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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.

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.

This study aims to investigate the flow and heat transfer of a hybrid nanofluid through an exponentially stretching/shrinking sheet along with mixed convection and Joule heating. The nanoparticles alumina (Al₂O₃) and copper (Cu) are suspended into a base fluid (water) to form a new kind of hybrid nanofluid (Al₂O₃-Cu/water). Also, the effects of constant mixed convection parameter and Joule heating are considered.

The governing partial differential equations are transformed into ordinary differential equations (ODEs) using appropriate similarity transformations. The transformed nonlinear ODEs are solves using the bvp4c solver available in MATLAB software. A comparison of the present results shows a good agreement with the published results.

Dual solutions for hybrid nanofluid flow obtained for a specific range of the stretching/shrinking parameter values. The values of the skin friction coefficient increases but the local Nusselt number decreases for the first solution with the increasing of the magnetic parameter. Enhancing copper volume fraction and Eckert number reduces the surface temperature, which intimates the decrement of heat transfer rate for the first and second solutions for the stretching/shrinking sheet. In detail, the first solution results show that when the Eckert number increases as 0.1, 0.4 and 0.7 at λ = 1.5, the temperature variations reduced to 10.686840, 10.671419 and 10.655996. While in the second solution, keeping the same parameters temperature variation reduced to 9.750777, 9.557349 and 9.364489, respectively. On the other hand, the results indicate that the skin friction coefficient increases with copper volume fraction. This study shows that the thermal boundary layer thickness rises due to the rise in the solid volume fraction. It is also observed that the magnetic parameter, copper volume fraction and Eckert number widen the range of the stretching/shrinking parameter for which the solution exists.

In practice, the investigation on the flow and heat transfer of a hybrid nanofluid past an exponentially stretching/shrinking sheet with mixed convection and Joule heating is crucial and useful. The problems related to hybrid nanofluid have numerous real-life and industrial applications, such as microelectronics, manufacturing, naval structures, nuclear system cooling, biomedical and drug reduction.

In specific, this study focuses on increasing thermal conductivity using a hybrid nanofluid mathematical model. The novelty of this study is the use of natural mixed convection and Joule heating in a hybrid nanofluid. This paper can obtain dual solutions. The authors declare that this study is new, and there is no previous published work similar to the present study.

This paper aims to scrutinize the analysis of non-axisymmetric Homann stagnation point flow and heat transfer of hybrid Cu-Al₂O₃/water nanofluid over a stretching/shrinking flat plate.

The similarity transformation which fulfils the continuity equation is opted to transform the coupled momentum and energy equations into the nonlinear ordinary differential equations. Numerical solutions which are elucidated in the tables and graphs are obtained using the bvp4c solver.

Non-unique solutions (first and second) are feasible for both stretching and shrinking cases within the specific values of the parameters. First solution is the physical/real solution based on the execution of stability analysis. An upsurge of the ratio of the ambient fluid strain rate to the plate strain rate can delay the boundary layer separation, whereas a boost of the ratio of the ambient fluid shear rate to the plate strain rate only accelerates the separation of boundary layer. The heat transfer rate of hybrid nanofluid is greater for the stretching case than the shrinking case. However, for the shrinking case, the heat transfer rate intensifies with the increment of the copper (Cu) nanoparticles volume fraction, whereas a contrary result is found for the stretching case.

The present numerical results are original and new. It can contribute to other researchers on electing the relevant parameters to optimize the heat transfer process in the modern industry, and the right parameters to generate non-unique solution so that no misjudgment on flow and heat transfer features.

Nanofluid research has piqued the interest of scientists due to its intriguing applications in nanoscience, biomedical and electrical engineering, medication delivery, biotechnology, food processing, chemotherapy and other fields. This paper aims to inspect the behavior of the mixed convection magnetohydrodynamic flow and heat transfer induced by a nonlinear stretching/shrinking sheet in a nanofluid with a convective boundary condition. Tiwari and Das mathematical nanofluid model is incorporated in the analysis.

The mathematical model is initially transformed to a nondimensional form by using dimensionless variables. Then the nondimensional partial differential equations are further transformed to a set of similarity equations by using the similarity technique. These equations are solved numerically by the bvp4c function in MATLAB software.

For a certain range of the stretching/shrinking parameter, two solutions are obtained. The friction factor and the heat transfer rate escalate due to suction parameter with adding nanoparticles volume fraction by almost 27.15% and 0.153% for the upper branch solution, while the friction factor declines by almost 30.10% but the heat transfer rate augments by 0.145% for the lower branch solution. Furthermore, the behavior of the nanoparticle volume fractions on the heat transfer rate behaves differently in the presence of the mixed convection effect. The temperature of fluid augments with increasing Biot number for both solutions.

The present work considers the flow and heat transfer induced by a stretching/shrinking sheet in a nanofluid using the Tiwari–Das nanofluid model with a convective boundary condition, where the effect of the buoyancy force is taken into consideration. It is shown that two solutions are found for a certain range of the shrinking strength, while the solution is unique for the stretching case. This study is important for scientists working in the growing field of nanofluids to become familiar with the flow properties and behaviors of such nanofluids.

This paper aims to analyze the behavior of the stagnation-point flow and heat transfer over a permeable stretching/shrinking sheet in the presence of the viscous dissipation and heat source effects.

The governing partial differential equations are converted into ordinary differential equations by similarity transformations before being solved numerically using the bvp4c function built in Matlab software. Effects of suction/injection parameter and heat source parameter on the skin friction and heat transfer coefficients as well as the velocity and temperature profiles are presented in the forms of tables and graphs. A temporal stability analysis will be conducted to verify which solution is stable for the dual solutions exist for the shrinking case.

The analysis indicates that the skin friction coefficient and the local Nusselt number as well as the velocity and temperature were influenced by suction/injection parameter. In contrast, only the local Nusselt number, which represents heat transfer rate at the surface, was affected by heat source effect. Further, numerical results showed that dual solutions were found to exist for the certain range of shrinking case. Then, the stability analysis is performed, and it is confirmed that the first solution is linearly stable and has real physical implication, while the second solution is not.

In practice, the study of the steady two-dimensional stagnation-point flow and heat transfer past a permeable stretching/shrinking sheet in the presence of heat source effect is very crucial and useful. The problems involving fluid flow over stretching or shrinking surfaces can be found in many industrial manufacturing processes such as hot rolling, paper production and spinning of fibers. Owing to the numerous applications, the study of stretching/shrinking sheet was subsequently extended by many authors to explore various aspects of skin friction coefficient and heat transfer in a fluid. Besides that, the study of suction/injection on the boundary layer flow also has important applications in the field of aerodynamics and space science.

Although many studies on viscous fluid has been investigated, there is still limited discoveries found on the heat source and suction/injection effects. Indeed, this paper managed to obtain the second (dual) solutions and stability analysis is performed. The authors believe that all the results are original and have not been published elsewhere.

This study aims to investigate the heat transfer characteristic of the magnetohydrodynamic (MHD) hybrid nanofluid over the linear stretching and shrinking surface in the presence of suction and thermal radiation effects.

Mathematical equations are transformed into pairs of self-similarity equations using similarity transformation. Boundary value problem solver (bvp4c) in MATLAB was adopted to solve the system of reduced similarity equations. In this study, the authors particularly examine the flow and heat transfer properties for different values of suction and thermal radiation parameters using single-phase nanofluid model. A comparison of the present results shows a good agreement with the published results.

It is noticed that the efficiency of heat transfer of hybrid nanofluid (Cu-Al₂O₃/H₂O) is greater than the nanofluid (Cu/H₂O). Furthermore, it is also found that dual solutions exist for a specific range of the stretching/shrinking parameter with different values of suction and radiation parameters. The results indicate that the skin friction coefficient and the local Nusselt number increase with suction effect. The values of the skin friction coefficient increases, but the local Nusselt number decreases for the first solution with the increasing of thermal radiation parameter. It is also observed that suction and thermal radiation widen the range of the stretching/shrinking parameter for which the solution exists.

In practice, the investigation on the flow and heat transfer of MHD hybrid nanofluid through a stretching/shrinking sheet with suction and thermal radiation effects is very important and useful. The problems related to hybrid nanofluid has numerous real-life and industrial applications, for example microfluidics, manufacturing, transportation, military and biomedical, etc.

In specific, this study focused on increasing thermal conductivity using a hybrid nanofluid mathematical model. This paper is able to obtain the dual solutions. To the best of author’s knowledge, this study is new and there is no previous published work similar to present study.

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 study is to investigate the influence of the second order slip velocity on the boundary layer stagnation point flow of a nanofluid past a non-aligned stretching/shrinking sheet.

Proper similarity variables are used to transform the system of partial differential equations into a system of ordinary (similarity) differential equations. This system is then solved numerically using the bvp4c solver in MATLAB software. As in the papers by Kuznetsov and Nield (2010, 2013) and Fang et al. (2009), the authors considered the stretching/shrinking parameter λ, the first-order (a₁, a₂) and second-order (b₁) slip parameters and the Lewis number Le, Nb the Brownian parameter and Nt the thermophoresis parameter fixed at Le = 10, Nb = Nt = 0.5 when the Prandtl number Pr is fixed at Pr = 1.

Dual solutions are found as the sheet is shrunk in the horizontal direction. Stability analysis shows that the first solution is physically realizable, whereas the second solution is not practicable.

The present results are original and new for the study of fluid flow and heat transfer over a stretching/shrinking surface, as they successfully extend the problem considered by Wang (2008) and Lok et al. (2011) to the case of nanofluids.

The purpose of this paper is to study the laminar boundary layer cross flow and heat transfer on a rotational stagnation-point flow over either a stretching or shrinking porous wall submerged in hybrid nanofluids. The involved boundary layers are of stream-wise type with stretching/shrinking process along the surface.

Using appropriate similarity variables the partial differential equations are reduced to ordinary (similarity) differential equations. The reduced system of equations is solved analytically (by high-order perturbed field propagation for small to moderate stretching/shrinking parameter and low-order perturbation for large stretching/shrinking parameter) and numerically using the function bvp4c from MATLAB for different values of the governing parameters.

It was found that the basic similarity equations admit dual (upper and lower branch) solutions for both stretching/shrinking surfaces. Moreover, performing a linear stability analysis, it was confirmed that the upper branch solution is realistic (physically realizable), while the lower branch solution is not physically realizable in practice. These dual solutions will be studied in the present paper.

The authors believe that all numerical results are new and original and have not been published before for the present problem.