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This study conducts a thorough numerical analysis of a complex problem involving magnetohydrodynamic flow, radiation, activation energy, Soret, the Dufour effect, and heat transfer in a permeable channel for a Maxwell fluid. Governing equations for momentum, concentration, and heat are transformed into ordinary differential equations through a similarity transformation, enabling focused analysis. Utilizing a fourth-order Runge-Kutta scheme, we compute dimensionless velocity, temperature, and concentration fields for the steady motion of a Maxwell fluid in a channel. To ensure accuracy, our results are validated against existing research, showing excellent agreement. The study investigates the impact of various physical parameters on fluid motion, revealing insights such as decreased velocity and concentration with increased chemical reaction, and rising fluid temperature. Additionally, concentration distribution decreases with a chemical reaction and a higher Schmidt number. This research enhances understanding of complex fluid dynamics, providing valuable insights for engineering applications, particularly in industries requiring precise fluid behavior control.

This study investigates the intricate interplay among various factors, including melting heat, Brownian motion, activation energy, and thermophoresis in the Maxwell fluid along a stretching sheet. To model these complex phenomena, we have formulated mathematical equations that are subsequently transformed into a set of partial differential equations. Through a similarity renovation, these equations are transformed into ordinary differential equations, and we solve the resulting boundary value problem using numerical techniques, specifically the BVP4C method, known for its fourth-order accuracy. Utilizing the numerical capabilities of BVP4C, we delve into the influence of different parameters, including thermophoresis, melting effect, Brownian motion, and activation energy, on critical flow characteristics. These characteristics encompass the concentration profile and the Sherwood number profile. Our work has been validated against previously published research, showing good matching. The primary objective of this work is to provide plots depicting Nusselt number, skin friction, and Sherwood number profiles for different parameters. This examination aims to shed light on the interconnected effects of these complex phenomena, which have direct relevance across numerous industrial and engineering applications, spanning areas such as metallurgy, nanotechnology, and chemical processing. The primary finding from this research underscores that both the melting heat effect and activation energy contribute to an increase in the concentration profile.

This research delves into Maxwell fluid flow's intricate fluid dynamics, as well as heat-mass transfer features over a porous horizontal plate. The investigation encompasses magnetohydrodynamics, heat generation, radiation, chemical reactions, and the influence of activation energy. Furthermore, the study integrates slip conditions for temperature, velocity, and concentration gradients at the interface between the fluid and solid surfaces. After formulating the governing equation with the abovementioned parameters, the partial differential equations are changed into ordinary ones through a similarity revolution. The results of the bvp4c solver in MATLAB provide significant insights into how various physical parameters interact and influence the flow dynamics and heat-mass transfer characteristics. This research's primary aim and objective is to comprehensively compare plots generated for different parameters under both no-slip and slip conditions by considering fixed values as Q = 0-0.3, Sc = 1–4, Ec = 0–1, Pr = 1–4, σ = 0–2, E = 0.1-0.7, λ = 0–2. The primary discoveries of this study indicate that with an increase in the power index, there is a decline in the concentration profile. In contrast, an increase in the activation energy leads to an elevation in this profile. An important revelation from this research is the significant decrease observed in the concentration profile as the activation energy parameter is raised.

This study investigates the complex interplay between radiation and magneto-flow on a porous, perpendicular plate immersed in a viscoelastic Maxwell fluid. The significance lies in understanding how the Lorentz force shapes dynamics and energy transfer in such systems, which is crucial for various industrial applications. Utilizing a comprehensive model that accounts for heat generation and absorption, we transform partial differential equations into ordinary ones through flow similarity analysis, providing valuable insights into the system's behavior. Employing the bvp4c solver in MATLAB, we evaluate velocity and energy profiles in Maxwell fluid flow. Graphical representations allow for a thorough examination of the influence of thermophysical parameters. Our findings reveal an intensified velocity profile attributed to prolonged material relaxation times.Furthermore, temperature profiles progressively increase with higher radiation and porous parameters. These results hold significant implications for industrial applications, particularly in Maxwell fluids' heat transport studies. Insights from this research can inform the design and optimization of systems such as ovens, boilers, cross-flow heat exchangers, and solar panels, enhancing efficiency and performance. Overall, this study contributes novel insights that advance understanding beyond previous efforts in the literature, paving the way for further exploration in this field. The primary finding of this study indicates that as the Rayleigh number increases, the velocity profile experiences an increase, whereas the temperature profile shows a decrease.

This study aims to bridge the gap by conducting a numerical analysis of Maxwell fluid behaviour on a perpendicular plate within a porous medium, considering both chemical reaction and heat generation. The investigation also encompasses the study of energy and mass transfer within magnetohydrodynamic (MHD) Maxwell fluids. We utilise a transformation technique employing similarity variables to address the challenge posed by the nonlinear partial differential equations (PDEs). These transformed equations are subsequently solved via the bvp4c solver in MATLAB. The obtained results exhibit a high degree of agreement with the previously published work. The study systematically explores the influence of chemical reaction, energy generation, and Deborah number parameters on temperature and velocity, as well as concentration, presenting the outcomes graphically. In addition, we calculate local Sherwood numbers, Nusselt numbers, and skin friction coefficients to assess the impact of chemical reactions. Our findings notably indicate that Sherwood numbers and skin friction coefficients increase with higher levels of chemical reaction, while local Nusselt numbers decrease as chemical reactions become more pronounced. By studying Maxwell fluid flow over a perpendicular plate with chemical reactions, this research contributes to optimizing processes, enhancing product quality, and providing deeper insights into the behaviour of complex fluids in real-world scenarios.

Examining the behavior of non-Newtonian fluids in natural environments like rivers and groundwater flow is essential for evaluating their environmental impact. The Soret and Dufour impacts significantly influence the transportation of contaminants and heat within these systems. With this application in mind, this study investigates the impacts of steady-state Soret/Dufour and radiation on a porous perpendicular plate in the context of the Maxwell fluid model. Additionally, the study considers the impact of magnetohydrodynamics (MHD) and heat generation to provide a comprehensive modeling framework. The similarity technique is employed to transform the partial differential equations (PDEs) into nonlinear ordinary differential equations (ODEs). These ODEs are subsequently solved numerically using the inbuilt software bvp4c in MATLAB. The study presents graphical representations of Sherwood, skin friction, Nusselt number profiles, momentum, temperature, and concentration contours. These visual representations and tables illustrate the results for various fluid flow parameters, highlighting the influence of different flow characteristics. Key findings of this research include a rise in Nusselt and Sherwood numbers and skin friction with higher radiation values. Furthermore, there is an observed increase in temperature and concentration profiles as the Soret parameter value rises, indicating the significant impact of these parameters on the studied phenomena.

Abstract Heat transport in a 2D steady radiative boundary layer with Maxwell fluid flow and the influence of heat generation and MHD has been studied across a porous inclined vertical plate. Through similarity transformation, the mathematical modelling is converted to ODEs, and the built-in solver Bvp4c via MATLAB is used to solve. The linear movement of an inclined porous plate introduced the flow. The MHD (M), Prandtl number (Pr), radiation (Rd), Rayleigh number (Ra), local Nusselt number (Nu x ), angle of inclination (γ), and material relaxation time (β) have a considerable impact on the flow field as a result. The local Nusselt numbers and the skin friction coefficient are also given as numbers. The validation with the numerical solution is presented. The results are shown, and a thorough physical analysis has been done. The temperature of the fluid rises due to the greater electric field, increasing the heat transfer on the inclined plate. However, skin friction increases dramatically as the heat radiation parameter rises. The critical findings of this study are that the temperature profile increases and the velocity profile lower as the inclination angle increases. The Maxwell fluid parameter raises the velocity profile as well.

This study carries significant importance in its exploration of the consequences of thermal diffusion and diffusion-thermal energy on a Maxwell fluid near a stretching cylinder. It also considers the influence of Brownian motion and thermophoresis, alongside examining the impact of the Prandtl number and magnetohydrodynamics (MHD) and radiation into the analysis. The primary aim of this study is to investigate the impact of thermal diffusion, diffusion-thermal energy, and various influencing factors on a Maxwell fluid near a stretching cylinder. This investigation aimed to shed light on the behavior of critical parameters, particularly in the presence of Brownian motion, thermophoresis, and variations in the Prandtl number. Additionally, this study sought to adapt a Bvp4c inbuilt software in MATLAB to solve the governing equations, enabling the visualization of velocity, energy, and concentration contours. This study mainly investigated the results for the mentioned parameters for both flat plate (Curvature parameter αc = 0) and cylinder (αc = 1) on temperature, concentration, and velocity profiles. The major conclusions drawn from this study are the concentration profiles exhibited opposing behaviors as thermophoresis, Brownian motion and the Dufour effect increased. However, the temperature profile increases for these parameters. Secondly, concentration displayed an increasing behavior as the Soret number increased.

Energy transfer and boundary layer flow on a stretching cylinder are used in extrusion procedures, fiber technology, the manufacture of polymer sheets, and the creation of plastic films. In view of these applications, this article examines a transport model for a boundary layer flow with magnetohydrodynamics derived from a Maxwell fluid flow under the influence of radiation, Brownian motion, thermo phoresies and Stefan blowing on a stretching cylinder. Using MATLAB bvp4c function, the governing equations are numerically solved after being changed into ordinary differential equations via similarity renovations. Calculations and analyses determined how various emergent parameters affected the flow field, heat transfer and mass transfer. Some characteristics were discovered to affect the width of the boundary layer. The influence on MHD (M), radiation (Rd), the Schmidt number (Stc), Brownian motion (Nb), thermo phoresies (Nt) and the Stefan blowing parameter (Sb) on the velocity, energy and concentration profile have been explored. Excellent assessment is recognized over a tabular explanation to validate the adopted numerical technique. Significantly the Stefan blowing parameter rises the profiles on velocity and temperature. For an increasing value of the Maxwell fluid parameter, the skin friction coefficient decreases compared to the Stefan parameter.

This article evaluated the effects of heat generation and the chemical reaction of Maxwell fluid over a porous stretching cylinder. Partial differential equations are framed using the mentioned parameters. The partial differential equations for momentum, heat, and mass are changed using similarity transformations into highly nonlinear ordinary differential equations. These equations are answered numerically via the Runge-Kutta with the shotting scheme of the 4th order in MATLAB with the help of inbuilt software bvp4c. Results are offered graphically and in tabular format for essential parameters over a velocity, temperature, concentration, skin friction coefficient, Sherwood number, and Nusselt number. We checked our coding against an existing article and obtained a good match. The essential findings of this research are that velocity decreases as the permeability parameter rises. It has been observed that a viscoelastic fluid parameter induces a decreasing velocity profile. Due to the Maxwell fluid, porosity, heat generation, and chemical reaction parameters, the thermal profile has been improved. These results have several manufacturing applications, including metal turning, the fabrication of glass filaments, the formation of elastic sheets, wire drawing, the ejection of polymer sheets by oil companies, and the synthesis of polymers.

The current research paper aims to investigate the Maxwell fluid flow over a flat plate under slip conditions and suction/injection parameters. Radiation, velocity slip conditions, and heat generation are also considered for this study. The governing equations are changed into ordinary differential equations with the appropriate similarity transformations. Then these equations are answered using the shooting technique and the Runge-Kutta 4th-order method. The impact of Deborah number (De), suction/ injection (fw), thermal radiation (Rd), viscous dissipation (Ec), velocity slip parameter (λ) and heat generation (Q) are displayed through graphs and tables. The main target of this investigation is to discuss the impact of the Nusselt number profile for the suction parameter (fw) against momentum slip (λ), thermal slip conditions (β) and radiation parameter. Skin friction profile for suction parameter also discussed against Maxwell fluid parameter (De). Numerical findings are compared to previously published research using bvp4c in MATLAB programme. The critical solutions of this study are that the Nusselt number outline increased for increasing the value of momentum slip, radiation parameter and thermal slip conditions. Still, it got decreases for viscous dissipation (Ec) against radiation. Skin friction profile decreased for increasing value of suction parameter against Deborah number.

This study aims to investigate the influence of radiation, thermal conductivity and variable viscosity on natural convective flow on a semi-infinite perpendicular plate. Variable viscosity, thermal conductivity and thermal radiation are considered for the given study. The dimensional governing equations are framed with the use of the mentioned parameters and then these equations were converted into dimensionless equations by applying non dimensional quantities. The main aim of this study is to find the Nusselt number and skin friction for both air and water for considered parameters. Using the finite difference method through Fortran software, numerical solutions to the governing heat equations and dimensionless momentum equations were computed. The results for the parameters thermal conductivity, variable viscosity, radiation, and Prandtl number for both air and water are displayed via various graphs. The skin friction coefficients, Nusselt parameter, and local Nusselt numbers were discussed for both the air and water. The key conclusions of this study are that the succeeding velocity declines as the radiation's increases. By increasing the radiation value and the fluctuation time, the temperature distribution increases. Notably, the temperature profile increases significantly when the variable viscosity parameter decreases.

Maxwell fluids display viscous flow on a long timescale but exhibit additional elastic resistance during rapid deformations. Among various types of rate-type fluids, the Maxwell fluid has achieved prominence in numerous study fields. This viscoelastic fluid has viscous and elastic properties. Due to their reduced complexity, this Maxwell fluid is utilized used in the polymeric industries. We have established a mathematical model based on the applications. This article examines the mathematical and graphical analysis for steady-state magnetohydrodynamic flow in a horizontal flat plate of Maxwell viscoelastic fluid for a permeable medium with heat and thermal radiation. The non-dimensional and similarity transformation used to frame the partial differential equations with restored ordinary differential equations. The shooting technique is originated to find solutions to nonlinear boundary value problems with the help of MATLAB software via the Runge-Kutta Fehlberg method. The primary idea behind this strategy is to change the boundary conditions of boundary value problems into initial value problems. Several plots illustrate the leading parameters such as Prandtl number (Pr), Deborah number (De), Eckert number (Ec), heat generation (Q), radiation (Rd), Lewis number (Le), magnetic parameter (M), and thermal slip condition (β) on the velocity profile and energy transfer behaviour. We validated our results with published work. The most significant impact of this study is that the Nusselt number drops as the Eckert number rises and climbs when heat radiation increases. The skin friction coefficient increases as Deborah number increases.

Here, we employ computational methods to study the impacts of heat production and radiation fascination on the movement of a chemically reactive flow of Maxwell fluid in a magnetic field via a vertically slanted permeable plate. A boundary layer approximation is carried out to construct a flow that accurately captures the time-independent equations for momentum, energy and concentration. Partial differential equations describe the path taken by the electrically conducting fluid as it moves through the porous media. The RK45 method is then used to assess the solutions once appropriate dimensionless variables have been applied. In addition, tables and graphs are used to clarify the physical contribution of important parameters like the Prandtl number (Pr), the Maxwell fluid parameter (β), the Lewis number (Le), the Rayleigh number (Ra), the magnetic parameter (M), the inclined plate (γ), the buoyancy ratio (Nr), the heat generation parameter (Q), the radiation parameter (Rd) and the chemical reaction parameter (Kr), the skin friction formulas and the Nusselt number are calculated, both used to quantify heat and mass transmission. Finally, a graphical representation of the change in magnetic parameters across a range of Rd embedded factor values is shown.

This study examines the numerical representation of fluid flow on the Maxwell model in a double-diffusive boundary layer over a horizontal plate. The investigation incorporates slip conditions, encompassing momentum slip, thermal slip, and suction parameters. Moreover, the study includes the inspiration of thermal radiation, heat generation, and mass transfer. The governing partial differential equations (pertaining to momentum, continuity, energy transport, and mass transport) are transformed into ordinary differential equations (ODEs) using appropriate similarity transformations. To solve these equations in conjunction with suitable boundary conditions, the bvp4c inbuilt software is implemented. This is achieved through the shooting approach employed in MATLAB. A comprehensive agreement between the numerical technique and previously published findings demonstrates its efficacy. The outcomes are presented through graphical representations and tables, showcasing various parameters such as momentum slip, temperature slip, local Nusselt number, Sherwood number, and suction parameter. The primary motivation of this research lies in investigating the behaviour of Maxwell fluid flow in the absence of slip conditions. The study of Maxwell fluid flow over a flat plate with the combined effects of suction, thermal slip, and momentum slip conditions has a wide range of practical applications that span multiple industries, contributing to improved designs, efficiency, and understanding of fluid behaviour in various systems. The main aim of this study is to present streamlined results under varying conditions, explicitly investigating the influence of suction effects and slip conditions on the flow., Competing Interests: There are no conflicts to declare., (This journal is © The Royal Society of Chemistry.)

Heat transport in a 2D steady radiative boundary layer with Maxwell fluid flow and the influence of heat generation and MHD has been studied across a porous inclined vertical plate. Through similarity transformation, the mathematical modelling is converted to ODEs, and the built-in solver Bvp4c via MATLAB is used to solve. The linear movement of an inclined porous plate introduced the flow. The MHD (M), Prandtl number (Pr), radiation (Rd), Rayleigh number (Ra), local Nusselt number (Nu x ), angle of inclination (γ), and material relaxation time (β) have a considerable impact on the flow field as a result. The local Nusselt numbers and the skin friction coefficient are also given as numbers. The validation with the numerical solution is presented. The results are shown, and a thorough physical analysis has been done. The temperature of the fluid rises due to the greater electric field, increasing the heat transfer on the inclined plate. However, skin friction increases dramatically as the heat radiation parameter rises. The critical findings of this study are that the temperature profile increases and the velocity profile lower as the inclination angle increases. The Maxwell fluid parameter raises the velocity profile as well., (© 2023. The Author(s).)