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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.

The main purpose of this numerical study is to provide a solution for natural convection in a partially heated, wavy cavity filled with a nanofluid using Buongiorno’s nanofluid model.

The domain of interest is a two-dimensional cavity bounded by an isothermal left wavy wall, adiabatic horizontal flat walls and right flat wall with a partial isothermal zone. To study the behaviour of the nanofluid, a two-phase Buongiorno mathematical model with the effects of the Brownian motion and thermophoresis is used. The governing dimensionless partial differential equations with corresponding boundary conditions were numerically solved by the finite difference method of the second-order accuracy using the algebraic transformation of the physical wavy cavity in a computational rectangular domain. The study has been conducted using the following values of the governing parameters: Ra = 10⁴-10⁶, Le = 10, Pr = 6.26, Nr = 0.1, Nb = 0.1, Nt = 0.1, A = 1, κ = 1-3, b = 0.2, h_hs/L = 0.25, h₁/L = 0.0-0.75 and τ = 0-0.25.

It is found that an increase in the undulation number leads to a weak intensification of convective flow and a reduction of Nū because of more essential cooling of the wavy troughs where the temperature gradient decreases. Variations of the heater location show a modification of the fluid flow and heat transfer. The upper position of the heater reflects the minimum heat transfer rate, while the position between the bottom part and the middle section (h₁/L = 0.25) characterizes an enhancement of heat transfer.

The originality of this work is to analyse the natural convection in a partially heated wavy cavity filled by a nanofluid using Buongiorno’s nanofluid model. The results will benefit scientists and engineers to become familiar with the flow behaviour of such nanofluids, and the way to predict the properties of this flow for possibility of using nanofluids in advanced nuclear systems, in industrial sectors including transportation, power generation, chemical sectors, ventilation, air-conditioning, etc.

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.

The purpose of this paper is to reinvestigate the problem of multiple similarity solutions of the two-dimensional magnetohydrodynamic boundary-layer flow of an incompressible, viscous and electrically conducting fluid past a stretching/shrinking permeable surface studied by Aly et al. (2007).

The transformed ordinary (similarity) differential equation was solved numerically using the function bvp4c from MATLAB. The relative tolerance was set to 10^(−10).

Dual solutions were found and a stability analysis was performed to show which solutions are stable and which are not stable. On the other hand, Aly et al. (2007) have shown that for each value of the power index and magnetic parameter in the range and for any specific values of the stretching/shrinking parameter and suction parameter the problem has only a solution.

The paper describes how multiple (dual) solutions for the flow reversals were obtained. The stability analysis has shown that the lower solution branches are unstable, while the upper solution branches are stable.

This paper aims to consider the influence of the temperature and of an external magnetic field on the steady oblique stagnation-point flow for a Boussinesquian nanofluid past a stretching or shrinking sheet.

The flow is reduced through similarity transformations to an ordinary boundary value problem, which is solved numerically in MATLAB using the bvp4c function. The behavior of the solution is discussed physically, and some analytical considerations concerning existence of the solution and the occurrence of dual solutions are drawn.

The study of the influence of an external magnetic field on the oblique stagnation-point flow of a Buongiorno's Boussinesquian nanofluid is carried out. The fluid clashes on a vertical stretching or shrinking sheet. Dual solutions appear for suitable values of the parameters.

The present results are new and original.

This study aims to study the mixed convection flow and heat transfer of Al₂O₃-Cu/water hybrid nanofluid over a vertical plate. Governing equations for conservation of mass, momentum and energy for the hybrid nanofluid over a vertical flat plate are introduced.

The similarity transformation approach is used to transform the set of partial differential equations into a set of non-dimensional ordinary differential equations. Finite-deference with collocation method is used to integrate the governing equations for the velocity and temperature profiles.

The results show that dual solutions exist for the case of opposing flow over the plate. Linear stability analysis was performed to identify a stable solution. The stability analysis shows that the lower branch of the solution is always unstable, while the upper branch of the solution is always stable. The results of boundary layer analysis are reported for the various volume fractions of composite nanoparticles and mixed convection parameter. The outcomes show that the composition of nanoparticles can notably influence the boundary layer flow and heat transfer profiles. It is also found that the trend of the variation of surface skin friction and heat transfer for each of the dual solution branches can be different. The critical values of the mixed convection parameter, λ, where the dual solution branches joint together, are also under the influence of the composition of hybrid nanoparticles. For instance, assuming a total volume fraction of 5 per cent for the mixture of Al₂O₃ and Cu nanoparticles, the critical value of mixing parameter of λ changes from −3.1940 to −3.2561 by changing the composition of nanofluids from Al₂O₃ (5 per cent) + Cu (0%) to Al₂O₃ (2.5%) + Cu (2.5 per cent).

The mixed convection stability analysis and heat transfer study of hybrid nanofluids for a stagnation-point boundary layer flow are addressed for the first time. The introduced hybrid nanofluid model and similarity solution are new and of interest in both mathematical and physical points of view.

The purpose of this paper is to theoretically investigate the unsteady separated stagnation-point flow and heat transfer past an impermeable stretching/shrinking sheet in a copper (Cu)-water nanofluid using the mathematical nanofluid model proposed by Tiwari and Das.

A similarity transformation is used to reduce the governing partial differential equations to a set of nonlinear ordinary (similarity) differential equations which are then solved numerically using the function bvp4c from Matlab for different values of the governing parameters.

It is found that the solution is unique for stretching case; however, multiple (dual) solutions exist for the shrinking case.

The authors believe that all numerical results are new and original, and have not been published elsewhere.

The purpose of this paper is to examine the axisymmetric flow and heat transfer of a hybrid nanofluid over a permeable biaxial stretching/shrinking sheet.

In this study, 0.1 solid volume fraction of alumina (Al₂O₃) is fixed, then consequently, various solid volume fractions of copper (Cu) are added into the mixture with water as the base fluid to form Cu-Al₂O₃/water hybrid nanofluid. The hybrid nanofluid equations are converted to the similarity equations by using the similarity transformation. The bvp4c solver, which is available in the Matlab software is used for solving the similarity equations numerically. The numerical results for selected values of the parameters are presented in tabular and graphical forms, and are discussed in detail.

It is found that dual solutions exist up to a certain value of the stretching/shrinking and suction parameters. The critical value λ_c < 0 for the existence of the dual solutions decreases as nanoparticle volume fractions for copper increase. The temporal stability analysis is performed to analyze the stability of the dual solutions, and it is revealed that only one of them is stable and physically reliable.

The present problem is new, original with many important results for practical problems in the modern industry.

This study aims to numerically analyze natural convection of alumina-water nanofluid in a differentially-heated square cavity partially filled with a heat-generating porous medium. A single-phase nanofluid model with experimental correlations for the nanofluid viscosity and thermal conductivity has been considered for the description of the nanoparticles transport effect in the present study. Local thermal non-equilibrium approach for the porous layer with the Brinkman-extended Darcy model has been used.

Dimensionless governing equations formulated using stream function, vorticity and temperature have been solved by the finite difference method. The effects of the Rayleigh number, Ostrogradsky number, Nield number and nanoparticles volume fraction on nanofluid flow, heat and mass transfer have been analyzed.

It has been revealed that the dimensionless heat transfer coefficient at the fluid/solid matrix interface can be a very good control parameter for the convective flow and heat transfer intensity. The present results are original and new for the study of non-equilibrium natural convection in a differentially-heated nanofluid cavity partially filled with a porous medium.

The results of this paper are new and original with many practical applications of nanofluids in the modern industry.

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.