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This study aims to investigate the dual solutions for axisymmetric flow and heat transfer due to a permeable radially shrinking disk in copper oxide (CuO) and silver (Ag) hybrid nanofluids with radiation effect.

The partial differential equations that governed the problem will undergo a transformation into a set of similarity equations. Following this transformation, a numerical solution will be obtained using the boundary value problem solver, bvp4c, built in the MATLAB software. Later, analysis and discussion are conducted to specifically examine how various physical parameters affect both the flow characteristics and the thermal properties of the hybrid nanofluid.

Dual solutions are discovered to occur for the case of shrinking disk (λ < 0). Stronger suction triggers the critical values’ expansion and delays the boundary layer separation. Through stability analysis, it is determined that one of the solutions is stable, whereas the other solution exhibits instability, over time. Moreover, volume fraction upsurge enhances skin friction and heat transfer in hybrid nanofluid. The hybrid nanofluid’s heat transfer also heightened with the influence of radiation.

Flow over a shrinking disk has received limited research focus, in contrast to the extensively studied axisymmetric flow problem over a diverse set of geometries such as flat surfaces, curved surfaces and cylinder. Hence, this study highlights the axisymmetric flow due to a shrinking disk under radiation influence, using hybrid nanofluids containing CuO and Ag. Upon additional analysis, it is evidently shows that only one of the solutions exhibits stability, making it a physically dependable choice in practical applications. The authors are very confident that the findings of this study are novel, with several practical uses of hybrid nanofluids in modern industry.

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.

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 paper aims to analyse numerically the unsteady stagnation-point flow of Cu-Al₂O₃/H₂O hybrid nanofluid towards a radially shrinking Riga surface with thermal radiation.

The governing partial differential equations are transformed into a set of ordinary (similar) differential equations by applying appropriate transformations. The numerical computation of these equations including the stability analysis is conducted using the bvp4c solver.

Two solutions are possible within the allocated interval: shrinking parameter, unsteadiness decelerating parameter, electro-magneto-hydrodynamics (EMHD) parameter, nanoparticles volumetric concentration, radiation parameter and width parameter, whereas the stability analysis certifies that the first (upper branch) solution, which fulfills the boundary conditions is the physical/real solution. The EMHD parameter generated from the application of Riga plate enhances the skin friction coefficient as well as the heat transfer process. The width parameter d is also one of the factors in the deterioration of the skin friction coefficient and heat transfer rate. It is crucial to control the width parameter of the magnets and electrodes to obtain the desired outcome. The radiation parameter is not affecting the boundary layer separation because the critical values are unchanged. However, the addition of radiation and unsteadiness decelerating parameters boosts the thermal rate.

The results are novel and contribute to the discovery of the flow and thermal performance of the hybrid nanofluid subjected to a radially shrinking Riga plate. Besides, this work is beneficial to the other researchers and general audience from industries regarding the factors which contribute to the thermal enhancement of the working fluid.

This paper aims to report theoretical and numerical results for the problem of laminar axisymmetric flow of hybrid nanofluid over a permeable non-linearly stretching/shrinking sheet with radiation effect.

The numerical solutions of the arising boundary value problem are obtained using the function bvp4c from MATLAB for different values of the governing parameters.

It is found that the solutions of the ordinary (similarity) differential equations have two branches, upper and lower branch solutions, in a certain range of the stretching/shrinking and suction parameters. To establish which of these solutions are stable and which are not, a stability analysis has been performed.

To the best of the authors’ knowledge, 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 Mustafa et al. (2015) to the case of hybrid nanofluids.

The investigation of fluid flow over a rotating disk has been increasing due to the spread of machine technology. Because of this development, we scrutinized the Magnetohydrodynamic (MHD) flow of hybrid nanofluid caused by a decelerating rotating disk with Ohmic heating, Soret and Dufour effects. The disk's angular velocity is taken to be an inversely time-dependent linear function. Moreover, the temperature-dependent viscosity of hybrid nanofluid is incorporated in the present investigation. Methanol is considered as base fluid, while copper oxide (CuO) and magnesium oxide (MgO) are nanoparticles.

Estimated fundamental partial differential equations of flow problems are altered as a dimensionless system of ordinary differential equations using appropriate similarity transformation and solved using a numerical technique: BVP Midrich scheme in Maple software. The impression of emerging non-dimensional parameters is portrayed graphically. All outcomes are shown in the velocity, temperature and concentration profiles.

The developed flow problem involves a non-dimensional parameter (A) that reveals the deceleration of the disk. For larger values of A, the disk decelerates faster and for some fixed time, the fluid surrounding the disk revolves more rapidly than the disk itself. The radial velocity of fluid diminishes and axial velocity becomes uniform when the disk is subjected to wall suction velocity (B).

This analysis is significant in biomedical engineering, cancer therapeutic, manufacturing industries and nano-drug suspension in pharmaceuticals. The novelty of the current study is the hybrid nanofluid flow with Ohmic heating, Soret and Dufour effects on a decelerating rotating disk. To the best of the author's knowledge, no such consideration has been published in the literature.

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.

The purpose of study is to examine the dual nature of the branches for the problem of Darcy–Forchheimer porous medium flow of rotating nanofluid on a linearly stretching/shrinking surface under the field of magnetic influence. The dual nature of the branches confronts the uniqueness and existence theorem, moreover, mathematically it is a great achievement. For engineering purposes, this study applied a linear stability test on the multiple branches to determine which solution is physically reliable (stable).

Nanofluid model has been developed with the help of Buongiorno model. The partial differential equations in space coordinates for the law of conservation of mass, momentum and energy have been transformed into ordinary differential equations by introducing the similarity variables. Two numerical techniques, namely, the shooting method in Maple software and the three-stage Lobatto IIIA method in Matlab software, have been used to find multiple branches and to accomplish stability analysis, respectively.

The parametric investigation has been executed to find the multiple branches and explore the effects on skin friction, Sherwood number, Nusselt number, concentration and temperature profiles. The findings exhibited the presence of dual branches only in the case of a shrinking sheet.

The originality of work is a determination of multiple branches and the performance of the stability analysis of the branches. It has also been confirmed that such a study has not yet been considered in the previous literature.

Shrink fit is an important method used in mechanical transmission, but the design of these connections is not sufficiently precise due to lack of knowledge regarding the effects of many parameters. The paper aims to discuss these issues.

This paper presents the torque capacity and contact stress analysis for a new shrink disc by numerical analysis and experimental methods. Torque capacity analysis aims to predict the effectiveness of mechanical transmission, and to check the structural safety. Stress-strain curves are measured using the universal testing machine, and Bilinear Kinematic Hardening rate-independent plasticity model characterises the elastic-plastic response of the materials.

The numerical result shows that there is no plastic deformation at the interference regions, and the maximum equivalent strain happens on the inner ring. The stress of outer ring is decreased with radius increasing, and shows a periodic variation along the circumference, but stress concentration happens at the threaded holes. Finally, the torque capacity of the shrink disc system is measured through a developed test machine, and the torque capacity difference between numerical and experimental results is about 3.3 per cent.

This paper presents a numerical and experimental carrying capacity analysis for a shrink disc. Plasticity model characterises the materials’ elastic-plastic response based testing. A test machine is designed to measure the torque capacity; the error is about 3.3 per cent.

The purpose of this paper is to study the laminar boundary layer flow between a stationary nonporous disk and a porous rotating disk, both being immersed in large amount of fluid.

The governing nonlinear momentum equations in cylindrical polar coordinates together with relevant boundary conditions are reduced to a system of coupled nonlinear ordinary differential equations (NODEs) using similarity transformations. The resulting coupled NODEs are solved using computer-extended series solution and homotopy analysis method.

The analytical solutions are explicitly expressed in terms of recurrence relation for determining the universal coefficients. The nature and location of singularity which restricts the convergence of series is analyzed by using Domb–Sykes plot. Reversion of series is used for the improvement of series. The region of validity of series is extended for much larger values of Reynolds number (R), i.e. R = 6 to 15.

The resulting solutions are compared with earlier works in the literature and are found to be in good agreement.