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 The purpose of this study is to develop an exact analytical framework for boundary layer flow of Nano-Encapsulated Phase Change Material (NePCM)-enhanced fluids over stretching and shrinking surfaces, accounting for key physical effects such as velocity slip, wall transpiration and thermal radiation. Design/methodology/approach The governing Navier–Stokes equations are formulated for NePCM suspensions and solved analytically to obtain closed-form exponential and algebraic solutions for the velocity, temperature and concentration fields. Numerical computations are further used to validate the analytical solutions and to ensure the reliability of the computational framework for benchmarking purposes. Findings The analysis reveals the existence of dual and triple solution branches for the skin friction coefficient, thermal gradient and species gradient, highlighting the strong nonlinearity of the system. A critical turning point in the transpiration parameter $(s_c)$ is identified, beyond which no physically admissible solutions exist for the shrinking sheet. The results demonstrate that NePCMs significantly suppress temperature gradients, while thermal radiation and the Stefan number play a decisive role in regulating the thermal boundary layer thickness. Originality/value To the best of the authors’ knowledge, this work presents the first exact analytical treatment of NePCM-enhanced boundary layer flow over stretching and shrinking surfaces with slip, transpiration and radiation effects. It provides fundamental physical insight into complex flow transitions and establishes a rigorous theoretical foundation for the design of advanced thermal management systems in microelectronics and aerospace applications.

This study discovers the mass and heat transfer characteristics of a magnetohydrodynamic GO-Ag-CuO-Al₂O₃/EG nanofluid streaming past a shrinking sheet under the force of various physical effects. The Powell-Eyring fluid model is engaged to account for the effect of a magnetic field, a stagnation point, viscous dissipation, radiation, Joule heating, and suction. Through similarity transformations, the governing partial differential equations are reduced to a system of ordinary differential equations, which are then resolved numerically using the bvp4c solver in MATLAB. The results affirmed that raising the heat transfer complements the thermal boundary layer, whereas reductions in the skin friction coefficient contribute to a reduction in drag force. Moreover, the velocity profile rises because of the shrinking effect, while the temperature profile decreases. The enhanced thermal conductivity provided by the quaternary nanoparticle suspension (GO-Ag-CuO-Al₂O₃) suggests that this fluid can maintain lower surface temperatures under high-heat flux conditions compared to mono or hybrid nanofluids. Consequently, these characteristics are particularly advantageous for electronic device cooling and heat exchangers in renewable energy systems where rapid heat dissipation is critical. Furthermore, the observed reduction in drag force under the influence of the magnetic field provides a theoretical basis for optimizing energy efficiency in magnetohydrodynamic pumps and metallurgical processing. These findings offer specific design parameters for boosting thermal management in industrial applications involving tetra-hybrid nanofluids.

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.

Nanofluids, formed by dispersing nanoparticles into conventional base fluids, have gained significant attention as advanced heat transfer media in various sectors, including automotive, aerospace, nuclear reactors, biomedical systems, etc. Their enhanced thermophysical properties and particularly improved thermal conductivity, which make them superior to traditional fluids for high-efficiency thermal management. To accurately estimate the thermal conductivity of nanofluids, several theoretical models have been proposed, among which the Hamilton-Crosser (HC) model is widely adopted. This model extends Maxwell's classical theory by incorporating particle shape factors, providing more accurate predictions for fluids containing non-spherical or clustered nanoparticles. In the field of fluid mechanics, boundary layer flows influenced by magnetohydrodynamics (MHD) and thermal radiation are of increasing relevance in systems subjected to magnetic fields and high heat loads, such as in magnetic cooling, plasma processing and solar energy applications. According to this problem statement, this paper provides a numerical solution for the mathematical modelling of the MHD radiative water-based nanofluid flow. This nanofluid contains a combination of four nanoparticles: alumina, copper, graphene and silicon dioxide, which represents a quaternary hybrid nanofluid system. The mathematical modelling was initiated with the partial differential equations (PDEs) and finalised by ordinary differential equations (ODEs). The transformations from PDEs to ODEs simplify the mathematical model for numerical analysis, which can be done by using similarity transformations. The resulting ODEs are solved using MATLAB's built-in bvp4c solver. Key physical quantities for the profiles (velocity and temperature) and the quantities measured for the flow and thermal characteristics (skin friction coefficient and local Nusselt number) are graphically illustrated and described. The results show that the quaternary case produces the highest heat transfer rate compared to the nanofluid case, which has the lowest number of nanoparticle types (nanofluid case). In addition, the cylindrical shape is proven to have the highest local Nusselt number value for the stretching boundary case, while the spherical shape produces the highest local Nusselt number value for the shrinking boundary. Meanwhile, the magnetic field, thermal radiation, slip condition and projected angle at the sheet influence the distribution of the profile and physical parameters.

Carbon nanotubes (CNTs) spark interest due to their inimitable characteristics, leading to a multitude applications across various sectors. Thus, a mathematical model is developed for hybrid carbon nanotubes flow towards stagnation zone on an exponentially permeable cylinder. The flow is unsteady under stretching conditions. Nanoparticle geometry and thermal source are the physical manifestations of thermal energy. The basic formulation that defines the mathematical description of unsteady flow is recast into highly nonlinear differential equations through a new self-similarity variable. To produce observational data, a numerical tool (bvp4c) in Matlab utilized. The responses to flow factors are physically depicted through graphical illustration. The calculation yielded a non-unique nature in both (elongation/contraction) zones, with clearly spotted the respective critical points. The key results indicate that an exalts exponential parameter lead to prolong the onset of turbulent flow. The presence of hybrid carbon nanotubes reduced the range of solutions. Further, noticeable linear drop found in thermal source and shape factor with the strength of hybrid carbon nanotubes. It is also found that the temperature profile of platelet nanoparticle exhibits the highest recorded values among all considered cases.

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 presents a mathematical and statistical analysis of hybrid carbon nanotube (CNT) nanofluid flow in the boundary layer across a wedge, emphasising the optimisation of heat transfer rates under the influence of hydromagnetic effects. The novelty of the study lies in the integrated use of Response Surface Methodology (RSM) to statistically maximise the thermal performance of this complex hybrid fluid system (single-and multi-walled CNTs in water), offering practical design parameters that extend beyond a numerical simulation. Using the bvp4c numerical method to solve the transformed ordinary differential equations (ODEs), we found that increasing the wedge and magnetic parameters significantly enhances both skin friction and heat transfer coefficients. Furthermore, the RSM analysis, optimised via the desirability function, successfully identified the best heat transfer conditions, concluding that the CNT volume fraction is the most influential factor and that the hybrid CNT nanofluid is superior to its mono-CNT counterpart for efficient heat transfer applications.

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.

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.

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.

This study analyzes the steady hybrid nanofluid flow over a permeable, non-isothermal cone and wedge. The heat transfer analysis considers the effects of thermal radiation and convective boundary conditions. Non-linear ordinary differential equations, derived through similarity transformation of the governing partial differential equations and boundary conditions, are solved numerically using the bvp4c solver. The resulting triple solutions are then subjected to a stability analysis. It is confirmed that only the first solution is stable and physically meaningful, while the other solutions are unstable. The physical quantities of interest, namely the local skin friction coefficient and local Nusselt number, are found to be higher for assisting mixed convection flow than for opposing flow. Compared to the wedge geometry, hybrid nanofluid flow over the cone exhibits a lower local skin friction coefficient but a higher local Nusselt number. Furthermore, optimization results from the response surface methodology (RSM) indicate that the maximum local Nusselt number, corresponding to the highest heat transfer rate at the cone/wedge surface, can be achieved at high values of the Biot number, radiation parameter, and wall temperature parameter.

In electromagnetism, a magnetic dipole is a tiny loop of electric current or a pair of magnetic poles. As the loop size decreases to zero while maintaining a constant magnetic moment, it forms a magnetic dipole. Composed by the magnetic particles, ferromagnetic fluids align with magnetic fields and when a magnetic dipole interacts with such fluids, the particles magnetize the fluid and influence the dipole's field. Hence, this study investigates the magnetic dipole and velocity slip on ternary hybrid ferrofluid flow past a shrinking surface. The model considers three magnetic nanoparticles - iron oxide (Fe3O4), cobalt ferrite (CoFe2O4), and copper (Cu) - dispersed in a base fluid. The similarity transformation technique is applied to derive mathematical models, which were solved numerically using bvp4c program in MATLAB. The analysis reveals that ferrohydrodynamic interaction reduces the skin friction coefficient and heat transfer rate but enhances velocity and temperature profiles. Additionally, the ternary hybrid ferrofluid is also shown to outperform both conventional ferrofluid and hybrid ferrofluid in fluid flow characteristics. Response Surface Methodology (RSM) is employed to identify the optimal combination of parameters, suggesting that the highest ferrohydrodynamic parameter and viscous dissipation, along with minimal Cu-nanoparticle concentration, maximize the heat transfer rate. Contour and surface plots illustrate these optimal conditions. This study highlights an innovative application of ternary ferrofluid with a magnetic dipole and employs RSM to optimize parameters for enhanced heat transfer performance, addressing a gap in existing literature and providing the way for further advancements in this field.

This work presents a novel numerical and statistical framework for analyzing tetra hybrid nanofluid flow over a shrinking disk with the influence of magnetohydrodynamic (MHD), thermal radiation, and suction effects, integrating both computational and optimization techniques. The fluid comprises water-based tetra hybrid nanofluid containing aluminium oxide (Al2O3), copper (Cu), silicon dioxide (SiO2), and titanium dioxide (TiO2) nanoparticles. The governing partial differential equations (PDEs) are transformed into a system of ordinary differential equations (ODEs) using similarity transformations and solved numerically via the bvp4c solver in the Matlab software. Validation of the numerical approach is carried out by comparing results with existing literature, showing excellent agreement for limiting cases. Response Surface Methodology (RSM) with central composite design is applied to analyze interactions between the key parameters. The results reveal that increasing the magnetic parameter by 0.1 leads to a 17.3% increase in the skin friction coefficient and an 1.4% enhancement in heat transfer rate. Conversely, raising the radiation parameter by 0.1 reduces the Nusselt number by up to 9.5%, indicating weakened thermal gradients at the wall. In addition, with a 2% increase in suction parameter improving heat transfer and skin friction by approximately 26.6% and 8.8%, respectively. Analysis of variance (ANOVA) confirms that suction and magnetic parameters are the most influential on thermal and flow responses. Sensitivity analysis further highlights strong linear and interaction effects. These findings offer valuable guidelines for the design of advanced thermal management systems using multi-nanoparticle-enhanced fluids, particularly in applications such as rotating heat exchangers, MHD pumps, and electromagnetic cooling devices operating under radiative conditions.

The heat transfer optimization for magnetohydrodynamic unsteady stagnation-point flows and the thermal progress with the effect of heat generation is performed using the response surface methodology. The first step in this study involves reducing the mathematical model of partial differential equations and boundary conditions into non-linear ordinary differential equations via similarity transformations. Numerical solutions of the emerged system are obtained using the bvp4c solver. As observed from this study, the magnitude of the skin friction coefficient and heat transfer rate rises with the suction parameter. The statis-tical analysis and optimization done using the response surface methodology revealed that the suction parameter highly impacts the local Nusselt number. The maximum sensitivity of the heat transfer rate is towards the magnetic and suction parameters.

Carbon nanotubes (CNTs) have proven their value in diverse multidisciplinary applications. For this purpose, the current study sheds light on time-reliant properties of electrically conducting flow of hybrid carbon nanotubes, scenario involving joule dissipation at a permeable cylinder that can expand and shrink. To get a precise insight into numerical outcome, the unsteady governing momentum and energy equations in cylindrical coordinates rendered into pertinent ODEs via incorporating the rescaling technique, Thereafter, the rendered equations cracked numerically via a built-in function in MATLAB (BVP4C) package. Notably, the sundry parameters yield two distinct solutions in both assisting and opposing zones, so the flow separation is identified. The governing physical factors are well explored through various graphical forms with physical explanations. Graphical observations declare, that heightening the value of curvature and volume fraction parameters contributes to speed up the onset of turbulence flow, augmentation in skin friction rate is noted through unsteadiness, magnetic field, and curvature parameters. Additionally, the stability assessment clearly specifies the mathematical robustness of the first branch as time passes. This study stands out for it is an inimitable configuration that holds significant addition in optimization of modern heat transfer applications.

This work presents an in-depth analytical study of the flow and heat transfer characteristics of a nanotriple fluid system—comprising copper, alumina, and silver nanoparticles—over a permeable, elastic, and deformable surface, subject to magnetohydrodynamics (MHD) and velocity slip conditions. Unlike many emerging numerical treatments of water-based nanotriple fluids, the primary objective is to derive exact, closed-form solutions, providing a substantial contribution to the analytical understanding of such complex systems. A unique aspect of this investigation is the identification of multiple algebraic-type solutions for the stretching/shrinking sheet problem, yielding dual solutions under injection and a single solution under suction conditions. In addition, critical numbers are identified as thresholds delineating the boundaries for the existence or absence of solutions. It is found that the number of solutions increases as the magnetic force strength decreases. Dual solutions are observed for both skin friction and thermal gradient in the exponential and algebraic cases. These analytical findings are further reinforced by extensive numerical computations, which offer robust validation of the exact solutions derived. Additionally, stability analysis is carried out in order to determine the stability of solutions, where the first branch demonstrates stability, and the second branch is unstable, highlighting the distinct behaviors within the solution branches.

This study deals with three dimensional rotating nanofluid over a shrinking sheet with velocity slip. Similarity transformations have been used for reducing the partial differential equations into a system of ordinary differential equations. The transformed ordinary diffential equations are solved numerically using BVP4C. The effects of Prandtl number Pr, suction parameter S, shrinking parameter λ, rotation parameter ω and slip parameter K on the velocity and temperature fields are presented and discussed in detail. The change in Prandtl number only affects the temperature profile while changing the rotation parameter affects velocity profiles. As the suction parameter rises, it results an increased velocity profile while the increase of slip parameter leads to a reduction in velocity proΫiles. As the Prandtl number, suction parameter, shrinking parameter, rotation parameter, and slip parameter rises, there is a reduction in the boundary layer thickness. This study provides valuable guidance and insights for researchers and practitioners investigating the mathematical or experimental aspects of three‐dimensional rotating hybrid nanofluids with slip effects.

This investigation aims to solve the mathematical modelling of magnetohydrodynamic (MHD) stagnation flow and heat transfer of GO-TiO2-Ag/water nanofluid towards a shrinking surface problem. The model contains the impacts of suction/injection, radiation, magnetic field intensity and heat source/sink parameters. By converting the governing equations into ordinary differential equations with applying similarity conversion and employing the bvp4c built-in solver in MATLAB software, numerical results are attained. Furthermore, the existence of GO as the ternary particles improves more the temperature profile but declining the velocity profile. This investigation promotes substantial understandings into heat transference achievement in MHD stagnation flow towards a shrinking surface, specifically with the interaction of GO particles and heat source/sink influences.