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The purpose of this paper is to present a new approximate analytical procedure to obtain dual solutions of nonlinear differential equations arising in mixed convection flow in a semi-infinite domain. This method, which is based on Padé-approximation and homotopy–Padé technique, is applied to a model of magnetohydrodynamic Falkner–Skan flow as well. These examples indicate that the method can be successfully applied to solve nonlinear differential equations arising in science and engineering.

Homotopy–Padé method.

The main focus of the paper is on the prediction of the multiplicity of the solutions, however we have calculated multiple (dual) solutions of the model problem namely, mixed convection heat transfer in a porous medium.

The authors conjecture here that the combination of traditional–Pade and Hankel–Pade generates a useful procedure to predict multiple solutions and to calculate prescribed parameter with acceptable accuracy as well. Validation of this conjecture for other further examples is a challenging research opportunity.

Dual solutions of nonlinear differential equations arising in mixed convection flow in a semi-infinite domain.

In this study, the authors are using two modified methods.

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.

This paper aims to present the steady dual solutions on three-dimensional flow and heat transfer of nanofluid over a permeable non-linearly shrinking surface with two-order velocity slips conditions. Boundary layer assumption is considered in the mathematical modelling. Authors comprehend from previous studies and papers that the shrinking surfaces are extremely important in current engineering and environmental systems.

Using appropriate similarity variables, the full partial differential equations (PDF) are modified into a specific set of ordinary (similar) differential equations (ODE). The resulting non-linear ordinary differential system is then solved both analytically for some particular cases and numerically for the general case using the function bvp4c from MATLAB for characteristic values of the parameters which govern the equations. The transformed mathematical model is analysed using the bvp4c procedure. Based on the given assumptions, this study is able to produce multiple solutions of the problem.

The ordinary (similarity) differential equations have two branches solutions, upper and lower branch solutions, given some interval of shrinking and velocity slip parameters. The authors consider here a temporal stability analysis, as they want to establish which of the solutions are stable and which are not. In a distinct paragraph, the authors discuss in detail and present in a graphical manner the effects of shrinking and second-order slip flow model on the skin friction coefficient, surface wall heat flux and dimensionless velocity and temperature profiles. The analysis reveals that the second order slip has a big influence on the flow and heat transfer characteristics.

The present discoveries are unique and truly new for the research of the three-dimensional stretching/shrinking forced convection flow and heat transfer nanofluids. The nanofluid is a water-based nanofluid (H₂O), which contains one type of nanoparticles, namely, copper (Cu). Of course, the analysis can be further extended considering other types of nanoparticles such as alumina (Al₂O₃). The authors assume that the thermal equilibrium is reached for the base fluid together with the suspended nanoparticles and that the nanoparticles are uniform in dimension and form.

This paper aims to study a non-Newtonian micropolar fluid flow over a bidirectional flexible surface for multiple exact solutions of momentum boundary layer and thermal transport phenomenon subject to wall mass flux, second-order slip and thermal jump conditions.

The coupled equations are transformed into ordinary differential equations using similarity variables. Analytical and numerical techniques are used to solve the coupled equations for single, dual or multiple solutions.

The results show that the stretching flow, shrinking flow, the wall drag, thermal profile and temperature gradient manifest large changes when treated for special effects of the standard parameters. The role of critical numbers is definitive in locating the domains for the existence of exact solutions. The nondimensional parameters, such as mass transfer parameter, bidirectional moving parameter, plate deformation strength parameter, velocity slips, material parameter, thermal jump and Prandtl number, are considered, and their physical effects are presented graphically. The presence of governing parameters exhibits special effects on the flow, microrotation and temperature distributions, and various exact solutions are obtained for the special parametric cases.

The originality and value of this work lie in its exploration of non-Newtonian micropolar fluid flow over a bidirectional flexible surface, highlighting the multiple exact solutions for momentum boundary layers and thermal transport under various physical conditions. The study provides insights into the effects of key parameters on flow and thermal behavior, contributing to the understanding of complex fluid dynamics.

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 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.

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.

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.

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.

The purpose of this paper is to investigate the steady mixed convection boundary layer flow from a vertical frustum of a cone in water-based nanofluids. The problem is formulated to incorporate three kinds of nanoparticles: copper, alumina and titanium oxide. The working fluid is chosen as water with the Prandtl number of 6.2. The mathematical model used for the nanofluid incorporates the particle volume fraction parameter, the effective viscosity and the effective thermal diffusivity. The entire regime of the mixed convection includes the mixed convection parameter, which is positive for the assisting flow (heated surface of the frustum cone) and negative for the opposing flow (cooled surface of the frustum cone), respectively.

The transformed non-linear partial differential equations are solved numerically for some values of the governing parameters. The derivatives with respect to? were discretized using the first order upwind finite differences and the resulting ordinary differential equations with respect to? were solved using bvp4c routine from Matlab. The absolute error tolerance in bvp4c was 1e-9.

The features of the flow and heat transfer characteristics for different values of the governing parameters are analysed and discussed. The effects of the particle volume fraction parameter \phi, the mixed convection parameter \lambda and the dimensionless coordinate? on the flow and heat transfer characteristics are determined only for the Cu nanoparticles. It is found that dual solutions exist for the case of opposing flows. The range of the mixed convection parameter for which the solution exists increases in the presence of the nanofluids.

The paper models the mixed convection from a vertical truncated cone using the boundary layer approximation. Multiple (dual) solutions for the flow reversals are obtained and the range of existence of the solutions was found. Particular cases for ?=0 (full cone), ? >>1 and (free convection limit) \lambda>>1were studied. To the authors best knowledge this problem has not been studied before and the results are new and original.

The purpose of this paper is to theoretically investigate the steady two‐dimensional boundary‐layer flow past a moving semi‐infinite flat plate in a water‐based nanofluid containing three different types of nanoparticles: copper (cuprum) Cu, alumina (aluminium oxide) Al₂O₃, and titania (titanium dioxide) TiO₂. The effects of moving parameter λ as well as solid volume fraction parameter φ on the flow and heat transfer characteristics are studied. Taking into account the rising demands of modern technology, including chemical production, power stations and microelectronics, there is a need to develop new types of fluids that will be more effective in terms of heat exchange performance.

A similarity transformation is used to reduce the governing partial differential equations to a set of nonlinear ordinary differential equations which are then solved numerically using Keller‐box method.

There is a region of unique solutions for λ>0, however, multiple (dual) solutions exist for λ_c<λ≤0 and no solutions for λ<λ_c<0. A reverse flow is formed when λ<0.

The solutions can be obtained up to a certain value of the moving parameter (critical value or turning point). The boundary layer separates from the plate beyond the turning point hence it is not possible to get the solution based on the boundary‐layer approximations after this point. To obtain further solutions, the full Navier‐Stokes equations have to be solved.

The present results are original and new for the boundary‐layer flow and heat transfer of a moving flat plate in a nanofluid. Therefore, this study would be important for the scientists and engineers in order to become familiar with the flow behaviour and properties of such nanofluids, and the way to predict the properties of this flow for the process equipments.

The purpose of this paper is to study the effects of MHD, suction, second-order slip and melting on the stagnation-point and heat transfer of a nanofluid past a stretching/shrinking sheet.

Using appropriate variables, the governing partial differential equations were transformed into ordinary (similarity) differential equations, which are then solved numerically using the function bvp4c from Matlab.

It is found that dual (upper and lower branch) solutions exist for some values of the governing parameters. From the stability analysis, it is found that the upper branch solution is stable, while the lower branch solution is unstable. The sample velocity, temperature and concentration profiles along both solution branches are graphically presented.

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

The purpose of this paper is to theoretically study the problem of mixed convection boundary layer flow and heat transfer past a vertical needle with variable wall temperature using nanofluids. The similarity equations are solved numerically for copper nanoparticles in the based fluid of water to investigate the effect of the solid volume fraction parameter of the fluid and heat transfer characteristics. The skin friction coefficient, Nusselt number, and the velocity and temperature profiles and are graphically presented and discussed.

The transformed system of ordinary differential equations was solved using the function bvp4c from Matlab. The relative tolerance was set to 1e-10. For the study of the stability the authors also used the bvp4c function in combination with chebfun package from Matlab.

It is found that the solid volume fraction affects the fluid flow and heat transfer characteristics. The numerical results for a regular fluid and forced convection flow are compared with the corresponding results reported by Chen and Smith. The solutions exists up to a critical value of λ, beyond which the boundary layer separates from the surface and the solution based upon the boundary-layer approximations is not possible

The paper describes how multiple (dual) solutions for the flow reversals are obtained. A stability analysis for this flow reversal has been also done showing that the lower solution branches are unstable, while the upper solution branches are stable.