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.     

Effects of heat and mass transfer on MHD nonlinear free convection non-Newtonian fluids flow embedded in a thermally stratified porous medium

This communication examines heat alongside mass transport in a nonlinear free convection magnetohydrodynamics (MHD) non-Newtonian fluid flow with thermal radiation and heat generation deep-rooted in a thermally stratified penetrable medium. The Casson and Williamson fluid considered in this communication flos simultaneously across the boundary layer and are mixed together. The model of heat alongside mass transport is set up with chemical reaction and thermal radiation alongside heat generation to form a system of partial differential equations (PDEs). Appropriate similarity variables are used to simplify the PDEs to obtain systems of coupled ordinary differential equations. An efficiently developed numerical approach called the spectral homotopy analysis method was used in providing solutions to the transformed equations. A large value of Casson term is observed to degenerate the velocity plot while the Williamson parameter enhances the velocity profile. The parameter of thermal stratification is found to enhance the rate of heat transport within the boundary layer. An incremental value of the magnetic parameter declines the velocity of the fluid and the entire boundary layer thickness. The present result was compared with previous studies and was seen to be in good agreement.     

Magnetized impacts of Brownian motion and thermophoresis on unsteady squeezing flow of nanofluid between two parallel plates with chemical reaction and Joule heating

Present research article investigate the heat and mass transfer characteristics of unsteady magnetohydrodynamic Casson nanofluid flow between two parallel plates under the influence of viscous dissipation and first order homogeneous chemical reaction effects. The impacts of thermophoresis and Brownian motion are accounted in the nanofluid model to disclose the salient features of heat and mass transport phenomena. The present physical problem is examined under the presence of Lorentz forces to investigate the effects of magnetic field. Further, the viscous and Joule dissipation effects are considered to describe the heat transfer process. The non‐Newtonian behaviour of Casson nanofluid is distinguished from those of Newtonian fluids by considering the well‐established rheological Casson fluid model. The two‐dimensional partial differential equations governing the unsteady squeezing flow of Casson nanofluid are coupled and highly nonlinear in nature. Thus, similarity transformations are imposed on the conservation laws to obtain the nonlinear ordinary differential equations. Runge‐Kutta fourth order integration scheme with shooting method and bvp4c techniques have been used to solve the resulting nonlinear flow equations. Numerical results have been obtained and presented in the form of graphs and tables for various values of physical parameters. It is noticed from present investigation that, the concentration field is a decreasing function of thermophoresis parameter. Also, concentration profile enhances with raising Brownian motion parameter. Further, the present numerical results are compared with the analytical and semianalytical results and found to be in good agreement.     

Magnetized squeezed flow of time‐dependent Prandtl‐Eyring fluid past a sensor surface

The present numerical investigation describes the influence of a transverse magnetic field on the heat and mass transfer characteristics of time‐dependent squeezed flow of Prandtl‐Eyring fluid past a horizontal sensor surface. The current physical problem is modeled based on the considered flow configuration. Also, the present problem is analyzed under the influence of Lorentz forces, to explore the impact of a magnetic field on the flow behaviour. The considered physical problem in the present study gives highly nonlinear coupled time‐dependent, two‐dimensional partial differential equations. The governing flow equations are reduced to the system of nonlinear ordinary differential equations by imposing the suitable similarity transformations on the laws of motion. Due to the inadequacy in the analytical methods, the present problem is solved by using the Runge‐Kutta fourth order integration scheme with shooting method. The flow and heat transfer behaviour of various control parameters are studied and presented in terms of graphs and tables. From the current investigation it is noticed that, the increasing magnetic parameter enhances the velocity field and diminishes the temperature profile in the flow region. Also, the magnifying permeable velocity parameter decreases the temperature field. The present similarity solutions are found to be in good agreement with previously published results.     

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.     

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

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

Impact of porosity and radiation on two-dimensional unsteady magnetohydrodynamics heat transfer stagnation point flow with viscous dissipation

The current study aims to investigate the influence of porosity in the presence of radiation, and viscous dissipation on two-dimensional unsteady magnetohydrodynamics mixed convection heat and mass transfer flow at the stagnation point. The governing time-dependent nonlinear partial differential equations are converted into a nonlinear ordinary differential equation by utilizing similarity transformations. The transformed equations are solved by employing the bvp4c technique, a well-known numerical approach. The influence of nondimensional factors on fluid velocity, temperature, and species concentration profiles is explored and graphically represented. For varied values of the Prandtl number, magnetic field, and Schmidt number, the friction factor, Nusselt, and Sherwood numbers are also explored and provided in tabular format. As increasing the porosity parameter, the temperature profile, concentration profile is growing, and velocity profile diminishes. The conclusions of this study are widely accepted by the scientific community.     

Bioconvective Casson nanofluid flow toward stagnation point in non-Darcy porous medium with buoyancy effects,chemical reaction,and thermal radiation

The emphasis on non-Newtonian fluid encountered in biomedical, pharmaceuticals, mining, food, chemical, and plastics industries and in noticeably enormous diverse industrial applications influenced this article. This study is accomplished in a non-Darcy porous stretching surface to investigate the stagnation point of bioconvective Casson nanofluid. Chemical reaction, applied consistent magnetic field, radiative heat transfer, and buoyancy force consequences are studied for numerical examination. Composed of nonlinear partial differential equations for the above presumptions and reforming them into ordinary differential equations by means of compatible transformations are enforced. Adopting the fifth order Runge–Kutta Felhberg method with the shooting technique obtained a numerical solution. Obtained solutions are authenticated by comparing previous solutions. The major finding includes the reduction of the Casson parameter on the skin friction coefficient.     

Differential transform solution for Hall and ion‐slip effects on radiative‐convective Casson flow from a stretching sheet with convective heating

Magnetohydrodynamic (MHD) materials processing is becoming increasingly popular in the 21st century as it offers significant advantages over conventional systems, including improved manipulation of working fluids, reduction in wear, and enhanced sustainability. Motivated by these developments, the present work develops a mathematical model for Hall and ion‐slip effects on non‐Newtonian Casson fluid dynamics and heat transfer toward a stretching sheet with a convective heating boundary condition under a transverse magnetic field. The governing conservation equations for mass, linear momentum, and thermal (energy) are simplified with the aid of similarity variables and Ohm's law. The emerging nonlinear‐coupled ordinary differential equations are solved with an analytical technique known as the differential transform method. The impact of different emerging parameters is presented and discussed with the help of graphs and tables. Generally, aqueous electroconductive polymers are considered, for which a Prandtl number of 6.2 is employed. With increasing Hall parameter and ion‐slip parameter, the flow is accelerated, whereas it is decelerated with greater magnetic parameter and rheological (Casson) fluid parameter. Skin friction is also decreased with greater magnetic field effect, whereas it is increased with stronger Hall parameter and ion‐slip parameter values.     

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.     

Nonlinear convection flow of dissipative Casson nanofluid through an inclined annular microchannel with a porous medium

The nonlinear convection study on the flow of a dissipative Casson nanofluid through a porous medium of an inclined micro-annular channel is presented. The cylindrical surfaces were conditioned to temperature increase and velocity slip effects. A uniform magnetic field strength was applied perpendicular to the cylinder surface. The heat source and Darcy number influence are explored in the examination of the blood rheological model (Casson) through the annular cylinder. Appropriate dimensionless variables are imposed on the dimensional equations encompassing Casson nanofluid rheology through an annular microchannel. The resulting systems of equations were solved and computed numerically via Chebyshev-based collocation approach. Thus, the solutions of flow distributions, volumetric flow rate, and other flow characteristics were obtained. The result shows that both nonlinear convection parameters decrease the nanoparticle volume fraction, whereas they increase the energy and momentum distributions. Moreover, the volumetric flow rate is upsurged significantly by a wider porous medium, annular gap, a higher Casson parameter, and nonlinear convection influence.     

A generalized perspective of magnetized radiative squeezed flow of viscous fluid between two parallel disks with suction and blowing

In this research article, the electrically conducting magnetized radiative squeezed flow of two‐dimensional time‐dependent viscous incompressible flow between two parallel disks with heat source/sink and Joule heating effects under the presence of an unsteady homogeneous first order chemical reaction is demonstrated numerically. The considered physical problem is studied under the influence of Lorentz forces to describe the effect of an applied magnetic field. Heat dissipation due to viscosity and Joule heating are considered in the energy equation to demonstrate the behavior of the thermal profile. Also, the thermodynamic behavior of temperature field is described by considering the concept of heat source/sink in the energy equation. The mass transport characteristics of a viscous fluid are described through the time‐dependent chemical reaction of first order type with homogenous behavior. Thus, the considered physical problem gives the time‐dependent, highly nonlinear coupled partial differential equations, which are reduced to a system of ordinary differential equations by invoking the suitable similarity transformations. The discretized first order ordinary differential equations are solved by using the Runge‐Kutta fourth order integration scheme with the shooting technique (RK‐SM) and bvp4c Matlab function. Flow sensitivity of various emerging control parameters are described with the help of tables and graphs. The axial velocity field enhanced for the suction case and suppressed for the blowing case for the increasing values of suction/injection parameter. Also, an excellent comparison between the present solutions and previously published results shows the accuracy and validity of the present similarity solutions and used numerical methods.     

Effect of Prandtl number on leading edge accretion and ablation: A numerical study of unsteady boundary layer flow over a flat plate

An efficient numerical method, namely, the Runge‐Kutta fourth order integration scheme with shooting technique is employed to give a suitable solution for the unsteady magnetohydrodynamic boundary layer flow of viscous incompressible fluid with accretion or ablation effects over a flat plate under the influence of homogenous first order chemical reaction. When compared to the other numerical techniques such as perturbation methods, this approach provides the accurate numerical results valid uniformly for all nondimensional time. The unsteady behavior of chemically reacting magnetohydrodynamic boundary layer flow is investigated by analyzing the nature of buoyancy and magnetic parameters in the momentum equation. Also, results are extended to the energy and concentration equations by considering the viscous dissipation, Joule heating and chemical reaction effects. With the help of suitable similarity transformations, the highly nonlinear, coupled, time‐dependent partial differential equations are reduced to ordinary differential equations. Furthermore, the numerical solutions in terms of velocity, temperature and concentration profiles within the boundary layer are presented for the various values of control parameters. Also, the impact of physical parameters on the flow, heat and mass transfer characteristics are examined thoroughly. The present investigation reports that, the increasing magnetic parameter increases the temperature field and decreases the velocity field. Also, Eckert number enhance the thermal field whereas, the chemical reaction parameter decays the concentration field. Before concluding the considered problem, present results are validated with the previous results and are found to be in good agreement.     

Heat and mass transfer of magnetohydrodynamic Casson fluid flow over a wedge with thermal radiation and chemical reaction

A numerical computation to analyze the heat and mass transfer mechanism of a magnetohydrodynamic radiative Casson fluid flow over a wedge in the presence of Joule heating, viscous dissipation, and chemical reaction is carried out in this study. The flow-governing partial differential equations are transformed as ordinary differential equations by relevant similarity transformations and subsequently resolved by Runge–Kutta numerical approach with a shooting technique. The characteristics of momentum, thermal, and concentration border layers due to various influencing parameters are graphically outlined and numerically computed by MATLAB software. We present comparative solutions to construe the relative outcomes of Casson fluid versus Newtonian fluid. Computational outcomes of friction factor and Nusselt and Sherwood numbers are tabulated with suitable interpretations. An increase in skin friction values is noted due to an increment in the thermal Grashof number, whereas a decrease is observed due to the chemical reaction parameter. The Casson fluid displays a superior heat transfer mechanism than the Newtonian fluid. Obtained outcomes are in good agreement with the prevailing literature in the limiting case.     

Heat and mass transfer effect on an unsteady MHD radiative chemically reactive Casson fluid over a stretching sheet in porous medium

The objective of the present study is to investigate the effects of the variable magnetic field, chemical reaction, thermal radiation, Soret effect, and variable heat absorption on the fluid flow and heat and mass transfer of an unsteady Casson fluid past a stretching surface in a saturated porous medium. Velocity slip near the plate and conjugate heating boundary conditions in heat and mass transfer have been considered in this study. Due to the complexities in boundary conditions, the analytic solution of the governing equations of the present model is not possible. Thus, to overcome these issues, the coupled partial differential equations of the model are converted into a set of ordinary differential equations using similarity transformation. These equations have then been solved numerically using the fourth-order Runge-Kutta technique via the shooting method. The effects of various pertinent flow parameters on the velocity, concentration, and temperature field have been studied graphically. For the field of engineering, to get an insight into the physical quantities, especially Nusselt number, Sherwood number, and skin friction, their numerical values have been estimated against various parameters and presented in tables. From the tabulated values, it has been perceived that the shear stress increases with an increase in magnetic parameter, unsteadiness parameter, Casson parameter, and heat source parameter, whereas the Biot number shows the reverse trend. The mixture of porous media has justified that the heat transport process over a stretching sheet results in averting heat loss and accelerating the process of cooling, which is a significant outcome of the study. Furthermore, it has also been revealed that with the increase in the Soret effect and magnetic field, there is a reduction in the fluid velocity and temperature near the plate, whereas there is an increase in the species concentration. It has also been mentioned that the effects of the variable magnetic field have been widely applied in various engineering applications like magnetohydrodynamic (MHD) propulsion forces, rate of cooling, MHD power generation, and so on.     

Thermal and solutal Marangoni stagnation point Casson fluid flow over a stretching sheet in the presence of radiation,Soret and Dofour effect with chemical reaction

The Marangoni flow is involved with microgravity and earth gravity, which causes undesirable effects in crystal growth experiments. Crystal growth experiments were designed in such a manner so as to appraise MIR (space station), which is one of the best platforms for protein crystallization and radiation experiments. In this article, a model is proposed with a stagnation point and a Casson fluid flow at the interface of the plate in the presence of Marangoni convection influenced by a magnetic field and chemical reaction. Furthermore, it is considered that both temperature and concentration surface tension vary linearly with the interface. It is important to choose similarity transformations for implementing nonlinear differential equations into linear ordinary differential equations. We solved the system of differential equations using fourth order Range‐Kutta method with suitable shooting techniques, and the results are displayed through graphs. A comparison is made with the earlier existing literature, and it shows a very good agreement. The results and a detailed discussion of velocity, temperature, and concentration have been shown graphically. The favorable and unfavorable buoyancy force to Marangoni flow, the features of temperature and concentration field, have been investigated.     

Influence of radiation and viscous dissipation on MHD heat transfer Casson nanofluid flow along a nonlinear stretching surface with chemical reaction

The present article investigates the influence of Joule heating and chemical reaction on magneto Casson nanofluid phenomena in the occurrence of thermal radiation through a porous inclined stretching sheet. Consideration is extended to heat absorption/generation and viscous dissipation. The governing partial differential equations were transformed into nonlinear ordinary differential equations and numerically solved using the Implicit Finite Difference technique. The article analyses the effect of various physical flow parameters on velocity, heat, and mass transfer distributions. For the various involved parameters, the graphical and numerical outcomes are established. The analysis reveals that the enhancement of the radiation parameter increases the temperature and the chemical reaction parameter decreases the concentration profile. The empirical data presented were compared with previously published findings.     

Application of heat transfer on MHD shear thickening fluid flow past a stretched surface with variable heat source/sink

The purpose of this study is to explore the viscous dissipation stimulus on the steady convective magnetohydrodynamic shear thickening liquid stream across a vertically stretched sheet. The impact of thermic heat, first-order velocity slip, and variable heat generation/absorption are considered and also ignored the effect of magnetic Reynold's number. We converted flow controlling equations into the set of dimensionless nonlinear ordinary differential equations by employing similarity variables to solve these coupled equations by R–K and shooting technique. The effect of different dimensionless variables on velocity, heat, friction factor, and local Nusselt numbers are presented through graphs and tables. Depreciation in velocity and growth in temperature distribution is detected when the Casson fluid parameter is increased. Temperature is the increasing function of the Eckert number.     

OHAM simulation of 3D transverse magnetic field in rotating flow of Maxwell nanofluid over a stretching sheet with chemical reaction and heat source/sink

In this article, we investigate the heat transfer characteristics of a Maxwell nanofluid along a stretching sheet with transverse magnetic field, considering the presence of heat source/sink and chemical reaction. We consider appropriate similarity transformation for transforming the governing nonlinear equations into nondimensional highly nonlinear coupled ordinary differential equations. The optimal homotopy analysis method is utilized for solving the resultant-coupled equations. The impact of all sundry parameters, like, Deborah number, Prandtl number, magnetic parameter, thermophoresis, rotation parameter, chemical reaction, velocity slip, Schmidt number, Brownian motion parameter, heat sources per sink, Biot number, and Eckert number, on the temperature, velocity, and concentration fields is reported, analyzed, and described through graphs and tables. It is noticed that higher values of magnetic parameter and Deborah number reduce the horizontal velocity field. Furthermore, it is observed that the Biot number and heat source/sink parameter enhance the temperature distribution.     

Entropy generation analysis of natural convective chemically reacting squeezing flow of Casson fluid between parallel disks with Hall and Ion slip currents

This paper concerns the second law analysis of free convective squeezing flow of a chemically reacting Casson fluid confined between two parallel disks with Hall and Ion slip effects. The upper one is impermeable and the lower disk is porous. The linear momentum, energy balance and mass partial differential equations converted as system of ODEs by similarity transformations and tackled with shooting method via fourth order Runge‐Kutta scheme. The impact of various dimensionless geometric and fluid parameters on the velocity fields, temperature and concentration fields, entropy generation and Bejan numbers are studied and presented in the form of pictorially. The present results are correlated with already published outcomes for viscous case and found to be good agreement. The Bejan number of the fluid enhances with Ion slip parameter, whereas the concentration profile of the fluid is decreases with increasing of the Casson fluid parameter. The entropy generation of the fluid is enhanced with Eckert number whereas the Bejan number is decreased with suction/blowing parameter