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This work aims to simulate the impacts of exothermic reaction and Soret–Dufour numbers on the double diffusion of Nano Enhanced Phase Change Materials (NEPCM) inside a porous annulus. The complex rectangular annulus contains two ellipses and two triangles on the walls’ vertical sides. The complex proposals of closed domains during heat/mass transfer of NEPCM can be used in energy savings, cooling electronic devices, and heat exchangers. The fractional-time derivative of the governing systems is solved numerically based on the ISPH method. The artificial neural network (ANN) is combined with the ISPH results to predict the average Nusselt number Nu ¯ and Sherwood number Sh ¯ . The main objective of establishing the ANN model in this investigation is to create a reliable predictive instrument capable of estimating the values of Nu ¯ and Sh ¯ . The results described the impacts of dimensionless Frank-Kamenetskii number (Fk = 0–1), Darcy number (Da = 10−2–10−5), Dufour number (Du = 0–0.1), buoyancy ratio (N = − 2 to 5), Rayleigh number (Ra = 103–106), Lewis number (Le = 1–20), Soret number (Sr = 0–0.2), fusion temperature (θf = 0.05–0.9), and fractional order parameter (α = 0.9–1) on thermosolutal convection of a suspension. The overall heat/mass transition as well as the velocity field are dramatically enhanced when Ra and N were boosted. The fractional time derivative helps reach a steady state in less time instants. The phase change material (PCM) is always changed when temperature distribution changes and is controlled by a fusion temperature. The porous struggled with nanofluid flow at a lower Darcy number. Frank-Kamenetskii number is a promising factor in enhancing the temperature distributions in an annulus. As a result, this work may be applied in various engineering and industrial fields because it contains significant terms in improving heat/mass transmission as well as a phase change material. The ANN model introduced a precise agreement of the prediction values with the actual values of Nu ¯ and Sh ¯ . Then, the present ANN model can accurately estimate the Nu ¯ and Sh ¯ values. [ABSTRACT FROM AUTHOR]

This investigation delves into the impact of double diffusion on the heat-generating hydro-magnetic transport of water-based nanofluids flowing past a moving permeable plate subjected to heat flux. The nanofluids considered incorporate copper (Cu) and titanium dioxide (TiO2) nanoparticles. The study is characterized by the formulation of dimensional quasi-linear partial differential equations (PDEs) with well-defined initial and boundary conditions, which are rendered dimensionless by introducing appropriate non-dimensional parameters. These equations are then resolved numerically using the implicit Crank-Nicolson finite difference scheme implemented in MATLAB. The research meticulously examines the influence of pertinent physical parameters on the velocity profile, temperature distribution, and concentration of reactive species, complemented by graphical illustrations. Furthermore, the study elucidates the impact of these physical variables on the wall shear stress, heat transfer rate, and mass transfer rate, presented in tabular format and subjected to comprehensive analysis. The findings underscore that an increase in the Soret parameter enhances fluid velocity and reactive species concentration, whereas the intensification of the magnetic field and heat flux parameter diminishes fluid velocity. The rate of thermal transfer is observed to escalate with higher values of the heat source, radiation-absorption, and radiative heat flux parameters; conversely, it diminishes with an increase in the Prandtl number. The study reveals superior thermal performance for Cu nanoparticles compared to TiO2. A comparative analysis with prior published results substantiates the accuracy of the computational framework, with the outcomes exhibiting exceptional concordance. [ABSTRACT FROM AUTHOR]

This paper examines the entropy generation and magneto-Soret effects on the convective-radiative heat transport of a chemically reactive fluid past an oscillatory surface in a permeable media with an internal heat source and suction at the porous wall. The perturbation scheme is adopted to convert the basic nonlinear partial differential equations into a set of linear ordinary differential equations that are solved analytically. It is found that the viscous dissipative force and the thermal radiation parameter both reduce the concentration distribution within the fluid. Physically, higher thermal radiation can increase the temperature within the fluid, leading to a decrease in concentration distribution. Conversely, an increase in the Soret parameter, which accounts for the effect of temperature gradients on mass diffusion, results in an opposite trend, enhancing the concentration distribution. The Soret effect, or thermal diffusion, causes mass to move from regions of high temperature to low temperature. Additionally, the total entropy of the fluid is examined across various values of the Reynolds number, Brinkman number, temperature difference parameter, and magnetic parameter. As the magnetic parameter increases, the entropy within the system rises, indicating enhanced irreversibilities and energy dissipation due to the presence of a magnetic field. [ABSTRACT FROM AUTHOR]

The aim of this research paper is to investigate the rotational flow of an unsteady magnetohydrodynamic heat and mass transfer flow due to convection over a vertical porous semi-infinite plate. The plate undergoes continuous circular motion, maintaining a constant velocity. To achieve this, we worked on both numerical methods and analytical techniques, particularly utilizing perturbation methods to solve the governing partial differential equations. Consequently, we derive an expression for the Nusselt and Sherwood numbers. We delve into the analysis of the velocity profile, temperature distribution, and concentration variation, exploring their behaviour under different physical parameters, including the magnetic field parameter, Grashof number, Soret number and Schmidt number, as well as the Prandtl number. Our findings reveal that the velocity increases with rising values of Grashof, modified Grashof and Soret numbers, whereas it decreases with declining values of the magnetic field parameter, Prandtl number and Schmidt number. Additionally, as rotation gradually intensifies, the fluid velocity closely follows the boundary and becomes negligible as it moves away from it. To facilitate a comprehensive examination of the fluid flow and heat and mass transfer characteristics, we employ graphical representations. Furthermore, this paper offers an indepth discussion of the underlying physical aspects and their implications. [ABSTRACT FROM AUTHOR]

This study investigates the heat and mass transmission behavior in an unsteady magnetohydrodynamic (MHD) movement of nanofluids over an inclined permeable surface, with applications in enhancing thermal management systems such as automotive cooling and industrial heat exchangers. The model specifically examines the consequence of thermal diffusion (Soret effect) and buoyant forces on Cu and TiO2 nanoparticles dispersed in engine oil. The governing equations, comprising velocity, energy, and concentration equations, are recast into nonlinear ODEs manipulating similitude adaptations. These ODEs are then solved through a standard perturbation method under appropriate boundary conditions. The key findings indicate that enhancing thermal radiation diminishes the velocity and temperature profiles, while raising chemical reaction rates decrease concentration levels. Additionally, higher Soret parameter values are associated with increased velocity and concentration. Quantitatively, TiO2-engine oil nanofluids exhibit a 15% higher velocity compared to Cu-engine oil nanofluids, highlighting the superior performance of TiO2 in dynamic thermal systems. Furthermore, numerical outcomes for the local skin contention, Nusselt numeral, and Sherwood digit are tabulated to illustrate the consequence of material properties. The outcomes of this study are particularly beneficial in optimizing the design of heat exchangers, improving fuel efficiency in automotive engines, and enhancing industrial processes where precise thermal control is critical.

This research offers an analysis of the mixed convective transient boundary film Casson fluid stream and thermal distribution through a vertical sheet. Viscous dissipation and the Soret effect are introduced to support the flow in stimulating heat capacity. Unlike typical investigations, the present flow-formulated model is done to capture an induced magnetic field. The Boussinesq approximation is used to describe the nonlinear formulated partial derivatives governing the heat transfer fluid that is non-dimensionalized using suitable dimensionless quantities. The transformed partial derivative model is numerically solved via the spectral Chebyshev technique and the results of shear stress, current density, Nusselt number, and mass gradient are tabulated. The role of numerous terms on dimensionless flow rate, induced velocity, and heat transfer with species distribution is discussed in a very effective way. Velocity and temperature of liquid decline as boosting the material parameter. Enhancing values G r , G r m favor the flow momentum and oppose the temperature profile. [ABSTRACT FROM AUTHOR]

AbstractUnderstanding and controlling the shape factors of nanoparticles in fluid flow problems is crucial for optimizing heat transfer, fluid dynamics, and various applications such as nanofluids, drug delivery systems, catalysis, and nanocomposites. By tailoring the shape of nanoparticles, researchers can manipulate their behavior, dispersion, and interaction with the fluid, thereby optimizing the desired outcomes in various fluid flow problems. The meticulous investigation uncovers the integration of entropy analysis in the investigation of the flow of convective magnetohydrodynamic hybrid nanofluid via a permeable flat surface, by considering the effects of Dufour, viscous dissipation, Soret, and thermal radiation. The governing equations are converted into a set of nonlinear ordinary differential equations using appropriate similarity transformations and then solved using the bvp4c solver, a MATLAB in-built function. We display the outcomes for three different shape factors: platelet, cylinder, and brick. The skin friction coefficient increases at a rate of 1.5089 for the brick shape, 1.5163 for the cylindrical shape, and 1.5212 for the platelet shape when the mixed convection parameter lies between 1 and 5. In this regard, it is important to point out that there is a direct connection between the increase in temperature transmission rate and a decline in the Dufour number (Du). It has been found that when 1≤Du≤2.4, the Nusselt number decreases by 0.4786, 0.4618, and 0.4512 for brick, cylinder, and platelet shapes, respectively. A decline in the Sherwood number is noticed with an upsurge in the Soret number (Sr). There is a noticeable increase of 0.0442 in the Sherwood number for brick shape, 0.0438 for cylinder form, and 0.0435 for platelet form when 1≤Du≤2.4. Furthermore, we have detected that an increase in the radiation parameter amplifies both the Bejan number and the profiles of entropy formation. [ABSTRACT FROM AUTHOR]

Here, we consider magnetohydrodynamic flow of an incompressible, time independent fluid past an elongated cylinder surrounded in a non-Darcian porous regime with magnetic flux supplied at an acute angle. The Soret/Dufour effects and the higher order chemical reactions are also included in the present study. The subsequent governing equations are resolved using the MATLAB-bvp4c method. The flow velocity appears to decrease with the growth of the Reynolds number, inertia parameter, magnetic field and angle of inclination of the magnetic flux, but improves with the Darcy number. The inertia parameter enhances the fluid temperature and skin friction. Further order of chemical reaction, Soret/ Dufour number plays a significant role in the system.

The present study investigates heat and mass transport phenomena associated with the MHD flow of micropolar fluid over a vertically stretched Riga plate under the action of a uniform magnetic field applied parallel to the plate. The objective of the study is to analyze Soret and Dufour effects on this physical situation in the presence of chemical reaction. The governing partial differential equations are converted into ordinary differential equations using suitable similarity transformations. The equations are solved numerically by developing programming codes in MATLAB for the very efficient shooting method along with the fourth order Runge-Kutta scheme. The velocity, microrotation, temperature and species concentration distribution are presented graphically for various emerging physical parameters like Hartmann number, material parameter, Soret number, Dufour number and other dimensionless parameters. It is found that the species concentration distribution profiles increase with increasing Soret number, whereas the temperature distribution profile decreases with an increasing Soret number. This work also provides solutions for shear stress at plates, the rate of heat and mass transfer in addition to those for velocity, microrotation, temperature and species concentration. Comparisons with previous studies are carefully examined, and it is found that they are generally in agreement. [ABSTRACT FROM AUTHOR]

Here, we consider magnetohydrodynamic flow of an incompressible, time independent fluid past an elongated cylinder surrounded in a non-Darcian porous regime with magnetic flux supplied at an acute angle. The Soret/Dufour effects and the higher order chemical reactions are also included in the present study. The subsequent governing equations are resolved using the MATLAB-bvp4c method. The flow velocity appears to decrease with the growth of the Reynolds number, inertia parameter, magnetic field and angle of inclination of the magnetic flux, but improves with the Darcy number. The inertia parameter enhances the fluid temperature and skin friction. Further order of chemical reaction, Soret/Dufour number plays a significant role in the system. [ABSTRACT FROM AUTHOR]

A steady, incompressible, and thermo-convective flow of micro-polar fluid over a stretching permeable sheet with heat and mass transfer under effects of radiation, Soret, Schmidt, and Dufour numbers have been analyzed. The modeled governing equations, of the classical Navier-Stokes, are coupled with micro rotation, temperature, and concentration equations, in the form of Partial Differential Equations, along with initial and boundary conditions, are transformed into a system of nonlinear coupled Ordinary Differential Equations by using an appropriate transformation. The numerical solution is obtained by using the Parametric Continuation Method. For the validity of the scheme, the results are compared with a numerical package bvp4c. It has been observed that both the results are in the best agreement with each other. The effects of associated parameters on the dimensionless velocity, micro-rotation, temperature, and concentration profiles are discussed and depicted graphically. It has been detected that the permeability parameter gives rise to the microrotation profile. [ABSTRACT FROM AUTHOR]

This research investigates how the presence of Dufour and Soret parameters affect the properties of hybrid nanofluid flow amidst two parallel plates with temperature dependent electrical conductivity. Heat transport procedure is studied using thermal radiation parameter and Eckert number. Equations that characterize the current problem are turned into a set of ordinary differential equations, which are then solved with the assistance of a built-in function in MATLAB called the bvp4c solver. Multiple linear regression is used to explain physical parameters of interest including Sherwood number. The main conclusions of this study are that the velocity decreases with increasing nanoparticle volume fraction parameter and the concentration profile minifies with the rise in Soret parameter. It is observed that the increase in Dufour number causes a corresponding increase in the fluid temperature. It is discovered that, when the magnetic field parameter (M) is set to 0≤M≤2.5, the decreasing rate in the skin friction parameter is 0.02215. It is observed that, when φ1 is set to 0≤φ1≤0.25, Nusselt number drops by 3.82891. It is detected that, when Sr is set to 0≤Sr≤2.5, Sherwood number increases at a rate of 0.037225.

The consequences of viscous, ohmic, thermal radiation, and chemical reaction on an incompressible, steady, magnetized nanofluid moving through a stretched surface are investigated for choosing two distinct Nanoliquids (Cuo/water and Silver/water). Through the similarity transformations, the controlling dimensional equations of momentum, energy, and concentration were simplified to non-dimensional forms. The solution to the resulting problem is obtained by Bvp4c. The velocity, temperature and concentration are significantly shown for various emerging factors. On both kinds of nanofluids, thermal is grown consistently with an increasing Eckart number. It does, however, lessen as the radiation term values increase. Moreover, if the Schmidt number ranges on the concentration fluid flow indicate that the H(η) profile is going to fall. The tendency for H(η) to boost as the Soret number raises is becoming increasingly noticeable. Ag/water has a higher heat flow rate than Cuo/water when the amounts of Eckart and Radiation numbers are changed. Tables and plots displayed an engineering physical parameter such as coefficient of skin friction, rate of heat transmission and mass transfer. [ABSTRACT FROM AUTHOR]

The purpose of the study is to discuss the comparison of the linear and nonlinear stretching sheets on the heat transfer over a curved surface spiraling in a circle with radial distance R. Additionally the consequences of Soret–Dufour in the presence of injection velocity and heat source/sink are investigated in the article. Silver and Copper Oxide are taken as hybrid nanoparticles with water as foundation fluid. Leading equations are converted to nonlinear ordinary differential equations taking some relevant transformations and solved with the combination of the shooting method and RK‐4th order method. MAPLE software is used to plot graphs and numerical charts. Flow features are examined for different pertinent parameters. Physical quantities like Sherwood number, Nusselt number, and so forth, are evaluated numerically for different nondimensional numbers. The result substantiates that improving the Dufour number boost the fluid flow and fluid temperature but the opposite behavior is seen in the concentration profile. It is observed that a linearly stretching sheet gives better fluid behaviors compared to nonlinear stretching sheet. [ABSTRACT FROM AUTHOR]

This work aims to study the nonlinear ordinary differential equations (ODEs) system of magnetohydrodynamic (MHD) past over an inclined plate using Levenberg-Marquardt backpropagation neural networks (LMBNNs). The stochastic procedures LMBNNs are provided with three categories of sample statistics, testing, training, and verification. The nonlinear MHD system past over an inclined plate is divided into three profiles, dimensionless momentum, species (salinity), and energy (heat) conservations. The data is applied 15%, 10%, and 75% for validation, testing, and training to solve the nonlinear system of MHD past over an inclined plate. A reference data set is designed to compare the obtained and proposed solutions for the MHD system. The plots of the absolute error (AE) are provided to check the accuracy and precision of the considered nonlinear system of MHD. The obtained numerical solutions of the nonlinear magnetohydrodynamic system have been considered to reduce the mean square error (MSE). For the capability, dependability, and aptitude of the stochastic LMBNNs procedure, the numerical performances are provided to authenticate the relative arrangements of MSE, error histograms (EHs), state transitions (STs), correlation, and regression. [ABSTRACT FROM AUTHOR]

This article inspects the effect of triple diffusion in a vertical conduit encapsulated with porous matrix and subjected to third kind boundary conditions. Third kind boundary condition is a combination of Dirichlet and Neumann boundary conditions which specifies a linear combination of function and its derivative values on the boundary. Homogeneous chemical reaction along with viscous and Darcy dissipation effects are included. Adapting the Boussinesq approximation, the soultal buoyancy effects due to concentration gradients of the dispersed components are taken into account. Applying suitable transformations, the conservation equations are reduced into dimensionless form and the dimensionless parameters evolved are thermal Grashof number (0≤Λ1≤20), solutal Grashof number (for species 1 and 2, 0≤Λ2,Λ3≤20), porous (2≤σ≤8) and inertial parameters (0≤I≤6), Biot numbers (at the left and right walls, 1≤Bi1,Bi2≤10), Brinkman number (0≤Br≤1), Schmidt numbers (0≤Sc1,Sc2≤6), Soret numbers (Sr1=Sr2=1) and temperature difference ratio (RT=1). Adopting perturbation technique, the analytical solutions which are applicable only when the Brinkman number is less than one is appraised. However for any values of the Brinkman number, Runge-Kutta shooting method is operated. The impact of selected parameters on the momentum, heat and dual species concentration fields are presented in the form of pictures. The solutions computed by numerical method are justified by comparing with the analytical method. The numerical and analytical solutions are equal in the absence of Darcy and viscous dissipations and the discrepancy advances as the Brinkman number expands. Further the solutions obtained are also justified by comparing the results with Zanchini [1] in the absence of chemical reaction for clear fluid. The thermal field is augmented with the Brinkman number for symmetric and asymmetric Biot numbers. However the profiles are highly distinct at the cold plate for unequal Biot numbers in comparison with equal Biot numbers. The conclusions are admissible to materials processing and chemical transport phenomena.

This study investigates the heat and mass transmission behavior in an unsteady magnetohydrodynamic (MHD) movement of nanofluids over an inclined permeable surface, with applications in enhancing thermal management systems such as automotive cooling and industrial heat exchangers. The model specifically examines the consequence of thermal diffusion (Soret effect) and buoyant forces on Cu and TiO2 nanoparticles dispersed in engine oil. The governing equations, comprising velocity, energy, and concentration equations, are recast into nonlinear ODEs manipulating similitude adaptations. These ODEs are then solved through a standard perturbation method under appropriate boundary conditions. The key findings indicate that enhancing thermal radiation diminishes the velocity and temperature profiles, while raising chemical reaction rates decrease concentration levels. Additionally, higher Soret parameter values are associated with increased velocity and concentration. Quantitatively, TiO2-engine oil nanofluids exhibit a 15% higher velocity compared to Cu-engine oil nanofluids, highlighting the superior performance of TiO2 in dynamic thermal systems. Furthermore, numerical outcomes for the local skin contention, Nusselt numeral, and Sherwood digit are tabulated to illustrate the consequence of material properties. The outcomes of this study are particularly beneficial in optimizing the design of heat exchangers, improving fuel efficiency in automotive engines, and enhancing industrial processes where precise thermal control is critical. [ABSTRACT FROM AUTHOR]

This research study discusses the flow of a magnetohydrodynamic Casson fluid under the influence of Soret, Dufour, and thermal radiation. Nonlinear partial differential equation (PDE) of governing equations is transformed into a dimensionless version of the modified PDEs presented in terms of dimensionless parameters. The solution of coupled PDEs is obtained by the finite difference method with a combination of the quasilinearization technique. The effects of various dimensionless parameters are shown graphically, such as buoyancy force (λ $\lambda $), concentration buoyancy force (λC) $({\lambda }_{{\rm{C}}})$, Casson parameter (β $\beta $), magnetic parameter (H $H$), thermal radiation (Rd $Rd$), Darcy parameter (K0 ${K}_{0}$), Forchheimer (fr), Dufour (Df ${D}_{f}$), Soret (Sor), Brownian motion (Nb $Nb$), thermopohersis (Nt $Nt$), and Lewis number (Le $Le$). Prevention of heat transfer in the industrial system is critical, the velocity behavior (F $F$), thermal variation (θ $\theta $), and concentration profile (ϕ $\phi $) are more prominent in the roles of coal, gas, and solar thermal collectors. [ABSTRACT FROM AUTHOR]

This paper is focused on the study of two dimensional steady magnetohydrodynamics heat and mass transfer by laminar free convection from a radiative horizontal circular cylinder in a non-Darcy porous medium by taking into account of the Soret/Dufour effects. The boundary layer equations, which are parabolic in nature, are normalized into non-similar form and then solved numerically with the well-tested, efficient, implicit, stable Keller–Box finite-difference scheme. Numerical results are obtained for the velocity, temperature and concentration distributions, as well as the local skin friction, Nusselt number and Sherwood number for several values of the parameters, namely the buoyancy ratio parameter, Prandtl number, Forchheimer number, magnetohydrodynamic body force parameter, Soret and Dufour numbers. The dependency of the thermophysical properties has been discussed on the parameters and shown graphically. Increasing the Forchheimer inertial drag parameter reduces velocity but elevates temperature and concentration. Increasing the Soret number and simultaneously reducing the Dufour number greatly boosts the local heat transfer rate at the cylinder surface. A comparative study of the previously published and present results in a limiting sense is made and an excellent agreement is found between the results.

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. [ABSTRACT FROM AUTHOR]

The main objective of this study is to investigate the effects of the Casson fluid parameter on an incompressible, magnetohydrodynamic boundary layer flow of a Casson fluid past a moving porous inclined plate in the presence of heat source and first-order chemical reaction. The governing partial differential equations are converted into ordinary differential equations using similarity transformation and then are solved numerically, adopting bv4pc method. The effects of relevant parameters on the velocity, temperature and concentration profiles are analyzed graphically. Also, tabular form is used to present skin friction, heat transfer and mass transfer. This investigation reveals that the Casson fluid parameter enhances the fluid velocity, skin friction and Sherwood number, while the Nusselt number decreases. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The ISPH method is utilized for analyzing the influences of a magnetic field on thermosolutal convection of solid particles spreading within a finned cavity. The nanofluid-filled cavity is containing four vertical fins and an embedded square shape involved solid particles over a square blockage. The governing equations of nanofluid flow and spreading of the solid particles were solved jointly by the ISPH method. The vertical walls, four fins, and an inner square blockage are maintained at Tc and Cc. The embedded square shape with gates and the plane walls are adiabatic. The solid particles are kept at Th and Ch for the duration of the simulation. The propagating of the solid particles and contours of the isothermal, isoconcentration, and streamlines below the impacts of the pertinent parameters are examined. The accomplished simulations recalled that the solid particles propagating often move towards the upper area of a cavity. As a result, the fin on the upper wall acts as an effective tool for adjusting the spread of the solid particles. The blending between solid and fluid particles is affected by an increment in the Hartman parameter. The values of Soret and Dufour numbers (Sr=2&Du=0.03) are reducing the blending of the solid particles within a nanofluid flow.

An analysis has carried out to study effect of Soret number on MHD free convection flow of heat and mass transfer of an electrically conducting non-Newtonian fluid through a vertical moving porous plate. Set of governing non linear coupled differential equations are converted to a non dimensional set of partial differential equations by introducing the suitable variables. Galerkin finite element technique is applied on non dimensional velocity, temperature and concentration profiles to get numerical solutions. The effect of physical quantities chemical reaction, dufour, other flow parameters on profiles of velocity, temperature and concentration are illustrated through graphs.

The characteristics of unsteady motion of Casson fluid over a wavy surface with magnetic field is studied. The energy and diffusion equations are encompassed with radiation, heat source and thermo diffusion effects. The flow governing equations have been solved using regular perturbation technique. MATLAB bvp4c package is for discerning the impact of governing parameters on the usual flow profiles. various pertinent parameters on velocity, temperature and concentration profiles of the Casson fluid flow. The frictional factor, heat and mass transfer coefficients are tabulated for the same parameters. From this analysis, it is worth mentioning that a hike in fluid viscosity slowdowns the velocity profiles but the result is opposite with Soret effect. Also, accelerating values of radiation parameter uplifts the rate of mass and heat transport but controls the shear stress.

In this paper, we discuss the Soret and Dufour effects on an MHD micropolar fluid flow over a linearly stretching sheet, through a non-Darcy porous medium, where stretching velocity of the sheet varies linearly with distance from the origin, and, temperature and concentration vary non-linearly in the boundary layer region. By suitable similarity transformations, the governing boundary layer equations are transformed to ordinary differential equations. These equations are solved by numerical computations with bvp4c along with the shooting technique method. The effects of the magnetic parameter, Soret number and Dufour number on velocity profiles, microrotation profile, heat transfer, and concentration, skin-friction, Nusselt number and Sherwood number are computed, discussed and analysed numerically and presented through tables and graphs.

An asymptotic series solution to the problem of an MHD convective steady three dimensional flow past a semi-infinite porous vertical plate with sinusoidal suction in slip flow regime in presence of a uniform transverse magnetic field, taking into account the effects of thermal diffusion, diffusion-thermo, and thermal radiation is obtained. Momentum slip, thermal slip. and solutal slip at the plate are considered. The flow is three dimensional merely due to sinusoidal suction. Expressions for the velocity field, temperature field, concentration field, and skin friction, Nusselt number, and Sherwood number are obtained. The effects of the physical parameters entering into flow problem on transport properties, for instance the rates of momentum transfer, heat transfer, and mass transfer are mainly focused. [ABSTRACT FROM AUTHOR]

The motivation of the current article is to explore a numerical investigation on steady triply diffusive convection in a vertical channel. Heat is exchanged from the external fluid with the plates. The reference temperature is taken as equal and also as different for the external fluid. Solutions in the absence of viscous dissipation and buoyancy forces are also obtained as special cases. General solutions including the effects of viscous dissipation and buoyancy forces are obtained analytically using the method of perturbation. The analytical solutions can be used only if the Brinkman number is small. Hence to know the flow properties for all values of Brinkman number, we resort to numerical solutions. The effects of thermal Grashof number, solutal Grashof number, and the chemical reaction parameter on the flow field are evaluated numerically. The obtained results are validated against previously published results for special case of the problems. [ABSTRACT FROM AUTHOR]

The present examination emphasizes the effects of heat and mass transfer on the peristaltic flow of Jeffery fluid through a non-uniform channel with variable viscosity and thermal conductivity. The porous walls are considered to make more realistic approximations to the flow characteristics of biological fluids. Further, the convective boundary conditions and wall properties have been employed in the analysis. The mathematical formulation is established on the grounds of low Reynolds number and long wavelength approximations. Perturbation solution is obtained for the resulting nonlinear differential equation of energy for small values of variable thermal conductivity, whereas the exact solution is found for the velocity and concentration fields. The MATLAB software is utilized to generate the graphical representation of the variables used in the model. From the examination, it is seen that a rise in the value of variable viscosity upgrades the velocity, Nusselt number, and temperature fields, though the contrary conduct is seen for concentration profiles. Besides, the rise in volume of the trapped bolus is noticed for an increase in the value of porous and Jeffery parameters. [ABSTRACT FROM AUTHOR]

An investigation has been undertaken as Soret and Schmidt outcomes on the mixed convection flow using Robin boundary conditions. The results use a vertical channel being kept at constant cold temperature and concentration at the left wall and hot temperature and concentration at the right wall. The exchange of heat is done by help of plates with a fluid. We consider the external fluid with equal and different temperatures. This physical problem is solved by using nondimensional parameters with the corresponding boundary conditions. To find analytical solution, the regular perturbation series method is used, and for finding the numerical solution, the well‐known Runge–Kutta method with shooting technique is employed. Comparison of the current study is favorable with the previous published results. The obtained results depend on the governing parameters such as thermal Grashof number, solutal Grashof number, Biot numbers, symmetric and asymmetric wall temperatures, Schmidt number, Soret number, and Brinkman number. An influence of these parameters on the fields of velocity, temperature, and concentration is reported. Further, the numerical results for the Nusselt number, mean value of the velocity, dimensionless bulk temperature, skin friction, and molecular diffusion coefficient are tabulated for different parametric conditions and explained. For small value of Brinkman number, the obtained values agree with other published results for all considered cases. [ABSTRACT FROM AUTHOR]

Abstract This paper reports a theoretical analysis of heat and mass transfer in the microchannels partially filled with porous materials and used in thermos-chemical microreactors. A first order catalytic chemical reaction is considered on the internal surfaces of a parallel-plates microchannel. The local thermal non-equilibrium approach along with two well-established porous-fluid interface models is employed to investigate the heat transfer within the porous section of the microreactor. The analysis further accounts for the finite thickness of the surrounding solid walls of the microchannel. The dispersion equations in both porous and clear sections of the microchannel are coupled with the fluid temperature through considering the thermal diffusion of mass. In addition, to enhance heat transfer in the partially-filled microchannel, the base fluid is replaced by a nanofluid. The results show that inclusion of the finite thickness of the walls in the thermal analysis can majorly affect the temperature fields and Nusselt number (Nu). In particular, the optimal thickness of the porous insert for achieving the maximum Nu is found to be strongly influenced by the wall thickness. It is also shown that the specific porous-fluid interface model, thicknesses of the porous section and that of the walls, and the volumetric concentration of the nanoparticles can all impart significant effects upon the concentration of chemical species and their distribution across the microchannel. More specifically, the concentration field within the porous region is found to be considerably dependent on the implemented porous-fluid interface model. Graphical abstract Schematic configurations of the model microreactor, and variation of Nusselt number with porous thickness. Image 1 Highlights • Thermal analysis of partially porous microreactors are performed. • Thermal diffusion and two models of BCs for interfaces are included in the investigation. • The effect of walls thicknesses on the thermal performance of the microreactor is studied. • Effects of Damköhler and Soret numbers on concentration contours are shown. • Effects of incorporating nanoparticles in the base fluid were investigated. [ABSTRACT FROM AUTHOR]

A computational simulation of two-dimensional magnetic-Carreau fluid in a suspension of gyrotactic microorganisms past a slendering sheet with variable thickness is investigated for slenderness parameters varied in the range of –0.2 to 1.0. Owing to the noticeable implication in various engineering applications, the effects of multiple slip is considered in the present simulation along with the Soret and the Dufour effects for the heat and mass transfer controlling process. The numerical values of the velocity, temperature, concentration, and the density of the motile organisms are computed by the robust Runge–Kutta-based Newton's scheme. The thermal and concentration boundary layer are changed with the increase in the multiple slip parameters such as velocity slip, temperature slip, concentration slip, and diffusion slip parameters. With the rise in the Carreau fluid power index parameter, the velocity field increases while it declines with the velocity slip and magnetic field parameter. The increasing values of velocity slip, Dufour number, Soret number, and magnetic parameter boost up the density of the motile organism profiles for different slenderness parameter considered in the present study. The effect of the nondimensional factors on the skin friction, local Nusselt, local Sherwood, and the density numbers of the motile organisms are discussed with the assistance of the table for three different slenderness parameters. It is found that multiple slip parameters enable to control the heat and mass transfer rate. Finally, both the qualitative and quantitative comparisons of the present results with previous study are presented in the tabular form and are found to be in excellent agreement. [ABSTRACT FROM AUTHOR]

The effects of chemical reaction, thermal stratification, Soret and Dufour numbers on magneto-hydrodynamic free convective heat and mass transfer of a viscous, incompressible,and electrically conducting fluid on a vertical stretching surface embedded in a saturated porous medium are presented. A similarity transform is used to reduce the governing partial differential equations into a system of nonlinear ordinary differential equations, which is solved analytically by means of homotopy analysis method. The results are compared with numerical predictions obtained by fourth order Runge-Kutta method. Close agreement is observed between the two sets of results. The method is led to an expression, which is acceptable for all values of effective parameters and controls the convergence of the solution.

The present note investigates the effects of chemical reaction and Soret effects on hydromagnetic laminar viscous pulsating flow in a porous channel with slip and convective boundary conditions. The governing equations are solved analytically and the expressions for velocity, temperature, concentration, Nusselt number and Sherwood number distributions are obtained. The effects of various pertinent parameters on flow variables have been discussed numerically and explained graphically. Analysis indicates that the temperature distribution decreases for a given increase in heat transfer Biot number and Prandtl number while it increases with an increase in Hartmann number. Further, the concentration distribution increases with an increase in the mass transfer Biot number, Soret number and the heat transfer Biot number, while it decreases for a given increase in the chemical reaction parameter.

This present study investigates the effects of Soret and heat source on steady MHD mixed convective heat and mass transfer flow past an infinite vertical plate embedded in a porous medium in the presence of chemical reaction, viscous and Joules dissipation. The governing system of partial differential equations is transformed to dimensionless equations using dimensionless variables. The dimensionless equations are then solved analytically using perturbation technique. With the help of graphs, the effects of the various important parameters entering into the problem on the dimensionless velocity, dimensionless temperature and dimensionless concentration fields within the boundary layer are discussed. Also the effects of the pertinent parameters on the skin-friction coefficient and rates of heat and mass transfer in terms of the Nusselt and Sherwood numbers are presented numerically in tabular form.

In the present analysis, the influence of thermal radiation, chemical reaction and thermal-diffusion on hydromagnetic pulsating flow of Casson fluid in a porous channel is investigated. The fluid is injected from the lower wall and sucked out from the upper wall with the same velocity. The governing flow equations are solved analytically by employing the perturbation technique. The influence of various emerging parameters on flow variables has been discussed. The obtained results show that the temperature distribution increases when there is an increase in heat source, whereas there is a decrease in temperature with an increase in radiation parameter. The concentration distribution decreases with an increase in chemical reaction parameter, while it increases for a given increase in Soret number. Further, the results reveal that, for both the Newtonian and non-Newtonian cases, Nusselt number distribution decreases at the upper wall with increasing Hartmann number and radiation parameter. The mass transfer rate decreases at the upper wall with increasing chemical reaction parameter and Soret number.

This study presents a numerical analysis on the effects of Soret, variable thermal conductivity, viscous-Ohmic dissipation, non-uniform heat sources, on steady two-dimensional hydromagnetic mixed convective heat and mass transfer flow of a micropolar fluid over a stretching sheet embedded in a non-Darcian porous medium with thermal radiation and chemical reaction. The governing differential equations are transformed into a set of non-linear coupled ordinary differential equations which are then solved numerically by using the fifth-order Runge-Kutta-Fehlberg method with shooting technique. Numerical solutions are obtained for the velocity, angular velocity, temperature and concentration profiles for various parametric values, and then results are presented graphically as well as skin-friction coefficient, and also local Nusselt number and local Sherwood number for different physical parameters are shown graphically and in tabular form. A critical analysis with earlier published papers was done, and the results were found to be in accordance with each other.

The present work investigates the effects of thermal-diffusion and diffusion-thermo on MHD flow of viscous fluid between expanding or contracting rotating porous disks with viscous dissipation. The partial differential equations governing the flow problem under consideration have been transformed by a similarity transformation into a system of coupled nonlinear ordinary differential equations. An analytical approach, namely the homotopy analysis method is employed in order to obtain the solutions of the ordinary differential equations. The effects of various emerging parameters on flow variables have been discussed numerically and explained graphically. Comparison of the HAM solutions with the numerical solutions is performed.

The present paper deals with the influence of thermophoretic particle deposition on the MHD mixed convective heat and mass transfer flow in a vertical channel in the presence of radiative heat flux with thermal-diffusion and diffusion-thermo effects. The resulting nonlinear coupled equations are solved under appropriate boundary conditions using the homotopy analysis method. The influence of involved parameters on heat and mass transfer characteristics of the fluid flow is presented graphically. It is noted that fluid velocity is an increasing function of radiation parameter, Dufour number, Buoyancy ratio parameter and mixed convection parameter whereas the magnetic parameter, thermophoresis constant, Soret number and Schimidt number lead to suppress the velocity. The fluid temperature increases with increasing radiation parameter and Dufour number. The convergence of homotopy analysis method (HAM) solutions is discussed and a good agreement is found between the analytical and the numerical solution.