Numerical research has been done on the mathematical modelling of micropolar Cu-Al2O3/water nanofluid flow driven by a deformable sheet in a stagnation area with suction effect. Using appropriate similarity transformations, the governing partial differential equations are reduced to nonlinear ordinary differential equations, which are then solved numerically using bvp4c function in MATLAB. The hybrid nanofluids are made up of aluminium and copper nanoparticles dispersed in a base fluid called water. Due to variations in numerous relevant parameters, the given problem yields multiple solutions for both shrinking and stretching sheets. Interaction between these input parameters (hybrid nanoparticle, micropolar and suction) and their influences on heat transfer are assessed by a statistical Response Surface Methodology (RSM) model developed by Box-Behnken design approach. The Nusselt number is anticipated to have a maximum value of 18.2903. The RSM results indicate that the highest heat transfer coefficient is achieved when the suction parameter is at its maximum value and the hybrid nanoparticle and micropolar parameters are at their minimum values.

Aluminum alloys are widely used in the automotive and aerospace industries due to their lower mass-to-strength ratio than other metallic alloys. Apart from their inherent properties, aluminum alloys like other metallic alloys show a significant change in their mechanical properties according to the machining parameters. The research literature on obtaining optimum mechanical properties of aluminum alloys that undergo machining is very limited. Moreover, the combined effect of several parameters on the machinability of aluminum alloys has not yet been explored. In this paper, the effect of three machining parameters (Depth of Cut (DoC)), feed rate (FR), and cutting speed (CS) on the subsurface damage and fatigue life of aerospace-grade aluminum alloy (Al-6082-T6) is observed. Samples are prepared using a full fractional approach to effectively capture the effect of all input parameters. Thereafter, samples were subjected to surface roughness, micro-hardness, and fatigue life tests. Results of surface roughness and micro-hardness tests are compared with fatigue life. The general linear model was employed to capture the percentage effect of each input parameter on the output parameters. The results showed that DoC was the main contributing factor that caused subsurface damage, while surface roughness and fatigue life were mainly affected by FR and CS. Optical microscope images showed a white layer formation that had higher hardness than the base metal. Overall, this research work proposes the input parameters that can be used to achieve minimum surface damage and fatigue life.

A nonlinear permeable shrinking sheet refers to a boundary where the permeability changes nonlinearly, and the sheet itself is contracting or shrinking over time. Meanwhile, magnetohydrodynamics (MHD) models electrically conducting fluids as a single continuous medium. Hence, this research integrates magnetohydrodynamics (MHD) models electrically conducting fluids as a single continuous medium. The distribution of three nanoparticles—aluminum oxide (Al2O3), titanium dioxide (TiO2), and silver (Ag)—in water (H2O) is considered to represent the ternary nanofluid model. Two analyses are performed: (i) numerical analysis—to describe the mathematical model; and (ii) statistical analysis—to optimize the heat transfer rate using Response Surface Methodology (RSM). The numerical results are obtained through bvp4c scheme in MATLAB after applying similarity transformation to reduce the governing equations. The findings show dual solutions due to the shrinking parameter, while ternary nanoparticles enhance heat transfer more than mono- or hybrid nanofluids. The nonlinearity parameter increases temperature profiles, whereas heat generation decreases them. Further, Response Surface Methodology (RSM) is applied to optimize the highest heat transfer rate by identifying three optimal parameter values for nonlinearity, radiation, and heat generation. With a suggested desirability of 99.98

Purpose This study aims to investigate the behaviour of unsteady three-dimensional (3D) boundary layer flow and heat transfer generated by a non-axisymmetric stagnation point over a shrinking surface. The influence of thermal radiation is also included to represent realistic thermal environments. In particular, the work here focuses on how the shear-to-strain rate parameter and shrinking effects influence the flow structure, heat transfer rate, and stability of the solution. Design/methodology/approach The governing partial differential equations describing the mass, momentum, and energy transport are reduced into a system of ordinary differential equations using an appropriate similarity transformation for the non-axisymmetric stagnation point flow. The resulting boundary value problem is solved numerically using the bvp4c solver. In addition, stability analysis is carried out to distinguish physically meaningful solutions between dual solution branches. An asymptotic analysis is also developed for large values of the shear-to-strain rate parameter to provide analytical inputs into the limiting behaviour of the system. Findings The results show that the shear-to-strain rate parameter has a significant impact on the flow and thermal fields. An increase in this parameter leads to a reduction in both the reduced skin friction coefficients and the heat transfer rate at the surface. The presence of a shrinking surface generates dual solutions due to bifurcation behaviour, resulting in upper and lower solution branches. Stability analysis confirms that upper solution corresponds to a physically stable flow configuration. Furthermore, the asymptotic results indicate that both skin friction and heat transfer increase in proportion to the square root of the shear-to-strain rate parameter for large parameter values, which corresponds to thinner momentum and thermal boundary layers. Practical implications The findings provide information into controlling flow and heat transfer in systems involving stagnation point flow over shrinking or stretching surfaces. Understanding the stability characteristics helps in identifying physically realizable operating conditions, which is important in industrial processes such as aerodynamic surface design, cooling technologies, and material processing where non-uniform strain fields may occur. Originality/value This work extends the existing stagnation point flow studies by incorporating a non-axisymmetric 3D configuration with unsteady effects, thermal radiation, and shrinking surface conditions. The main originality of this work lies in the combined use of numerical simulation, stability analysis, and asymptotic analysis within a single non-axisymmetric stagnation point flow framework, which has not been simultaneously addressed in previous studies. The integration of these three approaches provides more understanding of the flow behaviour across different parameter regimes.

In coating and fiber manufacturing, fluid flow along wedge-shaped molds critically affects the uniformity and thickness of the final product. Micropolar fluid models, which account for micro-rotational effects, provide a more accurate description of such complex fluids, especially those with high viscosity or suspended particles. Additionally, incorporating second-order velocity slip better captures realistic fluid-solid interactions at the boundary. This study investigates the forced convection boundary-layer flow and heat transfer of a micropolar fluid past a wedge moving either leftward or rightward in a stationary fluid, considering the effects of the micropolar parameter (K), suction/injection (S), and wedge motion direction. The governing equations are reduced via similarity transformations and solved numerically using Matlab's bvp4c. Dual solutions are observed only for leftward wedge motion, with stability analysis confirming that the first solution is stable and the second is unstable. Results show that increasing K enhances fluid mobility and affects skin friction and Nusselt number differently depending on wedge motion, while suction (S > 0) consistently promotes heat transfer. The novelty of this work lies in the integration of numerical and analytical solutions, along with a systematic analysis of micropolar flow under second-order slip conditions and its stability, providing valuable insights for precision control in coating applications.

The present study focuses on analysing and improving the heat -transfer performance of a rotating ternary hybrid nanofluid over a vertical flat surface under opposing mixed convection conditions. By employing suitable similarity transformations, the governing boundary-layer equations are reduced to nonlinear ordinary differential equations and subsequently solved using a MATLAB-based numerical approach. Furthermore, Response Surface Methodology (RSM) is used to investigate the combined influence of key parameters and to determine the conditions that maximise heat transfer efficiency. Increasing the concentration of nanoparticles, especially copper can greatly improve heat transfer efficiency, with copper nanoparticles showing the greatest enhancement, followed by aluminium oxide, (Al2O3) and titanium dioxide (TiO2) nanoparticles. Furthermore, desirability-based optimisation reveals that the heat-transfer rate attains a maximum value of 0.442152 with 99.93% desirability when the coded parameters A, B, and C (nanoparticle volume fractions) are at their maximum levels. Meanwhile, the decrement in skin friction along the x-direction is primarily influenced by the increase in volume fraction of copper nanoparticles, followed by titanium dioxide (TiO2) and aluminium oxide nanoparticles. The findings provide significant insights into optimising heat-transfer in complex fluid dynamics systems, with potential applications in diverse industrial and engineering domains.

Effects of discrete heat sources along a vertical plate are of practical importance due their occurrence in electronic devices. For growing demand of electronic appliances and their advancement, cooling processes of them must be improved. As usual fluids have limited heat transfer, nanofluids made by dispersing nanoparticles into them are utilized to enhance thermal performance. However, flow characteristics and heat transfer of a nanofluid for discrete heat sources along a vertical plate need to be explored. For this reason, this study analyzes the natural convective heat transfer and flow behaviors of CuO-water nanofluid induced by discrete heat sources along a vertical plate. The influences of variable thermophysical properties of the nanofluid, thermal radiation, and magnetic field are also considered. Using the finite difference method, the nonsimilar governing equations have been solved. Results reveal that for increasing surface temperature and radiation parameters, the shear stress (ST) and rate of heat transfer (HT), and their average values are increased. However, the opposite is observed for the magnetic parameter. For increasing ambient temperature and nanoparticles' volume fraction, the ST increases and the rate of HT decreases. An increase in the magnetic parameter, nanoparticles' volume fraction, and ambient temperature leads to a decrease in the magnitude of stream function; however, it causes an increase in the momentum and thermal boundary layers. Interestingly, a comparison reveals that the model for constant properties of the nanofluid provides a decrease in heat transfer by an average of about 7.8% compared to that for variable properties.

Heat transfer is frequently employed in various industrial processes such as paper production, electronic device cooling, and the synthesis of new materials. Hence, this study aims to investigate the effect of Joule heating and magnetohydrodynamics (MHD) on the flow of a hybrid nanofluid with a power law heat flux past a shrinking sheet. The transformed governing equations are solved numerically using MATLAB’s bvp4c solver, and the results are validated against previously published data, showing excellent agreement (error < 0.01

Maxwell fluid has attracted attention because of their impressive thermal characteristics and use in polymer processing, metal spinning and rolling, biomedical engineering, and cooling systems. This study emphasizes the graphical investigation of the flow behavior and heat transfer of a Maxwell ternary nanofluid (Al2O3-Cu-Fe3O4/water) over a shrinking surface with the impacts of a magnetic field, suction, shrinking parameter, and heat source or sink. Using a set of similarity transformations, ordinary differential equations (ODEs) are generated through alteration of the leading equations. The resulting equations are solved with the assistance of a well-established finite difference method (FDM). For the exactness and justification of the present outcomes, an assessment is conducted between the current solutions and the data that is accessible, which demonstrates a good consistency. Graphical results show that the velocity rises and the temperature declines for increasing magnetic field, suction, and shrinking parameter. In contrast, larger nanoparticle volume fractions, Biot number, and heat source feature cause an enlargement in the temperature outline. Larger suction parameter, shrinking parameter, and magnetic field parameter result in an increase in the heat flux about 6

PurposeThis study aims to investigate a mixed convection stagnation-point flow and heat transfer of a ternary nanofluid over a vertical linearly stretching/shrinking sheet. The ternary nanofluid consists of aluminium oxide (Al2O3), copper oxide (CuO) and silver (Ag) nanoparticles dispersed in water (H2O). The effects of surface permeability and thermal radiation on flow and thermal characteristics are examined, with emphasis on heat transfer enhancement and parameter optimization.Design/methodology/approachThe governing partial differential equations describing momentum and energy transport are transformed into a system of ordinary differential equations using a similarity transformation approach. The resulting equations are solved numerically in MATLAB using the bvp4c solver. To further evaluate the parameter influence and optimize thermal performance, response surface methodology and normalized sensitivity analysis are employed. A predictive correlation relating key parameters to the heat transfer response is also developed.FindingsThe analysis shows that ternary nanofluid yields the highest heat transfer rate compared to hybrid and mono-nanofluids under identical operating conditions. Dual solutions are observed in the shrinking regime which indicates multiple flow states. Furthermore, statistical analysis reveals the relative influence of Al2O3, CuO and Ag volume fractions on the heat transfer rate, where Ag exhibits the strongest influence on the heat transfer rate due to its high thermal conductivity, and the optimized combination of these ternary nanoparticles leads to maximum heat transfer performance.Practical implicationsThe combined numerical and statistical framework offers a structured approach for improving thermal transport in nanofluid-based systems. The findings may assist in the design of enhanced cooling and thermal regulation applications where improved heat transfer efficiency is required, as well as contributes to energy efficiency efforts that aligned with sustainable development goals (SDG 7).Originality/valueThis work integrates similarity-based numerical modelling with statistical optimization and normalized sensitivity analysis for a ternary nanofluid system under mixed convection stagnation-point flow. The development of a predictive correlation and systematic parameter ranking also provides further information into multi-nanoparticle heat transfer behaviour.

This study investigates the steady two-dimensional boundary layer flow of a couple stress hybrid nanofluid past a shrinking Riga plate. The flow is controlled by an externally powered Riga plate capable of producing Lorentz forces, strengthening flow stabilization. Unlike prior works that focused individually on couple stress fluids, hybrid nanofluids, or Riga plates, this work integrates all three within the framework of a shrinking surface. This unique combination offers new insights into boundary layer control, particularly in advanced microfluidic and cooling applications, where precise regulation of both momentum and heat transfer is crucial. The hybrid nanofluid considered in this study consists of aluminium alloys (AA7072) and (AA7075) nanoparticles dispersed in methanol (CH3OH). The governing system of partial differential equations is converted to ordinary differential equations using similarity transformations, which are then solved numerically using MATLAB's bvp4c solver. The results reveal that the inclusion of hybrid nanoparticles enhances heat transfer and improves thermal retention compared to the base fluid. Increasing suction significantly stabilizes the boundary layer, suppressing velocity profiles and reducing thermal boundary layer thickness. Conversely, higher values of the Riga plate parameter reduce velocity gradients while causing a slight increase in the thermal boundary layer thickness, indicating a trade-off between flow retardation and heat retention.

The fundamental mission of this study is to formulate and solve the mathematical model of the stagnation point flow of a hybrid nanofluid with the insertion of second-order velocity slips, magnetohydrodynamic (MHD), and radiation effects over a shrinking sheet, which are critical for enhancing thermal performance in industrial cooling and heat treatment processes. The model is transcribed into non-dimensional formulations using similarity variables and is solved numerically using the bvp4c solver in MATLAB. Dual solutions are executed, and the stable solution is validated via the stability analysis. In certain conditions, the comparison of current and prior findings demonstrates good agreement with nearly 0% relative error. The findings reported that the critical point is extended, and the bifurcation of the boundary layer is prevented by the boost in the magnitude of second-order velocity slips and copper volume fraction. The efficiency of heat transfer improves as the radiation effect and the copper volume fraction increase, particularly when the sheet is shrunk. The boost of copper volume fraction is also simulated to lessen the temperature and the thermal boundary layer thickness. Thus, the present model in this study has proven that the utilization of a hybrid nanofluid could increase the thermal performance of a system, and it could be used as a coolant for a heat treatment process.

Researchers are motivated to understand the behavior and properties of hybrid nanofluids due to their wide range of applications. For example, unsteady hybrid nanofluid flow can occur in marine propellers, hydrofoil flutters, rotor blades, and turbomachines. This study examines the unsteady mixed convection flow of a hybrid nanofluid over a radially shrinking disk. The time-dependent governing partial differential equations and associated boundary conditions are formulated and transformed into a system of non-linear ordinary differential equations using similarity transformations. These equations are solved numerically using MATLAB’s bvp4c function. Two solutions are obtained, and a stability analysis confirms that only the first solution is stable. In this flow problem, increasing both the Biot number and the mixed convection parameter increases the local Nusselt number and local skin friction coefficient. Increasing the mixed convection parameter from its lowest to highest considered values leads to increases of 113 α =-0.7, Bi=0.7, and λ =1.5 ). Meanwhile, the local skin friction coefficient is minimized when these parameters are at their lowest levels (i.e., α =-0.7, Bi=0.3, and λ =0.5 ). At these optimal conditions, the local sensitivity analysis suggests that the local Nusselt number is most sensitive to the Biot number, whereas the local skin friction coefficient is most sensitive to the mixed convection parameter.

This study introduces a novel integrated computational and statistical approach for analyzing steady boundary layer flow and heat transfer in a tetra hybrid nanofluid containing alumina, copper, silica and titania nanoparticles dispersed in water past a convectively heated moving plate with internal heat generation. The governing nonlinear equations are reduced via similarity transformations and solved using Matlab's bvp4c solver to obtain highly accurate velocity, temperature, skin friction, and Nusselt number distributions. Response Surface Methodology (RSM) and sensitivity analysis are employed to quantify and rank the influence of heat generation, Biot number, and suction parameter. The numerical results reveal that increasing the Biot number may enhance the heat transfer rate by approximately 42.6 for opposing flow case and 53.6% for assisting flow case, whereas stronger wall suction improves heat transfer by about 0.5-6.2%. However, higher heat generation slightly weakens the heat transfer rate up to 3.6% and 0.2%, for opposing and assisting flow cases, respectively. The response surface methodology and sensitivity analysis also reveal that the Biot number exerts the dominant influence on heat transfer, followed by suction strength and heat generation rate. The principal novelty lies in the exclusive integration of numerical simulation with statistical optimization for a tetra hybrid nanofluid under convective heating, an area rarely addressed in the literature. The proposed framework not only identifies the most influential parameters but also determines optimal ranges for maximizing thermal performance. These findings establish a benchmark for designing advanced thermal management systems in high-temperature industrial and energy applications.

A new theoretical ternary hybrid nanofluid, by suspending three types of nanoparticles with different physical and chemical bonds in a porous square cavity, is proposed in this paper. The ternary hybrid nanofluid is formed by suspending three types of nanoparticles with different physical and chemical bonds into a base fluid. In this study, the nanoparticles alumina (Al2O3), Copper, and Titania (TiO2) are suspended into water thus forming the combination. The system ofgoverning PDE, are numerically solved using finite element formulation based on the Galerkin along with ADINA software (Adina v 9.20) method. The average Nusselt number is computed for three values of the Rayleigh number: 10, 100, and 1000, with the results of other authors from the open literature. An excellent agreement, and therefore, we are deeply confident that the numerical results obtained are correct and very accurate. We wish to point out that the numerical results of the present paper are completely new and original with very important results for practical applications of the ternary hybrid nanofluid in the modern industry. To our best of knowledge, the results of the present paper were never published by any researcher.

Efficient heat transfer is a major challenge in systems like MHD pumps, electromagnetic cooling units and rotating heat exchangers especially under extreme conditions. The combined effects of magnetic fields, Joule heating and multiple nanoparticles create complex behaviors that are hard to solve using analytical methods. Therefore, reliable numerical methods and statistical optimization tools are needed to analyze and improve these systems. Hence, this work highlights ternary hybrid nanofluid flow over a permeable moving surface with the influence of magnetohydrodynamic (MHD), Joule heating and suction effects, integrating both computational and optimization techniques. The fluid comprises water-based ternary hybrid nanofluid containing aluminum oxide (Al2O3), copper (Cu) and titanium dioxide (TiO2) nanoparticles. The governing partial differential equations are transformed into a system of ordinary differential equations using similarity transformations and solved using the bvp4c solver (MATLAB). Validation of the numerical approach is carried out by comparing results with existing literature which showing excellent agreement for limiting cases. Response surface methodology (RSM) is applied to analyze interactions between the key parameters. Results indicate that both magnetic parameter and titania concentration significantly enhance the velocity profile due to the induced Lorentz force and altered thermal gradient. Analysis of variance (ANOVA) confirms that magnetic parameter and titania concentration are the most influential on thermal and flow responses. Sensitivity analysis further highlights strong linear and interaction effects.

Nanofluids have gained significant attention in industrial and engineering applications due to their enhanced thermal conductivity, making them suitable for cooling systems, heat exchangers, and electronic devices. Despite extensive research on boundary layer flow in nanofluids, the influence of thermodiffusion (Soret effect) and diffusion-thermo (Dufour effect) in the presence of second-order slip has not been thoroughly explored. This study aims to investigate the effects of second-order slip, Soret, and Dufour parameters on stagnation boundary layer flow over a stretching/shrinking sheet immersed in a Cu-water nanofluid. The primary objective is to analyze how these parameters influence skin friction, heat transfer, and mass transfer characteristics. The governing partial differential equations are transformed into ordinary differential equations using similarity transformations and numerically solved via the bvp4c solver in MATLAB. Results indicate that the presence of the first-order slip parameter broadens the solution region, whereas the second-order slip parameter narrows it. Additionally, the Soret effect enhances the heat transfer rate, while the Dufour effect increases mass transfer at the surface. The study also reveals the existence of dual solutions, necessitating a stability analysis to determine which solution is physically realizable. The findings provide valuable insights into optimizing nanofluid applications in industrial and engineering processes.

This study investigates the heat transfer and flow dynamics of a ternary hybrid nanofluid comprising alumina, copper, and silica/titania nanoparticles dispersed in water. The analysis considers the effects of suction, magnetic field, and Joule heating over a permeable shrinking disk. A mathematical model is developed and converted to a system of differential equations using similarity transformation which then, solved numerically using the bvp4c solver in Matlab software. The study introduces a novel comparative analysis of alumina-copper-silica and alumina-copper-titania nanofluids, revealing distinct thermal conductivity behaviors and identifying critical suction values necessary for flow stabilization. Dual solutions are found within a specific range of parameters such that the minimum required suction values for flow stability, with Sc=1.2457 for alumina-copper-silica/water and Sc=1.2351 for alumina-copper-titania/water. The results indicate that increasing suction by 1% enhances the skin friction coefficient by up to 4.17% and improves heat transfer efficiency by approximately 1%, highlighting its crucial role in stabilizing the opposing flow induced by the shrinking disk. Additionally, the inclusion of 1% silica nanoparticles reduces both skin friction and heat transfer rate by approximately 0.28% and 0.85%, respectively, while 1% titania concentration increases skin friction by 3.02% but results in a slight heat transfer loss of up to 0.61%. These findings confirm the superior thermal performance of alumina-copper-titania/water, making it a promising candidate for enhanced cooling systems, energy-efficient heat exchangers, and industrial thermal management applications.