing function of the temperature gradient far away from the plate whereas a decreasing function near the porous plate.     

Double‐Diffusive Convective Transport in a Nanofluid‐Saturated Porous Layer with Cross Diffusion and Variation of Viscosity and Conductivity

The onset of double‐diffusive nanofluid convection in a fluid‐saturated horizontal porous layer is studied with thermal conductivity and viscosity dependent on the nanoparticle volume fraction. The Darcy model has been used for the porous medium, while the nanofluid incorporates the effects of Brownian motion along with thermophoresis. The nanofluid is assumed to be diluted and this enables the porous medium to be treated as a weakly heterogeneous medium with variation in the vertical direction of conductivity and viscosity. In addition, the thermal energy equation includes regular diffusion and cross diffusion terms. The linear stability analysis is based on the normal mode technique, while for nonlinear analysis, minimal representation of the truncated Fourier series representation involving only two terms has been used. It is found that for the stationary mode the Soret parameter, Dufour parameter, viscosity ratio, and conductivity ratio have a stabilizing effect, while the solutal Rayleigh number destabilizes the system. For the oscillatory mode, the Soret parameter, Dufour parameter, and viscosity ratio have a stabilizing effect while the solutal Rayleigh number and conductivity ratio destabilize the system. For steady finite amplitude motions, the heat and mass transport decreases with an increase in the values of the Dufour parameter and solutal Rayleigh number. The Soret parameter enhances the solute concentration Nusselt number while it retards the thermal Nusselt number and concentration Nusselt number. The viscosity ratio and conductivity ratio enhances the heat and mass transports. We also study the effect of time on transient Nusselt numbers which is found to be oscillatory when time is small. However, when time becomes very large, all three transient Nusselt values approach a steady value. © 2013 Wiley Periodicals, Inc. Heat Trans Asian Res, 43(7): 628–652, 2014; Published online 11 November 2013 in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.21102     

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

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

Soret and Dufour effects on free convection boundary 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.     

Effects of chemical reaction,Soret and Lorentz force on Casson fluid flow past an exponentially accelerated vertical plate: A comprehensive analysis

The aim of the current study is to explore the effects of heat and mass transfer on unsteady chemically reacted Casson liquid flow over an exponentially accelerated vertical plate in a porous medium. It is assumed that the bounding plate has varying temperatures as well as concentrations in a porous medium under a uniform magnetic field. This phenomenon is modeled in the form of a system of partial differential equations (PDEs) with boundary conditions. The governing dimensionless PDEs are solved using Laplace transform method for velocity, temperature, and concentration. The impact of nondimensional parameters, which are controlling the flow like Casson parameter, Soret number, magnetic parameter, heat generation parameter, Prandtl number, radiation parameter, and Schmidt number is analyzed through graphs. The incremental values of the Casson fluid parameter lead to a reduction in velocity and discovered that for large values of the Casson parameter, the fluid is near to the Newtonian fluid. Also, the Sherwood number increases with enhancing dissimilar estimators of the Schmidt and Soret numbers. A comparison has been made with the published work (Kataria et al.) for a particular case, which was in good agreement.     

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.     

Chemically reacting mixed convective Casson fluid flow in the presence of MHD and porous medium through group theoretical analysis

This paper explores the consequences of chemically reacting magnetohydrodynamic mixed convective fluid substances driven by the porous medium, slippery, incompressible, and laminar vertical channel flow. Casson fluid model in a vertical channel is strengthened with mixed convection flow. The effects of the heat source-sink parameter, the suction-injection parameter, slips on the slide wall, and thermal radiation are also considered. A Lie group method is taken into consideration and nonlinear partial differential equations are converted into nonlinear ordinary differential equations (ODEs). The NDSolve command solves these ODEs and shows the action of the related parameters in the velocity, temperature, and concentration figures. The Casson fluid parameter increases the velocity profile but reduces the concentration profile. The parameter of suction-injection enhances the velocity, temperature, and concentration profiles. The variations in skin-friction coefficient in the heat and mass transfer rate are addressed in the diagrams. Moreover, streamlines are plotted for suction-injection parameter.     

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.