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This paper aims to probe the problem of an unsteady mixed convection stagnation point flow and heat transfer past a stationary surface in an incompressible viscous fluid numerically.

The governing nonlinear partial differential equations are transformed into a system of ordinary differential equations by a similarity transformation, which is then solved numerically by a Runge – Kutta – Fehlberg method with shooting technique and a collocation method, namely, the bvp4c function.

The effects of the governing parameters on the fluid flow and heat transfer characteristics are illustrated in tables and figures. It is found that dual (upper and lower branch) solutions exist for both the cases of assisting and opposing flow situations. A stability analysis has also been conducted to determine the physical meaning and stability of the dual solutions.

This theoretical study is significantly relevant to the applications of the heat exchangers placed in a low-velocity environment and electronic devices cooled by fans.

The case of suction on unsteady mixed convection flow at a three-dimensional stagnation point has not been studied before; hence, all generated numerical results are claimed to be novel.

This paper aims to investigate the flow and heat transfer characteristics of a hybrid nanofluid (Cu-Al₂O₃/water) in the presence of magnetohydrodynamics and thermal radiation over a permeable moving surface.

By choosing appropriate similarity variables, the partial differential equations are transformed into a system of linear equations which are solved by using the boundary value problem solver (bvp4c) in MATLAB. The implementation of stability analysis verifies the achievable result of the first solution which is considered stable while the second solution is unstable.

The findings revealed that the presence of a magnetic field and suction slows down the fluid motion because of the synchronism of the magnetic and electric field occurred from the formation of the Lorentz force. Also, the enhancement of the thermal radiation parameter escalates the heat transfer rate of the current study.

The present study is addressing the problem of MHD flow and heat transfer analysis of a hybrid nanofluid towards a permeable moving surface, with the consideration of the thermal radiation effect. The authors show that in both cases of assisting and opposing flow, there exist dual solutions within a specific range of the moving parameters. A stability analysis approved that only one of the solutions are physically relevant.

The purpose of this paper is to study the effects of vertical throughflow on the boundary layer flow and heat transfer of a nanofluid driven by a permeable stretching/shrinking surface.

Similarity transformation is used to convert the system of boundary layer equations into a system of ordinary differential equations. The system of governing similarity equations is then reduced to a system of first-order differential equations and solved numerically using the bvp4c function in Matlab software. The generated numerical results are presented graphically and discussed based on some governing parameters.

It is found that dual solutions exist in both cases of stretching and shrinking sheet situations. Stability analysis is performed to determine which solution is stable and valid physically.

Dual solutions are found for positive and negative values of the moving parameter. A stability analysis has also been performed to show that the first (upper branch) solutions are stable and physically realizable, while the second (lower branch) solutions are not stable and, therefore, not physically possible.

According to the previous research, bioconvection has been recognized as an important mechanism in current engineering and environmental systems. For example, researchers exploit this mechanism in modern green bioengineering to develop environmentally friendly fuels, fuel cells and photosynthetic microorganisms. This study aims to analyse how this type of convection affects the flow behaviour and heat transfer performance of mixed convection stagnation point flow in alumina-copper/water hybrid nanofluid. Also, the impact of a modified magnetic field on the boundary layer flow is considered.

By applying appropriate transformations, the multivariable differential equations are transformed into a specific sort of ordinary differential equations. Using the bvp4c procedure, the adjusted mathematical model is revealed. Once sufficient assumptions are provided, multiple solutions are able to be produced.

The skin friction coefficient is declined when the nanoparticle concentration is increased in the opposing flow. In contrast, the inclusion of aligned angles displays an upward trend in heat transfer performance. The presence of several solutions is established, which simply leads to a stability analysis, hence verifies the viability of the initial solution.

The current findings are unique and novel for the investigation of mixed bioconvection flow towards a vertical flat plate in a base fluid with the presence of hybrid nanoparticles.

The analysis of boundary layers is needed to reflect the behaviour of fluid flows in current industrial processes and to improve the efficacy of products. Hence, this study aims to analyse the flow and heat transfer performance of hybrid alumina-copper/water (Al₂O₃-Cu/H₂O) nanofluid with the inclusion of activation energy and binary chemical reaction effect towards a moving wedge.

The multivariable differential equations with partial derivatives are converted into a specific type of ordinary differential equations by using valid similarity transformations. The reduced mathematical model is elucidated in the MATLAB system by using the bvp4c procedure. This solution method is competent in delivering multiple solutions once appropriate assumptions are supplied.

The results of multiple control parameters have been studied, and the findings are verified to provide more than one solution. The coefficient of skin friction was discovered to be increased by adding nanoparticles volume fraction from 0% to 0.5% and 1%, by almost 1.6% and 3.2%. Besides, increasing the nanoparticles volume fraction improves heat transfer efficiency gradually. The inclusion of the activation energy factor displays a downward trend in the mass transfer rates, consequently reducing the concentration profile. In contrast, the increment of the binary reaction rate greatly facilitates the augmentation of mass transfer rates. There is a significant enhancement in the heat transfer rate, approximately 13.2%, when the suction effect dominates about 10% in the boundary layer flow. Additionally, the results revealed that as the activation energy rises, the temperature and concentration profiles rise as well. It is proved that the activation energy parameter boosts the concentration of chemical species in the boundary layer. A similar pattern emerges as the wedge angle parameter increases. The current effort aims to improve the thermal analysis process, particularly in real-world applications such as geothermal reservoirs, chemical engineering and food processing, which often encountered mass transfer phenomenon followed by chemical reactions with activation energy.

The present results are original and new for the study of flow and heat transfer over a permeable moving wedge in a hybrid nanofluid with activation energy and binary chemical reaction.

The purpose of this paper is to present both an analytical and a numerical analysis of the unsteady magnetohydrodynamic (MHD) rear stagnation-point flow over off-centred deformable surfaces.

The numerical MATLAB solver bvp4c suitable for routine boundary value problem is used for the set of ordinary differential equations reduced from the governing partial differential equations.

Multiple solutions are found for particular eigenvalues. The physical solution is computed by the help of a linear stability analysis. The authors have succeeded in discovering the second solutions, and it is suggested that these solutions are unstable and not physically realisable in practice. The current findings add to a growing body of literature on MHD stagnation-point flow problems. It is also found that the governing parameters have different effects on the flow characteristics.

Even though problems of steady MHD flows have been extensively studied for stagnation-point flows, limited findings can be found on the unsteady MHD rear stagnation-point flow over off-centred deformable surfaces.

The originality of this work is the application of a magnetic field on a time-dependent MHD rear stagnation-point flow over off-centred deformable surfaces.

This study aims to perform a stability analysis on a steady magnetohydrodynamic (MHD) mixed convection boundary-layer stagnation-point flow of an incompressible, viscous and electrically conducting fluid over a vertical flat plate. The effect of induced magnetic field is also considered.

The governing boundary layer equations are transformed into a system of ordinary differential equations using the similarity transformations. The system is then solved numerically using the “bvp4c” function in MATLAB.

Dual solutions are found to exist for a certain range of the buoyancy parameter for both the assisting and opposing flows. The results from the stability analysis showed that the first solution (upper branch) is stable and valid physically, while the second solution (lower branch) is unstable.

This problem is important in many metallurgical processes, namely, drawing, annealing and tinning of copper wires. The results obtained are very useful for researchers to determine which solution is physically stable, whereby mathematically more than one solution exists for the skin friction coefficient and the heat transfer characteristics.

The present results of the stability analysis are original and new for the problem of MHD mixed convection stagnation-point flow of viscous conducting fluid over a vertical flat plate, with the effect of induced magnetic field.

The investigation of fluid flow and heat transfer is incredibly significant in the present era, particularly in the engineering and manufacturing industries. Hence, this study aims to concern with analysing the unsteady stagnation point flow towards a permeable stretching/shrinking Riga plate of Al₂O₃-Cu/H₂O. The effect of thermal radiation on the boundary layer flow is also taken into account.

The multi-variable differential equations with partial derivatives are transformed into third-order and second-order differential equations by applying appropriate transformations. The reduced mathematical model is solved in the MATLAB system by using the bvp4c procedure. This solution approach is capable of producing multiple solutions once the necessary assumptions are provided.

The results of various control parameters were analysed, and it has been observed that raising the solution viscosity from 0% to 0.5% and 1% improves the coefficient of skin friction and thermal conductivity by almost 1.0% and 1.9%. Similar response and observation can be witnessed in the addition of modified Hartmann number where the highest values dominate about 10.7% improvement. There is a substantial enhancement in the heat transfer rate, approximately 1.8% when the unsteadiness parameter leads around 30% in the boundary layer flow. In contrast, the increment in thermal radiation promotes heat transfer deterioration. Further, more than one solution is proven, which invariably leads to a stability analysis, which validates the first solution’s feasibility.

The present results are new and original for the study of flow and heat transfer on unsteady stagnation point flow past a permeable stretching/shrinking Riga plate in Al₂O₃-Cu/H₂O hybrid nanofluid with thermal radiation.