In the present study, the entropy generation and heat transfer characteristics of an Ellis hybrid nanofluid flow over a rotating disk embedded in a Darcy–Forchheimer porous medium under the influence of electromagnetic hydrodynamics (EMHD), nonlinear thermal radiation and heat source/sink effects have been studied. The motivation for the analysis is based on major engineering and industrial applications such as cooling systems, rotating machinery, energy devices, thermal processing equipment, biomedical engineering and chemical transport systems. The governing nonlinear partial differential equations are converted into ordinary differential equations using appropriate self-similar transformations and are solved numerically by the Runge-Kutta fourth order approach together with the shooting technique. The obtained results are confirmed by comparison with the prior published studies and demonstrate very good agreement. The effect of important physical parameters on velocity, temperature, entropy generation and Bejan number are thoroughly explained and visually shown. The results show that improving the Ellis fluid parameter and Darcy-Forchheimer parameter boosts the azimuthal velocity and declines the radial velocity. The temperature profile is sharply increased due to the higher effects of thermal radiation and heat source. Moreover, the entropy generation enhances with the enhancement of the temperature ratio parameter, and the Bejan number increases with the effect of the electric field parameter. The present results provide significant insights into the optimization of the thermal performance, energy losses and heat transfer efficiency in advanced industrial and technological applications.

In engineering and technology, the micropolar-Sutterby fluid serves as a favorable alternative to conventional non-Newtonian fluids due to its practical applicability, robustness, and computational efficiency. This analysis focuses on the thermophysical characteristics of the magnetized flow of a micropolar-Sutterby fluid bounded by an exponentially stretching surface, taking into account Cattaneo–Christov heat flux, thermal radiation, activation energy, and heat source, which have wide industrial applications. The surface is influenced by Darcy-Forchheimer effects and is situated within a porous medium. The impacts of mobile microorganisms with magnetic flux are also probed. Similarity transformations are used to convert the governing PDEs into coupled ODEs. The resulting ordinary differential equations are solved using the collocation-based MATLAB built-in solver bvp4c. Tables, graphs, and literature comparisons are used to demonstrate the impact of different parameters on the involved profiles. Quantitative results obtained from the numerical simulations reveal that increasing the thermal radiation parameter Rd from 0.5 to 0.9 enhances the local Nusselt number by 24.3

The present work explores the simultaneous mass and heat transfer occurring in the unsteady magnetohydrodynamic boundary-layer flow of a Jeffrey nanofluid across a stretching surface, while accounting for the influence of Stefan blowing. The model accounts for viscoelastic behavior through the Jeffrey fluid constitutive equation and includes thermal radiation, chemical reactions, internal heat generation or absorption, and cross-diffusion phenomena. Radiative heat transfer is modeled through the Rosseland diffusion approximation, while Buongiorno’s nanofluid model is applied to capture nanoparticle transport driven by Brownian diffusion and thermophoresis. The governing unsteady partial differential equations are transformed into a system of strongly nonlinear ordinary differential equations using suitable similarity transformations, which are then solved under physically relevant boundary conditions. The boundary-value problem obtained is solved numerically by employing the bvp4c solver in MATLAB. The results indicate that both thermal radiation and internal heat generation substantially enhance the temperature distribution, whereas the viscoelastic memory characteristics exert a notable influence on the momentum boundary layer. In addition, thermophoresis and Brownian motion play a dominant role in regulating heat transfer and nanoparticle concentration. The findings demonstrate that viscoelasticity, Stefan blowing, and cross-diffusion effects expressively regulate momentum, thermal, and concentration boundary layers. These outcomes provide boosted insight for controlling heat and mass transport in energy systems, polymer manufacturing, and biomedical applications involving nanofluids.

Owing to the exceptional thermal properties of non-Newtonian nanofluids, nanomaterials have reported widespread use in numerous fields such as mechanical engineering, cooling technologies, solar energy systems and industrial processes. This work aims to scrutinize the thermal behavior of Jeffrey nanofluids under the impact of bioconvection caused by gyrotactic microorganisms, with specific attention on the flow over an elongating cylinder. The inclusive innovation of this study lies in the incorporation of non-Fourier heat conduction modeled using the Cattaneo-Christov heat formulation instead of Fourier's law. Appropriate similarity functions are applied to renovate the governing equations into transformed ordinary differential equations. Numerical simulations are carried out using bvp4c solver via MATLAB, and the outcomes are compared with published studies to authenticate accuracy. Graphical illustrations are used to estimate the inspiration of key fluid parameters such as magnetic parameter 0.2 <= M <= 2.2, curvature parameter 0 <= gamma <= 0.8, Jeffrey fluid 0.5 <= beta <= 2.5, Lewis number 1.2 <= Le <= 2.0, buoyancy ratio parameter 0.5 <= Nr <= 2.5, thermophoresis 0.1 <= Nt <= 1.0 and bioconvective Lewis number 0.2 <= Lb <= 1.0. It is witnessed that the velocity distribution rises with the curvature parameter and decreases with the Jeffrey, buoyancy ratio parameter and magnetic parameters. The thermal profile drops with growing thermal relaxation parameters, while it rises with both the magnetic parameter M and thermophoresis parameter Nt. The concentration and density profiles reduce with improving values of the Lewis number Le and bioconvective Lewis number Lb, respectively.

The thermal characteristics of nanofluids, arising from progressive mechanisms, present an intriguing phenomenon with significant implications for energy production, cooling processes, and heat transfer devices. This study investigates the magnetohydrodynamic combined convection of Maxwell nanofluids, focusing on heat transport properties over an exponentially stretching sheet. The effects of activation energy, nonlinear thermal radiation, Cattaneo-Christov heat flux, suction/injection, Joule heating, solutal energy, and viscous dissipation in the presence of swimming microorganisms are incorporated. To elucidate these phenomena, the study examines the impacts of bioconvection, magnetic fields, and thermophoresis under extended boundary conditions. The partial differential equations (PDEs) of the problem related to momentum, energy, concentration, and density are transformed into ordinary differential equations (ODEs) through the application of similarity variables. The resulting dimensionless nonlinear ODEs are solved using the shooting method. Numerical results for key parameters are presented in the form of tabular and graphical trends for both steady and unsteady flow cases, utilizing MATLAB for computational analysis. Notable improvements in the velocity profile are observed with increasing values of the Maxwell parameter. Conversely, a rise in the mixed convection parameter leads to a deterioration in both the temperature and concentration fields.

This study numerically investigates magnetohydrodynamic Casson nanofluid flow over a curved stretching sheet under the influence of Joule heating, thermal radiation, activation energy, and chemical reaction. The Buongiorno model incorporating Brownian motion and thermophoresis describes nanoscale transport, while gyrotactic motile microorganisms stabilize the nanoparticle suspension through bioconvection. A systematic comparison between Newtonian and non-Newtonian (Casson) fluid models highlights the yield stress effects on transport characteristics. The governing equations of the Casson nanofluid flow are transformed into coupled ordinary differential equations by using similarity variables. The obtained system is solved by using the built-in bvp4c method of MATLAB. Key findings reveal that the Casson nanofluid exhibits lower velocity but higher temperature and concentration profiles than the Newtonian fluid due to enhanced viscous resistance and internal friction. Velocity decreases with increasing magnetic parameter (M) and buoyancy ratio (Nr), while it increases with mixed convection (λ). Temperature rises with higher thermal radiation (Rd), Brownian motion (Nb), thermophoresis (Nt), and thermal Biot number (β₁). Concentration enhances with increasing Brownian motion, mass Biot number (β₂), and activation energy (E). Microorganism density increases with motile Biot number (β₃) and curvature (A), but decreases with Peclet number (Pe) and bioconvection Lewis number (Lb). Skin friction rises with magnetic and Casson parameters, while the Nusselt, Sherwood, and motile density numbers show strong Biot number dependence. Results agree excellently with existing literature. This comparative analysis demonstrates that non-Newtonian behavior significantly alters thermal and transport characteristics, making the Casson model suitable for biomedical and industrial applications involving yield-stress fluids such as blood and polymer solutions.

The study of fluid flow in stenosed arteries is important due to its direct relevance to cardiovascular disorders and biomedical applications. In the present work, the flow and heat transfer behaviour of a hybrid nanofluid in a narrowed arterial geometry is examined under the influence of key physical parameters. The mathematical formulation is developed by considering steady, laminar, and incompressible flow conditions, together with appropriate boundary conditions that represent the physical situation. The governing nonlinear equations are transformed into a system of ordinary differential equations using suitable similarity transformations. These equations are solved numerically using a boundary value approach. In addition, an artificial neural network (ANN) model is employed to predict the flow and temperature profiles using the generated numerical dataset. The results show that variations in the governing parameters directly influence the velocity and temperature distributions within the flow domain. The ANN predictions are found to closely match the numerical results, with error values of the order of 10^-6 to 10^-7 , indicating good agreement between both approaches. This agreement supports the use of the ANN model as a reliable tool for predicting flow and heat transfer behaviour in stenosed arteries. The outcomes of this study provide a useful framework for analyzing complex fluid flow problems in biomedical systems and may be extended to related applications involving hybrid nanofluids.

This numerical study addresses the flow and heat transfer characteristics of a power-law non-Newtonian fluid across a rotating disk with the presence of porous media, induced magnetic fields, non-linear thermal radiation and Cattaneo-Christov heat flux model. The coupled non-linear governing equations are solved numerically using the fourth order Runge-Kutta method. The effect of significant parameters on the radial, azimuthal and axial velocities, temperature profile, induced magnetic field components, entropy production, Bejan number, skin friction and Nusselt number is studied. The results reveal that the non-Newtonian fluid index has a considerable effect on the velocity distribution. The effective viscosity of shear-thinning fluids is less, and hence the axial velocity is higher. The magnetic damping and porous resistance decrease the radial and azimuthal velocities. Increasing the disk rotation increases the azimuthal motion and magnetic induction; on the other hand, larger magnetic fields suppress the fluid flow by Lorentz forces. The Eckert number, thermal radiation and thermal relaxation parameter play a significant role in thermal behavior changing temperature profiles and entropy formation in the boundary layer. The combined impacts of rotation, porosity, magnetic field and viscous dissipation considerably affect the surface transport parameters such as skin friction and heat transmission. Induced magnetic field components decrease with a high reciprocal magnetic Prandtl number, and the Bejan number falls with a high Brinkman number but increases with heat radiation. These results provide a full understanding of the interaction of rheological, magnetic and thermal components in rotating disk systems, yielding insights that may help to build and improve thermal-fluid and magnetohydrodynamic devices.

The primary goal of this work is to concentrate on a numerical investigation of thermal radiation and heat effects on the squeezed flow of a tangent hyperbolic fluid induced by sensor sheet. The impacts of magnetics field and time depended thermal conductivity are taken into account. The investigation is motivated by applications in various engineering and industrial processes where precise control and understanding of fluid flow and heat transference phenomena are fundamental. The governing PDEs of two dimensional squeezing flow model are transformed into couple of ODEs via suitable similarity approximations. Matlab platform via shooting approach is employed to solve couple of developing equations numerically. The outcomes of prominent parameters on involved profiles are computed via graph and tables. Additionally, the outcomes of skin friction and Nusselt number are computed for motivation. This investigation explores the act of controlling physical parameters in governing flow and heat transfer performance. The research has been able to establish that ANN is an effective surrogate model in highly complicated non-Newtonian fluid flows. The fact that BVP4C and ANN are very well correlated confirms the effectiveness of machine learning methods in computational fluid dynamics, which can be applied in fast optimization of parameters of the industrial process with the use of Tangent Hyperbolic Fluids.

This paper examined magnetohydrodynamic flow and heat transfer of an upper-convected Maxwell nanofluid with bioconvection over a bidirectional stretchable surface. The overall objective was to develop a comprehensive model capable of simultaneously capturing viscoelastic effects, nanoparticle transport, heat absorption, nonlinear thermal radiation, and motile microorganism dynamics under the sway of a magnetic field. The originality of this work lay in combining Maxwell fluid elasticity with bioconvection and nonlinear radiation in a complete three-dimensional stretching configuration, which had rarely been investigated collectively. The governing nonlinear PDEs were transformed into a set of coupled ODEs using suitable similarity functions and were solved numerically using the MATLAB bvp4c solver. The sway of key dimensionless parameters, including Brownian motion parameter, magnetic parameter, radiation parameter, Deborah number, thermophoresis parameter, and bioconvection parameters, on velocity, temperature, nanoparticle concentration, and microorganism density were analyzed. The findings indicated that magnetic and elastic effects suppressed velocity while enhancing thermal and concentration boundary layers. Motile microorganisms expressively altered heat and mass transfer characteristics. These results were found to be valuable for applications in thermal management, polymer processing, and bioengineering systems.

Energy storage devices in thermal solar plants play a crucial role in controlling the energy and power demand. Their performance is significantly influenced by the thermal capacity of the materials used. Motivated by the growing need for enhanced thermal energy efficiency, a Williamson ternary hybrid nanofluid is used to examine the non-steady magnetohydrodynamic (MHD) flow through a porous stretching cylinder containing gyrotactic microorganisms. Physics-informed neural network (PINN) with GaussSwish hybrid activation function is utilized in this study. The network minimizes the residuals of the governing equations together with boundary constraints using automatic differentiation and the NADAM optimizer until it converges to the optimal loss. The effects of different flow parameters on temperature, momentum, concentration, and motile density are analyzed. Magnetic and electric field parameters show a drop in the momentum profile, whereas an inverse trend is noticed in the temperature profile. Weissenberg number, curvature, and heat sink parameters contribute to elevate the temperature. Schmidt number lowers the concentration profile; on the other hand, the curvature parameter exhibits an opposite relation. Peclet and bioconvection Lewis number cause the motile microorganism density to decline. Ternary hybrid nanofluid achieves up to 24.3% greater heat transfer, 29.7% mass transfer, and 34.1% higher motile microorganisms density than the hybrid nanofluid, confirming its potential for advanced thermal energy storage systems. The results further show the effectiveness of physics-informed neural networks in handling complex fluid problems.

This study investigates the three-dimensional magnetohydrodynamic (MHD) flow of an upper-convected Maxwell nanofluid over a bidirectional stretching surface, incorporating motile microorganisms, nonlinear thermal radiation, chemical reaction, and heat absorption. A comparative analysis between non-Newtonian (Maxwell) and Newtonian nanofluids is performed. The governing nonlinear partial differential equations are reduced to ordinary differential equations via similarity transformations and solved numerically using the bvp4c solver in MATLAB. Key findings demonstrate that increasing the magnetic parameter and Deborah number reduces both axial and transverse velocity components. Higher thermal radiation and thermophoresis parameter elevate the temperature profile. An increase in the chemical reaction parameter and Lewis number significantly reduces nanoparticle concentration. The Brownian motion parameter enhances temperature while reducing concentration. The Nusselt number increases with Prandtl number and decreases with thermophoresis, whereas the Sherwood number declines with both parameters. Increasing the Peclet number reduces the motile microorganism density. Compared to Newtonian nanofluids, the non-Newtonian Maxwell nanofluid exhibits higher temperature yet lower heat transfer rate.

This paper is an exploratory work that examines the magneto-thermal flow of a Casson tri-hybrid nanofluid of nanoparticles suspended in blood on top of a rotating disk. The mathematical model takes into consideration the effects of applied magnetic field, resistance of the porous medium, nonlinear thermal radiation and Cattaneo-Christov theory of heat flux. The ensuing nonlinear governing equations are reduced to ordinary differential equations and are solved numerically by use of a fourth order Runge Kutta method in combination with a shooting technique. The effect of major physical parameters on radial and azimuthal velocities, temperature distribution, generated magnetic field, entropy generation, Bejan number, skin friction, and Nusselt number is investigated. The findings indicate that, as the Casson parameter is increased fluid motion is suppressed as temperature and entropy production are augmented by the augmented internal resistance. The rotation of a disk enhances the velocity of the azimuthal and magnetic induction and the presence of the higher magnitude of magnetic fields slows down the velocity of the flow using the Lorentz forces. The increase in porosity reduces the transport of momentum, whereas the effects of thermal relaxation cause the decrease of the temperature in the boundary-layer. Thermal gradient and increment of temperature ratio are considerably enhanced by thermal radiation and temperature ratio. It has been identified that rotation, magnetic effects, porosity, and viscous dissipation are very sensitive in terms of the values of skin friction and heat-transfer rates. The findings, in general, shed some light on coupled thermal, magnetic, and rheological processes controlling the tri-hybrid nanofluid flow, and its possible uses in the biomedical cooling and hyperthermia therapy, as well as in the complex heat-transfer technology.

In recent years, bioconvection patterns have garnered considerable interest due to their vital role in various pharmaceutical, environmental, and industrial engineering applications. Motivated by these practical implications and the broader scope of inclusive innovation, the current work discovers the bioconvective hydromagnetic flow of Ree-Eyring non-Newtonian nanofluid through a vertically elongating surface in a Darcy–Forchheimer porous medium. The flow mechanism is influenced by multiple physical, systems, comprising a chemical reaction, Cattaneo–Christov heat flux, uniform magnetic field, activation energy, thermophoresis, viscous dissipation, Brownian motion and mobility of gyrotactic microorganisms. The governing PDEs of Buongiorno-Ree-Eyring model are transmuted into a system of nonlinear ordinary differential equations via similarity functions. These reduced differential equations are tackled numerically using bvp4c-function of MATLAB, and the outcomes are compared with those obtained from ANN model and earlier published literature to confirm accuracy. The detailed graphical and tabular analysis are carried out to calculate the impact of diverse fluid parameters on involved profiles and physical quantities. The results reveal that the microorganism concentration profile drops with rising values of the Peclet and bioconvective Lewis number. The outcomes of this investigation offer useful perceptions for improving thermofluidic mechanism and optimizing heat transport efficiency in nanofluid-based bioconvective devotions.

This paper investigates the influence of chemical reactions and variable magnetic field on three dimensional Oldroyd B micropolar nanofluids subjected to exponentially stretching sheet in the presence of motile microbes. The study incorporates several significant physical phenomena, including Cattaneo-Christov heat, thermal radiation, chemical reaction kinetics, and Darcy-Forchheimer effects. A particularly novel aspect of PST (prescribed surface temperature) and PHF (prescribed heat flux) are taken into account. The governing nonlinear PDEs of Oldroyd B fluids with thermophoretic diffusion and Brownian motion are transformed in to nonlinear ODEs via similarity functions. The resulting set of nonlinear ODEs are solved numerically via MATLAB platform and compared the results with published literature through bvp4c built-in code for better agreement. The results of on different parameters like Peclet number, Forchheimer number, thermal relaxation time, chemical reaction, Prandtl number, Schmidt number, porosity parameter, heat source coefficient and magnetic parameter on Skin friction, Nusselt number, Sherwood number and motile density number are discussed in detail through graphs, tables and literature. It is declared that Skin friction coefficients decline for developed values of magnetic parameter M, porosity parameter K1.and viscoelastic parameter K2. The thermal boundary layer thickness decreases with growing value of Prandtl number. The findings have significant implications for industrial and engineering processes where heat transfer is major issue.

This investigation explore the thermal properties of electromagnetics Jeffrey tri-hybrid nanofluid with entropy generation subjected to Riga plate. The tri-hybrid nanofluid model is established using three different nanoparticles: titania (TiO2), cupric oxide (CuO) and alumina (Al2O3), engine oil performing as the base fluid. While dust particles are integrated to scrutinize two-phase flow phenomena. For motivation of problem, the Cattaneo-Christov heat flux theory and modified magnetic field are the important aspect of this investigation. Similarity framework is used to renovate the partial differential equations of dynamical system into a set of nonlinear ordinary differential equations. These dynamical equations are integrated numerically using bvp4c tool in MATLAB with shooting method. The results indicated that the modified magnetic field destroys the speed of fluid but raises temperature distribution, while the Cattaneo-Christov model diminishes thermal slopes associated to usual Fourier's law. An increase in the modified magnetic field strength and temperature ratio leads to enhanced entropy generation. This investigation gives significance application for enhancing thermal management system like consumer electronics, engine cooling, aerospace, automotive, power electronics and medical devices, where diminishing entropy generation is crucial for energy productivity.

ABSTRACT The three‐dimensional model of Darcy‐Forchheimer flow in a convection system consisting of Ti‐alloy nanoparticles (TiO2) suspended in paraffin oil is mathematically constructed using fluid mechanics and partial differential equations (PDEs). A novel aspect of this study is the application of similarity transformation techniques to convert complex PDEs into a system of ordinary differential equations (ODEs), which are then solved using the Runge–Kutta 4th order method with the shooting technique. This unique approach provides deeper insights into the effects of magnetohydrodynamics (MHD), porosity, heat source, stretching surface, and radiation on bi‐directional velocity and temperature profiles. The results demonstrate that Ti‐alloy nanoparticles significantly enhance the thermal conductivity of the base fluid, leading to a 34.7% increase in temperature profiles compared to conventional fluids. The presence of a magnetic field induces a Lorentz force, reducing the bi‐directional velocity by 18.5% while increasing fluid temperature by 22.9%. An increase in the porosity parameter results in a 15.3% reduction in velocity due to higher resistance, whereas the temperature profile shows a corresponding rise of 26.1%. Furthermore, an increase in the Forchheimer parameter reduces velocity by 21.6%, while the radiation parameter enhances heat transfer by 29.4%. These findings highlight the superior heat transfer efficiency of Ti‐alloy‐based nanofluids, making them highly suitable for applications in thermal energy storage, solar energy systems, and industrial cooling technologies.

The recent study is related to Cattaneo-Christov model of Electromagnetohydrodynamic (EMHD) tri-hybrid nanofluid flow in the occurrence of gyrotactic microorganisms, heat source/sink, Arrhenius activation energy, chemical reaction, natural convection, and thermal radiation on two dissimilar geometries i.e., wedge and cone. The tri-hybrid nanofluid flow with Electromagnetohydrodynamic (EMHD) has many uses in industrial and engineering fields. Most medicinal and organic uses require a study into the insight process in nanofluid consisting of microorganism suspension. The system of partial differential equations is changed into a set of ordinary differential equations by using similarity transformation. The semi-analytical technique HAM is used to attain the solution to the problem. The influence of substantial constraints on temperature, velocity, motile density microorganisms, and concentration, are displayed through graphs. Velocity of tri-hybrid nanofluid (THNF) is increased for rising in electric parameter. Results show that the velocity of THNF rises by approximately 8.5% with a rise in the electric parameter E1 = 0.1 -* 0.4. The temperature profile rises by approximately 10% with a rise in the magnetic parameter M = 0.5 -* 2.0 and by about 7% for an upsurge in the electric parameter E1 = 0.1 -* 0.4. The concentration profile reduces by approximately 9% with a rise in the Arrhenius activation parameter E2 = 0.2 -* 0.8. The motile microorganism density diminishes by nearly 11% for growing Peclet number Pe = 0.3 -* 1.0 and by 8% with a greater bio-convection Lewis number Lb = 0.5 -* 0.8. The numerical outcomes of local density number, Nusselt number, Sherwood number, and skin friction are shown in the tables. The addition of silver, copper and aluminum oxide increases the thermal physical behavior of the base fluid considerably.

In numerous engineering and biological systems, appreciative the flow characteristics of non-Newtonian fluids is important for cooling and heat transfer system. The present work aims to analyze the behavior of two-dimensional Williamson fluid with convective boundary induced by nonlinear stretchable curve surface. The role of chemical reaction, heat sink/source, variable thermal conductivity, suction/injection and thermal radiation is also considered. By utilizing the corresponding similarity variables, the system of governing PDEs of curve surface model is converted into to a system of nonlinear ODEs. The resultant equations are tackled numerically using MATLAB platform via bvp4c package. To check the reliability and legality of applied bvp4c code and upcoming outcomes a comparison with surviving literature has been prepared. The impact of prominent parameters on involved profiles are discussed via graphs and literature. It is concluded that both mass and thermal buoyancy forces are directly linked with concentration and temperature distribution. The outcomes expose that rising the Williamson fluid parameter leads to a notable decline (up to 28 %) in the velocity field, while developed magnetic parameter contribute to further velocity suppression due to Lorentz force. Further, the surface drag force, heat and mass transfer effects are all influenced by the material fluid properties.This research holds significant importance in the areas of chemical engineering and biology where heat transfer is crucial.