The hybrid nanofluid under the influence of magnetohydrodynamics (MHD) is a new interest in the industrial sector due to its applications, such as in solar water heating and scraped surface heat exchangers. Thus, the present study accentuates the analysis of an unsteady three-dimensional MHD non-axisymmetric Homann stagnation point flow of a hybrid Al₂O₃-Cu/H₂O nanofluid with stability analysis. By employing suitable similarity transformations, the governing mathematical model in the form of the partial differential equations are simplified into a system of ordinary differential equations. The simplified mathematical model is then solved numerically by the Matlab solver bvp4c function. This solving approach was proficient in generating more than one solution when good initial guesses were provided. The numerical results presented significant influences on the rate of heat transfer and fluid flow characteristics of a hybrid nanofluid. The rate of heat transfer and the trend of the skin friction coefficient improve with the increment of the nanoparticles’ concentration and the magnetic parameter; however, they deteriorate when the unsteadiness parameter increases. In contrast, the ratio of the escalation of the ambient fluid strain rate to the plate was able to adjourn the boundary layer separation. The dual solutions (first and second solutions) are obtainable when the surface of the sheet shrunk. A stability analysis is carried out to justify the stability of the dual solutions, and hence the first solution is seen as physically reliable and stable, while the second solution is unstable.

Unsteady stagnation point flow in hybrid nanofluid (Al₂O₃-Cu/H₂O) past a convectively heated stretching/shrinking sheet is examined. Apart from the conventional surface of the no-slip condition, the velocity slip condition is considered in this study. By incorporating verified similarity transformations, the differential equations together with their partial derivatives are changed into ordinary differential equations. Throughout the MATLAB operating system, the simplified mathematical model is clarified by employing the bvp4c procedure. The above-proposed approach is capable of producing non-uniqueness solutions when adequate initial assumptions are provided. The findings revealed that the skin friction coefficient intensifies in conjunction with the local Nusselt number by adding up the nanoparticles volume fraction. The occurrence of velocity slip at the boundary reduces the coefficient of skin friction; however, an upward trend is exemplified in the rate of heat transfer. The results also signified that, unlike the parameter of velocity slip, the increment in the unsteady parameter conclusively increases the coefficient of skin friction, and an upsurge attribution in the heat transfer rate is observed resulting from the increment of Biot number. The findings are evidenced to have dual solutions, which inevitably contribute to stability analysis, hence validating the feasibility of the first solution.

The flow of fluids over the boundaries of a rotating disc has many practical uses, including boundary-layer control and separation. Therefore, the aim of this study is to discuss the impact of unsteady magnetohydrodynamics (MHD) hybrid ferrofluid flow over a stretching/shrinking rotating disk. The time-dependent mathematical model is transformed into a set of ordinary differential equations (ODE’s) by using similarity variables. The bvp4c method in the MATLAB platform is utilised in order to solve the present model. Since the occurrence of more than one solution is presentable, an analysis of solution stabilities is conducted. Both solutions were surprisingly found to be stable. Meanwhile, the skin friction coefficient, heat transfer rate—in cooperation with velocity—and temperature profile distributions are examined for the progressing parameters. The findings reveal that the unsteadiness parameter causes the boundary layer thickness of the velocity and temperature distribution profile to decrease. A higher value of magnetic and mass flux parameter lowers the skin friction coefficient. In contrast, the addition of the unsteadiness parameter yields a supportive effect on the heat transfer rate. An increment of the magnetic parameter up to 30% reduces the skin friction coefficient by 15.98% and enhances the heat transfer rate approximately up to 1.88%, significantly. In contrast, the heat transfer is rapidly enhanced by improving the mass flux parameter by almost 20%.

This numerical study was devoted to examining the occurrence of non-unique solutions in boundary layer flow due to deformable surfaces (cylinder and flat plate) with the imposition of prescribed surface heat flux. The hybrid Al${}_2$O${}_3$-Cu/water nanofluid was formulated using the single phase model with respective correlations of hybrid nanofluids. The governing model was simplified by adopting a similarity transformation. The transformed differential equations were then numerically computed using the efficient bvp4c solver with the ranges of the control parameters $0.5\%\le {\varphi }_1,{\varphi }_2\le 1.5\%$ (Al${}_2$O${}_3$ and Cu volumetric concentration), $0\le K\le 0.2$ (curvature parameter), $2.6<S\le 3.2$ (suction parameter) and $-2.5<\lambda \le 0.5$ (stretching/shrinking parameter). Dual steady solutions are presentable for both a cylinder $(K>0)$ and a flat plate $(K=0)$ with the inclusion of only the suction (transpiration) parameter. The real and stable solutions were mathematically validated through the stability analysis. The Al${}_2$O${}_3$-Cu/water nanofluid with ${\varphi }_1=0.5\%$ (alumina) and ${\varphi }_2=1.5\%$ (copper) has the highest skin friction coefficient and heat transfer rate, followed by the hybrid nanofluids with volumetric concentrations $({\varphi }_1=1\%,{\varphi }_2=1\%)$ and $({\varphi }_1=1.5\%,{\varphi }_2=0.5\%)$, respectively. Surprisingly, the flat plate surface abates the separation of boundary layer while it enhances the heat transfer process.

The behaviour of magnetic impact on the unsteady separated stagnation-point flow of hybrid nanofluid with the influence of viscous dissipation and Joule heating is investigated numerically in this study. A new mathematical hybrid nanofluid model is developed, and similarity solutions are obtained in the form of ordinary differential equations (ODEs). The bvp4c approach in MATLAB is used to determine the reduced ODEs’ estimated solutions. The influence of various physical parameters is scrutinised. The findings revealed that the skin friction coefficient increases with the increment of the nanoparticle volume fraction and the unsteadiness parameter. This observation is also applied to the heat transfer rate of the fluid. Additionally, the presence of the magnetic and acceleration parameter provides a significant result in the heat transfer performance. The addition of the Eckert number increased the temperature profile distribution, thereby spontaneously decreasing the heat transfer rate. The first solution is declared stable by the analysis of solution stability.

Previous research has recognised the study of stagnation point flow by focusing Maxwell nanofluid on a stretching sheet surface. Motivated by this research idea, our main objective is to formulate and analyse a new mathematical model of stagnation point flow in Maxwell fluid that highlights the dual types of fluid known as hybrid nanofluids. The effects of thermal radiation and heat transfer are also considered. The partial differential equations (PDEs) are converted into ordinary differential equations (ODEs) via similarity variables that generate similarity solutions. Following that, the bvp4c approach is employed to discover the approximate solutions of the reduced ODEs. The significance of various parameters is graphically presented and considers the physical quantities of interest. A remarkable observation found in this study is the enhancement of the heat transfer rate in Maxwell hybrid nanofluids, which is steadily amplified in contrast to traditional fluids. Indeed, the Maxwell parameter in hybrid nanofluids embarks on a substantial increment of the heat transfer rate. The current study succeeds in establishing more than one solution along the stretching/shrinking sheet. Thus, the stability analysis is conducted to confirm the sustainability of the solutions.