Two-dimensional magnetohydrodynamic (MHD) stagnation point flow of incompressible Casson fluid over a shrinking sheet is studied. In the present study, homogeneous-heterogeneous reactions, suction and slip effects are considered. Similarity variables are introduced to transform the governing partial differential equations into non-linear ordinary differential equations. The transformed equations and boundary conditions are then solved using the bvp4c solver in MATLAB. The local skin friction coefficient is tabulated for different values of suction and shrinking parameters. The profiles for fluid velocity and concentration for various parameters are illustrated. It was found that two solutions were obtained at certain ranges of parameters. Then, the bvp4c solver was used to perform stability analysis on the dual solutions. Based on the results, the first solution was more stable and physically meaningful than the other solution. The skin friction coefficient increased when suction increased, but decreased when the magnitude of shrinking parameter increased. Meanwhile, the velocity and concentration profile increased in the presence of a magnetic field. It is also noted that the higher the strength of the homogeneous-heterogeneous reactions, the lower the concentration of reactants.

A mathematical model for the unsteady, two-dimensional mixed convection stagnation point flow over a Riga plate is presented in this study. Convective boundary conditions, time-dependent derivatives, mixed convection, radiation effects, and the Grinberg term were all incorporated into the formulation of the governing equations and boundary conditions. By incorporating similarity transformations, ordinary differential (similarity) equations (ODEs) are derived from the partial differential equations (PDEs) of the flow model. The boundary value problem of the fourth-order accuracy code (bvp4c) was implemented in MATLAB (2017b, The MathWorks, Inc., Natick, MA. USA, 2017) to solve the mathematical model numerically. Due to the plate’s shrinking motion, two (dual) solutions are possible (first and second solutions). Based on the stability analysis, it was found that the first solution is stable and physically realizable in practice, while the second solution is not stable and not physically realizable in practice. It was found that the increase in the mixed convection parameter, modified Hartmann number, and unsteadiness parameter improved the hybrid nanofluid’s temperature profile. In addition, increasing the unsteadiness parameter decreased the velocity profile and the skin friction coefficient. Thus, the numerical results suggested that the augmentation of the modified Hartmann number, mixed convection parameter, and unsteadiness parameter can enhance the heat transfer performance in this flow model. This study offers valuable insight into fundamental transport phenomena such as the transmission of momentum, heat, or mass. Hence, it provides valuable information on the gradients of essential factors to control the boundary layer flow pattern.

The purpose of this paper is to study the flow and heat transfer of a hybrid nanofluid, Cu–Al₂O₃/water, past a permeable stretching/shrinking sheet. The effects of Brownian motion and thermophoresis are considered here. Similarity transformations are used to reduce the governing partial differential equations to a system of ordinary (similarity) differential equations. A MATLAB solver called the bvp4c is then used to compute the numerical solutions of equations (12) to (14) subject to the boundary conditions of equation (15). Then, the effects of various physical parameters on the flow and thermal fields of the hybrid nanofluid are analyzed. Multiple (dual) solutions are found for the basic boundary layer equations. A stability analysis is performed to see which solutions are stable and, therefore, applicable in practice and which are not stable. Besides that, a comparison is made between the hybrid nanofluid and a traditional nanofluid, Cu/water. The skin friction coefficient and Nusselt number of the hybrid nanofluid are found to be greater than that of the other nanofluid. Thus, the hybrid nanofluid has a higher heat transfer rate than the other nanofluid. However, the increase in the shrinking parameter reduces the velocity of the hybrid nanofluid. The present results are original and new for the study of the flow and heat transfer past a permeable stretching/shrinking sheet in Cu–Al₂O₃/water hybrid nanofluid.