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.

The implementation of hybrid carbon nanotubes (CNTs) represents a significant advancement in enhancing heat transfer efficiency compared to conventional fluids. This study investigates the role of hybrid CNTs in stretch/shrink cylindrical frameworks, considering the effects of thermal slip and radiation on velocity. The hybrid nanofluid flow model is formulated using partial differential equations (PDEs) with appropriate boundary conditions (BCs), which are then transformed into ordinary differential equations (ODEs) via similarity solutions and numerically solved using bvp4c in MATLAB. The analysis evaluates the impact of slip, curvature, and radiation on the local Nusselt number and skin friction coefficient, offering deeper insights into the underlying heat transfer mechanisms. The results indicate that hybrid CNTs enhance both skin friction and heat transfer efficiency compared to single-walled (SWCNT) and multi-walled (MWCNT) CNTs. Furthermore, response surface methodology (RSM) is employed to establish the relationship between the Nusselt number and governing equation parameters, providing a comprehensive understanding of the parametric influences on heat transfer optimization. These findings underscore the practical significance of hybrid CNTs in improving thermal performance for various engineering applications, such as in polymer extrusion, cooling of flexible materials, micro-scale heat exchangers, and advanced thermal management systems, where efficient control of heat and fluid flow is crucial.

This study aims to examine the two-dimensional boundary layer flow and heat transfer over a nonlinearly permeable stretching/shrinking cylinder at the stagnation point, utilizing water and kerosene as base fluids with single-walled and multi-walled carbon nanotubes. The analysis considers hydromagnetic effects in the flow. The partial differential equations (PDEs) governing the fluid flow model are transformed into ordinary differential equations (ODEs) through similarity transformation and are then analyzed using bvp4c solver in MATLAB software. The impacts of the magnetic field, curvature, nanoparticle volume fraction and nonlinear parameters, as well as the influence of varying these parameters on the skin friction coefficients, heat transfer rate on surface, velocity and temperature profiles are observed. The study reveals a duality of solutions within a specific range of shrinking case, whereas a unique solution is obtained for the stretching case. The impact of magnetic and nonlinear parameter increase the rate of heat transfer. It also demonstrated that single-walled carbon nanotubes and kerosene-based fluid contribute to higher skin friction coefficients and heat transfer on the surface. A response surface methodology (RSM) is conducted to develop a correlation between the heat transfer rate, which is the response and the parameter considered in this study. The findings by using RSM found that nanoparticle volume fraction positively impacts the rate of heat transfer. The heat transfer rate is estimated to be optimized at -0.0144420 .

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.

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.

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 aims to explore the consideration of boundary layer flow and heat transfer over a moving plate with the presence of the magneto-hydrodynamics at the surface in carbon nanotubes. The mathematical model for the boundary layer flow problem is obtained and solved using numerical techniques based on Haar wavelet collocation. The types of nanoparticles used in this research were single-walled carbon nanotubes and multi-walled carbon nanotubes with water and kerosene that were used as base fluid. The partial differential equations are transformed into nonlinearly ordinary differential equations by similarity transformation. Maple software is used to work on these equations. The results were represented in the formation of graphs including velocity and temperature profile, skin friction coefficient and local Nusselt number for different values of magnetic field, CNTs volume friction and moving parameter. The outcomes obtained are that the moving plate gives non-unique solutions. In addition, the increments of magnetic field into the flow will increase value of skin friction coefficient and the heat transfer coefficient.

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.

The study of stagnation point flow and heat transfer over a stretching/shrinking sheet in a hybrid nanofluid has numerous possible applications such as microelectronics cooling design and, heat transfer in atmospheric re-entry. This study needs to solve the governing mathematical model (partial differential equations) to explore the properties of fluid flow and heat transfer. The model is converted into a system of ordinary differential equations by using similarity variables. In order to solve these nonlinear ordinary differential equations, the study uses numerical collocation method of Haar wavelets. By employing this method, the study can forecast several significant fluid flow and heat transfer characteristics, such as the local Nusselt number, skin friction coefficient, and temperature and velocity profiles. The numerical results show that dual solutions can appear for certain values of the shrinking parameter. It is also noted that the slip parameter, suction/injection parameter, and nonlinear parameter contribute to widening the range of possible solutions to allow for more variations in the fluid behaviour under different conditions.

A steady flow of carbon nanotubes (CNTs) nanofluids and heat transfer past a horizontally moving thin needle are investigated under the influence of magnetohydrodynamic (MHD). Single-walled CNTs (SWCNTs) and multi-walled CNTs (MWCNTs) are the two main nanoparticles that represent CNTs. The slender needle moves relative to the flow with parallel velocity in either the same or opposite direction. Using the similarity method, a system of partial differential equations (PDEs) subject to boundary conditions is converted into nondimensional ordinary equations (ODEs). The ODEs are then reduced to a first-order ODEs system and solved using the MATLAB R2022b bvp4c solver. On a numerical scale, the impacts of varying potential parameters, such as the magnetic, CNTs’ volume fraction, and moving parameters, on the velocity and temperature profiles, the skin friction, and heat transfer coefficients are investigated. Utilising response surface methodology (RSM), the optimisation of the response based on numerical experimentation of physical quantities is performed. The outcomes are depicted using tables and a graphical approach. Results indicate the existence of dual solutions when the needle travels in the opposite direction. Moreover, the increase in the magnetic parameter by 100% in the flow will increase both the skin friction and heat transfer coefficients by nearly 30% and 4%, respectively. Furthermore, when the value of the CNTs volume fraction increases by 100%, the heat transfer rate increases substantially by almost 33%.However, doubling the size of the thin needle cansignificantly reduce the skin friction coefficient by nearly 32%. RSM results demonstrate that the maximal heat transfer coefficient is generated at the highest values of the magnetic and CNTs’ volume friction parameters and the lowest value of the needle size parameter. Findings also show that SWCTNs are superior to MWCNTs both in the skin friction and heat transfer coefficients. When comparing the performance of water and kerosene, we find that water is less effective as a base fluid than kerosene.

The focus of this research is to observe how suction/injection affects the stagnation point flow and heat transfer in a hybrid nanofluid over stretching/shrinking cylinder. Silver (Ag) and copper oxide (CuO) nanoparticles are dispersed in pure water to create a hybrid nanofluid. By using similarity transformations, the governing partial differential equations are turned into a ordinary differential equations which are then solved by implementing bvp4c function in MATLAB software. The influence of the nanoparticle volume fraction, magnetic parameter, curvature parameter and suction/injection parameter, on velocity and temperature profiles, local skin friction and local Nusselt number are discussed and presented in graphical forms. The results indicate that all of the problems have dual solutions for a given range of parameters. It is noticed that with the suction effect, energy losses is reduced, thus the heat transfer increases and decreases the boundary layer separation. Furthermore, the presence of curvature parameter and suction effect expand the range of dual solutions. In addition, the rate of heat transfer for hybrid nanofluid was higher than viscous fluid and nanofluid.

The present study aims to provide a numerical and optimisation analysis of the hybrid carbon nanotubes flow over a vertical thin needle under the suction effect. The thin needle is suspended in water-based hybrid nanofluids containing a combination of single- and multi-walled carbon nanotubes. The heat is transported using the mixed convection flow. A mathematical model of partial differential equations (PDEs) is developed subject to boundary conditions. The PDEs system is converted to non-dimensional ordinal differential equations (ODEs) by using the similarity solution method. Then, the first order of the ODEs system is solved in a MATLAB bvp4c function. We innovatively carry out an optimisation process using response surface methodology (RSM) to enhance the efficiency of the numerical experiment data and fill a gap in the optimised analysis. To observe the variation solutions for the reduced skin friction and heat transfer coefficients, several parameters, such as the mixed convection and suction parameters, are altered into several values. The solutions are essential for accurately forecasting the occurrence of boundary layer separation, particularly in the case of dual solutions. Our analysis reveals that the model generates multiple solutions for the opposing flow, and the boundary layer separates slowly due to the suction effect. To complete the research, RSM reveals that the highest value of the nanoparticle volume fraction and suction parameters, as well as the smallest value of the needle size, generate the maximum transmitting heat through the slender needle. Furthermore, both the numerical and RSM results show that hybrid carbon nanotubes perform better than mono-carbon nanotubes for better practical reference.

The research investigates the boundary layer flow and heat transfer of carbon nanotube (CNT) nanofluid over a stretching/shrinking sheet with the magnetohydrodynamic (MHD) effect. The purpose of constructing this model is to increase the understanding of CNT nanofluid flow and heat transfer characteristics since numerous models use metallic nanoparticles. We conduct this study using numerical and response surface methodology (RSM) approaches in MATLAB and Minitab, respectively. We formulate the mathematical formula by applying the non-linear partial differential equations (PDE). Next, we transform the PDE into non-dimensional ordinary differential equations (ODE) by exploiting the similarity variables method. We show that the model produces multiple solutions in the shrinking region. The magnetic parameter can widen the solutions and delay the boundary layer separation. Both numerical and RSM methods reveal that the maximum value of the magnetic parameter maximizes the heat transfer coefficient. Additionally, both methods demonstrate that single-walled CNT nanofluid is better than multi-walled CNT nanofluid in transmitting heat.

Hybrid nanofluids have demonstrated superior heat transfer performance in numerous applications. However, there remains a need for further research to broaden the scope of their potential applications. The unique behavior of hybrid nanofluids, driven by their potential for improved thermal efficiency, continues to be a focal point of investigation and exploration. This study focuses on the effects of Newtonian heating in MHD hybrid nanofluid near the stagnation point over a nonlinear stretching/shrinking sheet. The Tiwari and Das model, which is a single-phase model, was used to develop the mathematical model. The base fluid and the nanoparticles are assumed to be in thermal equilibrium; hence there is no thermal slip between them. The combination of metal (Cu) and metal oxide (Al2O3) nanoparticles with water (H2O) as the base fluid is used for the analysis. Furthermore, the governing equations are transformed using a similarity transformation technique into similarity equations, which are then solved numerically using a bvp4c function in MATLAB software. Numerical comparison with the published literature is conducted to validate the numerical results, and excellent agreement is found. The impact of physical parameters on the velocity, temperature, skin friction, and local Nusselt number is graphically deliberated. The outcomes suggest that non-unique solutions are found in a specific range of the shrinking parameter. It is also observed that increasing Cu (copper) nanoparticle volume fractions cause an increase in the skin friction coefficient and the local Nusselt number. The presence of magnetic and nonlinear parameters widens the range of solutions to exist while different observation is noticed with an increase in the volume fraction of Cu. Other than that, it has been shown that the Nusselt number increases as the magnetic parameter increases. Lastly, the rise of Newtonian heating contributes to an increase in the temperature profile. This investigation is crucial for understanding the thermal behavior of Cu-Al2O3/ H2O under the influence of physical factors like a magnetic field and Newtonian heating.

This study is to analyze the problem of slip flow via nonlinearly stretching/shrinking sheet in carbon nanotubes (CNTs) with suction/injection. The governing partial differential equations are transformed into nonlinear ordinary differential equations via the transformation of similarity. The bvp4c solver in Matlab is then used to resolve them numerically. Water is used as the base fluid together with single-wall and multi-wall CNTs. Both velocity slip and thermal slip are considered in this study. The flow parameters effect is investigated, shown in the form of a graph, and physically evaluated for the dimensionless velocity, temperature, skin friction, and Nusselt numbers. The results show that there are unique solutions for stretching sheets and non-unique solutions for shrinking sheets. In addition, increasing the velocity slip parameter , suction/injection parameter and nonlinear parameter widens the range of solutions, meanwhile increasing the thermal slip causing the rate of heat transfer to decrease.

The present study aims to offer new numerical solutions and optimisation strategies for the fluid flow and heat transfer behaviour at a stagnation point through a nonlinear sheet that is expanding or contracting in water-based hybrid nanofluids. Most hybrid nanofluids typically use metallic nanoparticles. However, we deliver a new approach by combining single- and multi-walled carbon nanotubes (SWCNTs-MWCNTs). The flow is presumptively steady, laminar, and surrounded by a constant temperature of the ambient and body walls. By using similarity variables, a model of partial differential equations (PDEs) with the magnetohydrodynamics (MHD) effect on the momentum equation is converted into a model of non-dimensional ordinary differential equations (ODEs). Then, the dimensionless first-order ODEs are solved numerically using the MATLAB R2022b bvp4C program. In order to explore the range of computational solutions and physical quantities, several dimensionless variables are manipulated, including the magnetic parameter, the stretching/shrinking parameter, and the volume fraction parameters of hybrid and mono carbon nanotubes. To enhance the originality and effectiveness of this study for practical applications, we optimise the heat transfer coefficient via the response surface methodology (RSM). We apply a face-centred central composite design (CCF) and perform the CCF using Minitab. All of our findings are presented and illustrated in tabular and graphic form. We have made notable contributions in the disciplines of mathematical analysis and fluid dynamics. From our observations, we find that multiple solutions appear when the magnetic parameter is less than 1. We also detect double solutions in the shrinking region. Furthermore, the increase in the magnetic parameter and SWCNTs-MWCNTs volume fraction parameter increases both the skin friction coefficient and the local Nusselt number. To compare the performance of hybrid nanofluids and mono nanofluids, we note that hybrid nanofluids work better than single nanofluids both in skin friction and heat transfer coefficients.

The aim of this study is to investigate and analyze the influence of hydromagnetic forces on the boundary layer flow and heat transfer of carbon nanotubes over a moving surface. Both types of carbon nanotubes such as single -wall carbon nanotubes and multi -wall carbon nanotubes are considered in this research with two types of base fluid, namely water and kerosene. The governing nonlinear partial differential equations are transformed into a nonlinear ordinary differential equations by using similarity transformations then solved numerically by implementing bvp4c package in MATLAB software. The results obtained were represented in table and graphically illustrated with different values of the moving parameter, magnetic parameter, and carbon nanotubes volume fraction parameter. Additionally, the graphs included in this research are skin friction coefficients and local Nusselt number besides the velocity profile and temperature profile. The results proved that duality solutions exist when the plate and the free stream move in opposite directions. Moreover, the availability of magnetic parameter causes the boundary layer thickness to become thinner and increase the heat transfer.

The emergence of hybrid carbon nanotubes presents a promising trend in enhancing heat transfer efficiency when compared to traditional fluids. This study delves into the innovation of employing hybrid carbon nanotubes in the context of a stretch/shrink sheet within the scope of MHD and slip parameter. To address this, the paper formulates partial differential equations governing the flow of a hybrid carbon nanotubes along with the pertinent boundary conditions, which are subsequently modified into ordinary differential equations. By employing similarity solutions and numerical techniques via bvp4c (MATLAB), the investigation examines the significances of MHD, both velocity and thermal slip, nanoparticle volume fraction and viscous dissipation influence on the local Nusselt number Nu(x)root Re-x and skin friction coefficient Cf and Nu(x)root Re(x )in comparison to single-wall and multi-wall structures. Accordingly, the response root Re-x. The findings reveal that the use of hybrid carbon nanotubes amplifies both Cf root Re(x)surface methodology is utilized to establish a correlation with the response of Nun root Re-x and selected governing parameters.

The flow, considering the influence of gravity past a vertical thin needle, is studied analytically and numerically under the suction effect. The thin needle is presumed to be submerged in a combination of single-walled carbon nanotubes (SWCNTs) and multi-walled carbon nanotubes (MWCNTs), and then they are incorporated together with water. The investigation was launched by assembling the thermal boundary layer equation within a system of partial differential equations (PDEs) and bringing together this system with a number of boundary conditions. To reduce the complexity of computation, the PDEs system was conversed to non-dimensional ordinal differential equations (ODEs) by using similarity variables. Then, the ODEs system was successfully solved in a MATLAB build-in function, called bvp4c. Several parameters, such as the suction parameter, the nanoparticles volume fraction parameter, and the assisting/opposing flow parameter, were shown graphically to show how they affected the computational results and physical quantity solutions. Based on our analysis, there existed multiple solutions for the opposing flow, and boundary layer was separated slowly due to the suction effect. Besides, it was revealed that carbon nanotubes (CNTs) hybrid nanofluid contributed a higher rate of heat transfer than CNTs nanofluids.