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The purpose of this paper is to theoretically study the problem of the unsteady boundary layer flow past a permeable curved stretching/shrinking surface in the presence of a uniform magnetic field. The governing nonlinear partial differential equations are converted into ordinary differential equations by similarity transformation, which are then solved numerically.

The transformed system of ordinary differential equations was solved using a fourth-order Runge-Kutta integration scheme. Results for the reduced skin friction coefficient and velocity profiles are presented through graphs and tables for several sets of values of the governing parameters. The effects of these parameters on the flow characteristics are thoroughly examined.

Results show that for the both cases of stretching and shrinking surfaces, multiple solutions exist for a certain range of the curvature, mass suction, unsteadiness, stretching/shrinking parameters and magnetic field parameter.

The paper describes how multiple (dual) solutions for the flow reversals are obtained. It is shown that the solutions exist up to a critical value of the shrinking parameter, beyond which the boundary layer separates from the surface and the solution based upon the boundary layer approximations is not possible.

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.

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.

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.

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.

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.

The purpose of this study is to implement a new class of similarity transformation in analyzing the three-dimensional boundary layer flow of hybrid nanofluid. The Cu-Al₂O₃/water hybrid nanofluid is formulated using the single-phase nanofluid model with modified thermophysical properties.

The governing partial differential equations are reduced to the ordinary (similarity) differential equations using the proposed similarity transformation. The resulting equations are programmed in Matlab software through the bvp4c solver to obtain their solutions. The features of the reduced skin frictions and the velocity profiles for different values of the physical parameters are analyzed and discussed.

The non-uniqueness of the solutions is observed for certain physical parameters. The dual solutions are perceived for both permeable and impermeable cases and being the main agenda of the work. The execution of stability analysis proves that the first solution is undoubtedly stable than the second solution. An increase in the mass transpiration parameter leads to the uniqueness of the solution. Oppositely, as the injection parameter increase, the two solutions remain. However, no separation point is detected in this problem within the considered parameter values. The present results are decisive to the pair of alumina and copper only.

The present findings are original and can benefit other researchers particularly in the field of fluid dynamics. This study can provide a different insight of the transformation that is applicable to reduce the complexity of the boundary layer equations.

This study aims to analyze the unsteady flow of hybrid Cu-Al₂O₃/water nanofluid over a permeable stretching/shrinking disc. The analysis of flow stability is also purposed because of the non-uniqueness of solutions.

The reduced differential equations (similarity) are solved numerically using the aid of bvp4c solver (Matlab). Two types of thermophysical correlations for hybrid nanofluid (Type 1 and 2) are adopted for the comparison results. Using correlation Type 1, the heat transfer and flow analysis including the profiles (velocity and temperature) are presented in the figures and tables with different values control parameters. Three sets of hybrid nanofluid are analyzed: Set 1 (1% Al₂O₃ + 1% Cu), Set 2 (0.5% Al₂O₃ + 1% Cu) and Set 3 (1% Al₂O₃ + 0.5% Cu).

The comparison of numerical values between present (Types 1 and 2 correlations) and previous (Type 2 correlations) results are in a good compliance with approximate percent relative error. The appearance of two solutions is noticed when the suction parameter is considered and the unsteady parameter is less than 0 (decelerating flow) for both stretching and shrinking disc while only one solution is possible for steady flow. The hybrid nanofluid in Set 1 can delay the separation of boundary layer but the hybrid nanofluid in Set 3 has the greatest heat transfer rate. Moreover, the inclusion of wall mass suction for stretching case can generate a significant increment of heat transfer rate approximately 90% for all fluids (water, single and hybrid nanofluids).

The present findings are novel and can be a reference point to other researchers to further analyze the heat transfer performance and stability of the working fluids.

This study aims to investigate the unsteady two-dimensional viscous flow and heat transfer over an unsteady permeable stretching/shrinking sheet (surface) with generalized slip velocity condition.

Similarity transformation is used to reduce the system of partial differential equations into a system of nonlinear ordinary differential equations. The resulting equations are then solved numerically using “bvp4c” function in MATLAB software.

Dual solutions are found for a certain range of the unsteady, suction and stretching/shrinking parameters. Stability analysis is performed, and it is revealed that the first (upper branch) solution is stable and physically realizable, whereas the second (lower branch) solution is unstable.

The results obtained can be used to explain the characteristics and applications of the generalized slip in boundary layer flow. Such condition is applied for particulate fluids such as foams, emulsions, polymer solutions and suspensions. Furthermore, the phenomenon of stretching/shrinking sheet can be found on the manufacturing of polymer sheets, rising and shrinking balloon or moving and shrinking polymer film.

The present numerical results are original and new for the study of unsteady flow and heat transfer over a permeable stretching/shrinking sheet with generalized slip velocity.

This paper aims to accentuate the behavior of second-grade hybrid Al₂O₃–Cu nanofluid flow and its thermal characteristics driven by a stretching/shrinking Riga plate.

The second-grade fluid is considered with the combination of Cu and Al₂O₃ nanoparticles. Three base fluids namely water, ethylene glycol (EG) and methanol with different Prandtl number are also examined. The formulation of the mathematical model of second-grade hybrid nanofluid complies with the boundary layer approximations. The complexity of the governing model is reduced into a simpler differential equations using the similarity transformation. The bvp4c solver is fully used to solve the reduced equations. The observation of multiple solutions is conducted for the assisting (stretching) and opposing (shrinking) cases.

The impact of suction parameter, second-grade parameter, electromagnetohydrodynamics (EMHD) parameter, velocity ratio parameter and the volumetric concentration of the alumina and copper nanoparticles are numerically analyzed on the velocity and temperature profiles, skin friction coefficient and local Nusselt number (thermal rate) of the second-grade Al₂O₃–Cu/water. The solution is unique when (static and stretching cases) while dual for a specific range of negative in the presence of suction effect. Based on the appearance of the first solution in all cases of, it is physically showed that the first solution is stable. Further examination reveals that the EMHD and suction parameters are the contributing factors for the thermal enhancement of this non-Newtonian working fluid. Meanwhile, the viscosity of the non-Newtonian fluid also plays a significant role in the fluid motion and heat transfer rate based on the finding that the EG base fluid produces the maximum heat transfer rate but the lowest critical value and skin friction coefficient.

The results are novel and contribute to the discovery of the hybrid nanoparticles’ performance in the non-Newtonian second-grade fluid. Besides, this study is beneficial to the researchers in this field and general audience from industries regarding the factors, which contributing to the thermal enhancement of the working fluid.

The purpose of this paper is to present the results of an analysis performed to study unsteady mixed convection at the stagnation point flow over a plate moving along the direction of flow impingement. The similarity transformations are used to transform the governing nonlinear partial differential equation to a system of an ordinary differential equation.

The transformed equations are then solved numerically by a shooting technique together with bvp4c function.

The numerical results are compared with the corresponding results from previous researchers. The effects of the unsteadiness Parameter A, Prandtl number Pr, mixed convection parameter λ for plane (m = 0) and axisymmetric (m = 1) flow on the shear stress or the skin friction and heat transfer coefficients, as well as the velocity and temperature profiles, are presented and discussed.

Dual solutions for the opposing flow and multiple solutions for the assisting flow are found.

This paper aims to explore dual solutions for the flow of a hybrid nanofluid over a permeable melting stretching/shrinking sheet with nanoparticle shape factor, second-order velocity slip conditions and viscous dissipation. The hybrid nanofluid is formulated by dispersing alumina (Al₂O₃) and copper (Cu) nanoparticles into water (H₂O).

The governing partial differential equations (PDEs) are first reduced to a system of ordinary differential equations (ODEs) using a mathematical method of similarity transformation technique. These ODEs are then numerically solved through MATLAB’s bvp4c solver.

Key parameters such as slip parameter, melting parameter, suction parameter, shrinking parameter and Eckert number are examined. The results reveal the existence of two distinct solutions (upper and lower branches) for the transformed ODEs when considering the shrinking parameter. Increasing value of Cu-volume fraction and the second-order velocity slip enhances boundary layer thicknesses, whereas the heat transfer rate diminishes with rising melting and suction parameters. These numerical results are illustrated through various figures and tables. Additionally, a stability analysis is performed and confirms the upper branch is stable and practical, while the lower branch is unstable.

The analysis of hybrid nanofluid flow over a shrinking surface has practical significance with applications in processes such as solar thermal management systems, automotive cooling systems, sedimentation, microelectronic cooling or centrifugal separation of particles. Both steady and unsteady hybrid nanofluid flows are relevant in these contexts.

While the study of hybrid nanofluid flow is well-documented, research focusing on the shrinking flow case with specific parameters in our study is still relatively scarce. This paper contributes to obtaining dual solutions specifically for the shrinking case, which has been less frequently addressed.

The purpose of this paper is to numerically investigate the hybrid nanofluid flow with the imposition of magnetohydrodynamic (MHD) and radiation effects alongside the convective boundary conditions over a permeable stretching/shrinking surface.

The mathematical model is formulated in the form of partial differential equations (PDEs) and are then transformed into the form of ordinary differential equations (ODEs) by using the similarity variables. The deriving ODEs are solved numerically by using the bvp4c solver in MATLAB software. Stability analysis also has been performed to determine the stable solution among the dual solutions obtain. For method validation purposes, a comparison of numerical results has been made with the previous studies.

The flow and the heat transfer of the fluid at the boundary layer are described through the plot of the velocity profile, temperature profile, skin friction coefficient and local Nusselt number that are presented graphically. Dual solutions are obtained, but only the first solution is stable. For the realizable solution at the shrinking surface, the proliferation of nanoparticle volume fraction (copper) and magnetic (magnetohydrodynamics) parameters can impede the boundary layer separation. Also, Biot number could enhance the temperature profile and the heat transfer rate at the shrinking surface region. The incrementation of 0.1% of Biot number has enhanced the heat transfer rate by approximately 0.1% and the incrementation of 0.5% volume fraction for copper has reduced the heat transfer rate by approximately 0.17%.

The presented model and numerical results are original and new. It can be used as a future reference for further investigation and related practical application. The main contribution of this investigation includes giving the initial prediction and providing the numerical data for the other researchers for their future reference regarding the impacts of nanoparticles volumetric concentration towards the main physical quantities of interest in the presence of magnetic and radiation parameters with the convective boundary conditions.