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.

Purpose The purpose of this paper is to numerically analyze the stagnation point flow of Cu-Al2O3/water hybrid nanofluid with mixed convection past a flat plate and circular cylinder. Design/methodology/approach The similarity equations that reduced from the boundary layer and energy equations are solved using the bvp4c solver. The duality of solutions is observed within the specific range of the control parameters, namely, mixed convection parameter λ, curvature parameter γ and nanoparticles volumetric concentration ϕ1 for alumina, while for copper ϕ2. The stability analysis is also designed to justify the particular solutions’ stability. Additionally, the idea to obtain the solution for large value of λ and γ is also presented in this paper. Findings Two solutions exist in opposing and assisting flows up to a critical value λc where λc lies in the opposing region. An upsurge of the curvature parameter tends to extend the critical value (delay the separation process), whilst increase the heat transfer performance of the working fluid. Meanwhile, the application of hybrid Cu-Al2O3/water nanofluid also can decelerate the separation of laminar boundary layer flow and produce higher heat transfer rate than the Cu–water nanofluid and pure water. Originality/value The results are new and original. This study benefits to the other researchers, specifically in the observation of the fluid flow characteristics and heat transfer rate of the hybrid nanofluid. Also, this paper features with the mathematical formulation for the solution with large values of λ and γ.

Purpose 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. Design/methodology/approach 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. Findings 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. Originality/value 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.

The impact of nanoparticle shapes (i.e., blade, brick, cylindrical, platelet, and spherical) on the MHD flow of Ag-MgO/water hybrid nanofluid over a stretching/shrinking sheet is scrutinized in this study. Chemical reaction and activation energy are included in the governing partial differential equations of the flow problem. On the boundary, velocity slip and zero mass flux conditions are considered. The simplification of the governing equations and boundary conditions into non-linear ordinary differential equations is done through similarity transformation. Then, the bvp4c solver in Matlab is deployed for computation, with the results generated in the form of numerical solutions and graphs. It is found that the usage of spherical-shaped nanoparticles produces the lowest magnitude of skin friction coefficient, and the implementation of blade-shaped nanoparticles in the hybrid nanofluid provides the highest enhancement of heat transfer rate. The increment in activation energy slows down the chemical reaction that raises the concentration profile of the hybrid nanofluid. However, the concentration profile decreases as the reaction rate increases.

The flow of electrically conducting micropolar fluid past an exponentially permeable shrinking sheet in the presence of a magnetic field and thermal radiation is studied. Similarity transformations are applied to the governing partial differential equations to form ordinary differential equations. The solution for the resultant equations, subject to boundary conditions, is then computed numerically using the bvp4c solver in MATLAB. The effects of several parameters on the local skin friction coefficient, couple stress, Nusselt number, velocity, microrotation and temperature of the fluid are analysed. Because the numerical computations for this problem result in triple solutions, stability analysis is carried out to ascertain the stability and significance of these solutions. The first solution is revealed to be stable, hence more physically meaningful than the other solutions. Meanwhile, it is found that the increase in magnetic and thermal radiation parameters reduces the fluid temperature.

Purpose This paper aims to study the stagnation point flow of Al2O3–Cu/H2O hybrid nanofluid over a radially shrinking disk with the imposition of the magnetic field, viscous-Ohmic dissipation and convective boundary condition. Design/methodology/approach Similarity variables are introduced and used in reducing the governing partial differential equations into a system of ordinary differential equations. A built-in bvp4c solver in MATLAB is then used in the computation of the numerical solutions for equations (7) and (8) subject to the boundary conditions (9). Then, the behavior of the flow and thermal fields of the hybrid nanofluid, with various values of controlling parameters, are analyzed. Findings The steady flow problem resulted in multiple (dual) solutions. A stability analysis performed to identify the stable solution applicable in practice revealed that the first solution is stable while the second solution is unstable. The skin friction coefficient and Nusselt number of the hybrid nanofluid are found to be greater than the Al2O3–H2O nanofluid. Thus, the hybrid nanofluid has a better heat transfer performance than the nanofluid. Besides that, the presence of the magnetic field, suction, convective boundary condition and the enhancement of nanoparticle volume fraction of Cu augments the skin friction coefficient and Nusselt number of the hybrid nanofluid. Meanwhile, the presence of viscous-Ohmic dissipation reduces the heat transfer performance of the fluid. Originality/value To the best of the authors’ knowledge, the present results are original and new for the study of the flow and heat transfer of Al2O3–Cu/H2O hybrid nanofluid past a permeable radially shrinking disk. Considerable efforts have been directed toward the study of the boundary layer flow and heat transfer over stretching/shrinking surfaces and disks because of its numerous industrial applications, such as electronic, power, manufacturing, aerospace and transportation industries. Common heat transfer fluids such as water, alumina, cuprum and engine oil have limited heat transfer capabilities due to their low heat transfer properties. In contrast, metals have higher thermal conductivities than these fluids. Therefore, it is desirable to combine the two substances to produce a heat transfer medium that behaves like a fluid but has higher heat transfer properties.

Dual solutions are discovered in the problem of magnetohydrodynamics (MHD) boundary layer flow of Carreau fluid over a permeable shrinking sheet with thermal radiation. Therefore, a stability analysis is carried out to identify the stable solution of this problem. For the stability analysis, the problem is considered to be unsteady with time derivative introduced into the governing equations. Next, time-dependent solutions are substituted into these equations to form linear eigenvalue equations. The smallest eigenvalue of these equations is then computed using the bvp4c solver in MATLAB. The results showed that the first solution is stable, while the second solution is unstable. The first solution is physically meaningful and realizable in practice, and thus significant to the problem. ABSTRAK Penyelesaian dual telah diperoleh dalam masalah aliran lapisan sempadan magnetohidrodinamik (MHD) bendalirCarreau terhadap permukaan telap mengecut dengan radiasi terma. Sehubungan dengan itu, analisis kestabilan dilakukan bagi mengenal pasti penyelesaian yang stabil dalam masalah ini. Bagi analisis kestabilan tersebut, masalah ini telah dipertimbangkan sebagai masalah yang tidak stabil dengan memperkenalkan terbitan masa ke dalam persamaan menakluk. Kemudian, penyelesaian bersandar masa digantikan ke dalam persamaan tersebut untuk menghasilkan persamaan nilai eigen linear. Nilai eigen terkecil bagi persamaan ini kemudiannya dihitung menggunakan solver bvp4c di MATLAB. Keputusannya menunjukkan bahawa penyelesaian pertama adalah stabil sementara penyelesaian kedua tidak stabil. Penyelesaian pertama adalah bermakna secara fizikal dan boleh direalisasikan dalam amalan sebenar. Oleh itu, penyelesaian pertama adalah penting bagi masalah ini.

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., 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.