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The boundary layer flows exhibit symmetrical characteristics. In such cases, the flow patterns and variables are symmetrical with respect to a particular axis or plane. This symmetry simplifies the analysis and enables the use of symmetry-based boundary conditions or simplifications in mathematical models. Therefore, by using these concepts, the governing equations of the radiative flow of a hybrid nanofluid past a stretched and shrunken surface with the effect of a magnetic dipole are examined in this paper. Here, we consider copper (Cu) and alumina (Al2O3) as hybrid nanoparticles and use water as a base fluid. The heat transfer rate is enhanced in the presence of hybrid nanoparticles. It is observed that the heat transfer rate is increased by 10.92% for the nanofluid, while it has a 15.13% increment for the hybrid nanofluid compared to the base fluid. Also, the results reveal that the non-uniqueness of the solutions exists for a certain suction and shrinking strength. Additionally, the ferrohydrodynamic interaction has the tendency to reduce the skin friction and the heat transfer coefficients for both solution branches. For the upper branch solutions, the heat transfer rate increased over a stretching sheet but decreased for the shrinking sheet in the presence of the radiation. It is confirmed by the temporal stability analysis that one of the solutions is stable and acceptable as time evolves. [ABSTRACT FROM AUTHOR]

Hybrid nanofluids may exhibit higher thermal conductivity, chemical stability, mechanical resistance and physical strength compared to regular nanofluids. Our aim in this study is to investigate the flow of a water-based alumina-copper hybrid nanofluid in an inclined cylinder with the impact of buoyancy force and a magnetic field. The governing partial differential equations (PDEs) are transformed into a set of similarity ordinary differential equations (ODEs) using a dimensionless set of variables, and then solved numerically using the bvp4c package from MATLAB software. Two solutions exist for both buoyancy opposing (λ < 0) and assisting (λ > 0) flows, whereas a unique solution is found when the buoyancy force is absent (λ = 0). In addition, the impacts of the dimensionless parameters, such as curvature parameter, volume fraction of nanoparticles, inclination angle, mixed convention parameter, and magnetic parameter are analyzed. The results of this study compare well with previously published results. Compared to pure base fluid and regular nanofluid, hybrid nanofluid reduces drag and transfers heat more efficiently. [ABSTRACT FROM AUTHOR]

The exclusive behaviour of hybrid nanofluid has been actively emphasized due to the determination of improved thermal efficiency. Therefore, the aim of this study is to highlight the stagnation point Al 2 O 3 -Cu/H 2 O hybrid nanofluid flow with the influence of Arrhenius kinetics and thermal radiation over a stretching/shrinking sheet. This particular work is distinctive because it presents a novel hybrid nanofluid mathematical model that takes into account the highlighted issue with a combination of multiple consequences that have not yet been addressed in prior literature. The bvp4c package embedded in MATLAB software is used to address the formulated ordinary differential equations and specified boundary conditions based on similarity solutions. The flow is assumed to be incompressible and laminar, and the hybrid nanofluid is made up of two different types of nanoparticles. The findings demonstrate the viability of dual solutions within the defined ranges of the physical parameters. As predicted, the hybrid nanofluid flow has been convincingly proved to enhance the skin friction coefficient and the heat transfer performance as opposed to viscous flow and nanofluid flow. The heat of reaction and radiation parameters also act as contributing factors in the progress of thermal enhancement. On the other hand, the reaction rate parameter unexpectedly displays a decreasing trend in the heat transfer rate of the current study. It is anticipated that this study will benefit future research into this potential heat transfer fluid, particularly in the areas of thermal systems and boundary layer analysis. [ABSTRACT FROM AUTHOR]

The utilization of hybrid nanofluids (HNs) to boost heat transfer is a promising area of study, and thus, numerous scientists, researchers, and academics have voiced their admiration and interest in this area. One of the main functions of nanofluids is their dynamic role in cooling small electrical devices such as microchips and associated gadgets. The major goal of this study is to perform an analysis of the buoyancy flow of a shrinking/stretching sheet, whilst considering the fascinating and practical uses of hybrid nanofluids. The influence of a nonlinear heat source/sink induced by a micropolar fluid is also inspected. Water-based alumina and copper nanoparticles are utilized to calculate the fine points of the fluid flow and the features of heat transfer. The governing equations are framed with acceptable assumptions and the required similarity transformations are used to turn the set of partial differential equations into ordinary differential equations. The bvp4c technique is used to solve the simplified equations. Dual solutions are presented for certain values of stretching/shrinking parameters as well as the mixed convective parameter. In addition, the shear stress coefficient in the first-branch solution (FBS) escalates and decelerates for the second-branch solution (SBS) with the superior impact of the magnetic parameter, the mass transpiration parameter, and the solid nanoparticles volume fraction, while the contrary behavior is seen in both (FB and SB) solutions for the larger values of the material parameter. [ABSTRACT FROM AUTHOR]

The use of hybrid nanoparticles to increase heat transfer is a favorable area of research, and therefore, numerous scientists, researchers, and scholars have expressed their appreciation for and interest in this field. Determining the dynamic role of nanofluids in the cooling of microscopic electronic gadgets, such as microchips and related devices, is also one of the fundamental tasks. With such interesting and useful applications of hybrid nanofluids in mind, the main objective is to deal with the analysis of the unsteady flow towards a shrinking sheet in a water-based hybrid ferrite nanoparticle in porous media, with heat sink/source effects. Moreover, the impact of these parameters on heat and mass transfers is also reported. Numerical results are obtained using MATLAB software. Non-unique solutions are determined for a certain shrinking strength, in addition to the unsteadiness parameter. The mass transfer and friction factor increase for the first solution due to the hybrid nanoparticles, but the heat transfer rate shows the opposite effect. [ABSTRACT FROM AUTHOR]

The mixed convection flow with thermal characteristics of a water-based Cu-Al2O3 hybrid nanofluid towards a vertical and permeable wedge was numerically and statistically analyzed in this study. The governing model was constructed using physical and theoretical assumptions, which were then reduced to a set of ordinary differential equations (ODEs) using similarity transformation. The steady flow solutions were computed using the Matlab software bvp4c. All possible solutions were presented in the graphs of skin friction coefficient and thermal rate. The numerical results show that the flow and thermal progresses are developed by enhancing the controlling parameters (wedge parameter, volumetric concentration of nanoparticles, and suction parameter). Moreover, the response surface methodology (RSM) with analysis of variance (ANOVA) was employed for the statistical evaluation and conducted using the fit general linear model in the Minitab software. From the standpoint of statistical analysis, the wedge parameter and volumetric nanoparticle concentration have a considerable impact on all responses; however, the suction parameter effect is only substantial for a single response. [ABSTRACT FROM AUTHOR]

The present study attempts to analyze the unsteady flow over a rotating disk in a hybrid nanofluid with suction and deceleration effects. The partial derivatives of multivariable differential equations are converted to ordinary differential equations using appropriate transformations. The bvp4c function in MATLAB software is employed to solve the mathematical model. The outcomes show that multiple solutions are verifiable in certain operating parameters. The stability of the multiple solutions over time is investigated. It is discovered that the first and the second solutions are stable and physically relevant, whereas the third solution is unstable as time evolves. Moreover, the stronger deceleration contributes to enhancing the skin friction coefficient in the radial direction Re r 1 / 2 C f and in the azimuthal direction Re r 1 / 2 C g , for the first and third solutions whereas the second solution reduces. The values of Re r 1 / 2 C f and Re r 1 / 2 C g for the third solution enhance in the presence of suction, while the opposite behaviors​ are observed for the first and second solutions. The enhancement of the local Nusselt number Re r − 1 / 2 N u r on all solutions is noticed with the imposition of suction on the surface and stronger deceleration strength. [ABSTRACT FROM AUTHOR]

Riga surface can be utilized to reduce the pressure drag and the friction of the submarine by stopping the separation of the boundary layer as well as by moderating turbulence production. Therefore, the current symmetry of the work investigates the slip impacts on mixed convection flow containing water-based hybrid Ag-MgO nanoparticles over a vertical expanding/contracting Riga wedge. In this analysis, a flat surface, wedge, and stagnation point are also discussed. A Riga surface is an actuator that contains electromagnetic where a span-wise array associated with the permanent magnets and irregular electrodes accumulated on a smooth surface. A Lorentz force is incorporated parallel to the surface produced by this array which eases exponentially normal to the surface. Based on the considered flow symmetry, the physical scenario is initially modeled in the appearance of partial differential equations which are then rehabilitated into a system of ordinary differential equations by utilizing the pertinent similarity variables. A bvp4c solver is engaged to acquire the numerical solution. The flow symmetry and the influences of pertaining parameters involved in the problem are investigated and are enclosed in graphical form. The findings confirm that the velocity reduces, and temperature enhances due to nanoparticle volume fraction. A modified Hartmann number increases the velocity and diminishes the temperature. Moreover, the suction parameter enhances the velocity profiles and reduces the dimensionless temperature profiles. The heat transfer gradually increases by diminishing the contracting parameter and increasing the expanding parameter. [ABSTRACT FROM AUTHOR]

The hybrid nanofluid has sparked new significance in the industrial and engineering sectors because of their applications like water heating in solar and analysis of heat exchanger surfaces. As a result, the current study emphasizes the analysis of heat transfer and Agrawal axisymmetric flow towards a rotational stagnation point incorporated via hybrid nanofluids imposing on a radially permeable shrinking/stretching rotating disk. The leading partial differential equations are refined into ordinary differential equations by using appropriate similarity variables. The bvp4c solver in MATLAB is then employed to solve the simplified system numerically. The current numerical procedure is adequate of generating double solutions when excellent initial guesses are implemented. The results show that the features of fluid flow along with heat transfer rate induced by hybrid nanofluid are significantly influenced. The Nusselt number and the tendency of the wall drag force can be improved as the concentration of nanoparticles and the suction factor are increased. Moreover, the results of the model have been discussed in detail for both solution branches due to the cases of rotating disk parameter as well as non-rotating disk parameter. Therefore, an extraordinary behavior is observed for the branch of lower solutions in the case of rotating disk parameter. In addition, the shear stress in the radial direction upsurges for the first solution but declines for the second solution with higher values of suction. Moreover, the rotating parameter slows down the separation of the boundary layer. [ABSTRACT FROM AUTHOR]

This study examines the Blasius flow with Cu-Al2O3 hybrid nanoparticles over a moving plate. Additionally, the effects of viscous dissipation and radiation are considered. Similarity transformation is employed to convert the respective model into similarity equations. The results are generated by using bvp4c in MATLAB. Findings reveal that two solutions are attained when both the free stream and the plate move in opposite directions. Moreover, the domains of the velocity ratio parameter are extended when suction is available. Besides, the upsurge of radiation and hybrid nanoparticles lead to the heat transfer enhancement. The rise in radiation heat energy incorporated in radiation parameter leads to the development of fluid temperature as well as the thermal boundary layer. Meanwhile, hybrid nanoparticles offer good thermal characteristics because of synergistic effects. However, the effects reduce with the rise in Eckert number. The first solution is stable and acceptable based on the temporal stability analysis. Furthermore, the critical/separation values of the physical parameters are also reported. With these findings, the optimized productivity will be achieved as well as the processes on certain products can be planned according to the desire output. This significant preliminary study provides future insight to the engineers and scientist on the real applications. [ABSTRACT FROM AUTHOR]

Recent nanotech advancements have created a tremendous platform for the development of a superior ultrahigh performance coolant referred to as nanofluid for several industrial and engineering technologies. In this research, the impact of thermophoretic and viscous dissipation on the radiative mixed convective flow comprising hybrid nanofluid through an inclined permeable moving flat plate with a magnetic field is examined numerically. A model of non-linear differential equations is derived based on some realistic assumptions and tackled numerically using the bvp4c technique. The impact of the specific set of distinguished parameters on the velocity profiles, shear stress, temperature distribution profiles, heat transfer, concentration distribution profile, and mass transfer for the two dissimilar branch solutions are discussed in detail. In addition, it has been discovered that double solutions exist in the case of an opposing flow, while a single solution is observed in the case of an assisting flow. The temperature distribution profile escalates with the radiation parameter, while decelerating the velocity and concentration profiles. [ABSTRACT FROM AUTHOR]

This study examines the stagnation point flow over a stretching/shrinking sheet in a hybrid nanofluid with homogeneous-heterogeneous reactions. The hybrid nanofluid consists of copper (Cu) and alumina (Al2O3) nanoparticles which are added into water to form Cu-Al2O3/water hybrid nanofluid. The similarity equations are obtained using a similarity transformation. Then, the function bvp4c in MATLAB is utilised to obtain the numerical results. The dual solutions are found for limited values of the stretching/shrinking parameter. Also, the turning point arises in the shrinking region (λ < 0). Besides, the presence of hybrid nanoparticles enhances the heat transfer rate, skin friction coefficient, and the concentration gradient. In addition, the concentration gradient is intensified with the heterogeneous reaction but the effect is opposite for the homogeneous reaction. Furthermore, the velocity and the concentration increase, whereas the temperature decreases for higher compositions of hybrid nanoparticles. Moreover, the concentration decreases for larger values of homogeneous and heterogeneous reactions. It is consistent with the fact that higher reaction rate cause a reduction in the rate of diffusion. However, the velocity and the temperature are not affected by these parameters. Fromthese observations, it can be concluded that the effect of the homogeneous and heterogeneous reactions is dominant on the concentration profiles. Two solutions are obtained for a single value of parameter. The temporal stability analysis shows that only one of these solutions is stable and thus physically reliable over time. [ABSTRACT FROM AUTHOR]

This study investigates the unsteady flow of Al2O3‐Cu/water hybrid nanofluid on a stagnation point of a permeable rigid surface. The similarity equations are obtained using similarity variables and then solved using the bvp4c solver. The outcomes show that dual solutions exist for the negative unsteadiness and some mass flux parameters. Besides, the heat transfer rate is intensified with the hybrid nanoparticles. Moreover, the increasing of the mass flux parameter tends to increase the heat transfer rate and the skin friction on the surface. Also, the reduction of the skin friction coefficient is observed, and a higher heat transfer rate is achieved for smaller values of the unsteadiness parameter. Finally, by a temporal stability analysis, it is found that the first solution is stable and reliable as time evolves. [ABSTRACT FROM AUTHOR]

This paper examines the behaviour of a hybrid nanofluid flow towards a stagnation point on a stretching or shrinking surface with second-order slip and melting heat transfer effects. Copper (Cu) and alumina (Al2O3) are considered as the hybrid nanoparticles while water as the base fluid. The governing equations are reduced to the similarity equations using similarity transformations. The resulting equations are programmed in MATLAB software through the bvp4c solver to obtain the numerical solutions. The results reveal that two solutions are possible for the shrinking case (λ < 0) , where the bifurcation of the solutions occurs in this region. Moreover, the heat transfer rate and the skin friction coefficient enhance with the rise of the melting parameter. Meanwhile, these quantities decrease for a smaller second-order slip parameter. The temporal stability analysis shows that only one of the two solutions is stable as time evolves. [ABSTRACT FROM AUTHOR]

The mixed convection flow and heat transfer over an exponentially stretching/shrinking vertical surface in a hybrid nanofluid is considered in this paper. Copper (Cu) and alumina (Al 2 O 3) are employed as the hybrid nanoparticles while water as the base fluid. It is shown that the heat transfer rate enhances in the presence of nanoparticles. However, the reduction in heat transfer rate is observed for the shrinking case, caused by the imposition of stronger suction at the boundary. It is also found that the bifurcation of the solutions occurs for both stretching and shrinking regions as well as for both buoyancy aiding and opposing flows. The temporal stability analysis shows that only one of the solutions is stable and thus physically reliable as time evolves. [ABSTRACT FROM AUTHOR]

This paper considers the effect of transpiration on hybrid nanofluid flow and heat transfer over a stretching/shrinking sheet for uniform shear flow. The similarity equations are attained from the governing equations by using similarity transformation technique, and their solutions are obtained using the bvp4c solver in Matlab software. The effect of the several governing parameters on the flow and heat transfer characteristics are presented and interpreted theoretically. Results elucidate that dual solutions exist up to a certain range of the stretching/shrinking parameter. Moreover, the dual solutions are observed to occur for both stretching or shrinking cases in the present of suction parameter. An increasing of copper nanoparticle volume fractions generates an enhancement in the heat transfer coefficient, however dual behaviour is observed in the skin friction coefficient for the upper branch solutions. The temporal stability analysis is conducted to determine the stability of the dual solutions, and it is discovered that only one of them is stable and physically reliable. [ABSTRACT FROM AUTHOR]

• The effect of TiO 2 -CoFe 2 O 4 nanoparticles on MHD flow past a shrinking cylinder with radiative heat flux are studied. • The similarity equations are solved numerically using the bvp4c function in Matlab software. • The thermal performance is improved in the presence of radiation, magnetic field, and hybrid nanoparticles. • The dual solutions are then analysed using stability analysis to determine their stability. The equipment in the modern electronic frequently encounters difficulties in thermal significance due to a decline in effective surface area for heat exclusion or improved heat production. This most intriguing problem can be solved by designing an optimum structure for systems in refrigeration or improving heat transferal features. Therefore, nanofluid performs well enough in reconciling these issues. The goal of the current exploration is to theoretically inspect the steady magneto nanofluid flow with radiative heat transfer caused by a shrinking cylinder by employing the Tiwari-Das model. The radiation, suction, and magnetic parameters effects are investigated. The governing equations are converted to similarity equations by the similarity transformations. Later, for dissimilar values of the pertaining parameters, these equations are solved numerically by employing Matlab's function bvp4c. Outcomes perceived that the shrinking case gives the dual (multiple) solutions, and the unique solution is found for the stretching case. Moreover, the thermal performance is improved in the presence of radiation, magnetic field, and hybrid nanoparticles. It is found that the heat transfer rate is boosted up to 2.06% for CoFe 2 O 4 /water nanofluid. It is intensified when hybrid nanoparticles are considered with 4.23% increment compared to the regular fluid case. [ABSTRACT FROM AUTHOR]

Colloidal suspensions of regular fluids and nanoparticles are known as nanofluids. They have a variety of applications in the medical field, including cell separation, drug targeting, destruction of tumor tissue, and so on. On the other hand, the dispersion of multiple nanoparticles into a regular fluid is referred to as a hybrid nanofluid. It has a variety of innovative applications such as microfluidics, heat dissipation, dynamic sealing, damping, and so on. Because of these numerous applications of nanofluids in minds, therefore, the objective of the current exploration divulged the axisymmetric radiative flow and heat transfer induced by hybrid nanofluid impinging on a porous stretchable/shrinkable rotating disc. In addition, the impact of Smoluchowski temperature and Maxwell velocity slip boundary conditions are also invoked. The hybrid nanofluid was formed by mixing the copper (Cu) and alumina (Al2O3) nanoparticles scattered in the regular (viscous) base fluid (H2O). Similarity variables are used to procure the similarity equations, and the numerical outcomes are achieved using bvp4c in MATLAB software. According to the findings, double solutions are feasible for stretching (λ > 0) and shrinking cases (λ < 0) . The heat transfer rate is accelerated as the hybrid nanoparticles increases. The suction parameter enhances the friction factors as well as heat transfer rate. Moreover, the friction factor in the radial direction and heat transfer enrich for the first solution and moderate for the second outcome due to the augmentation δ 1 , while the trend of the friction factor in the radial direction is changed only in the case of stretching for both branches. [ABSTRACT FROM AUTHOR]

This current study considers the hybrid nanofluid flow on a non-axisymmetric stagnation point region of a flat plate in a porous medium. By adopting the similarity variables, the equations that govern the flow are adapted to the similarity equations and then be solved by utilizing the bvp4c solver. Some samples of the friction factor and heat transfer rate, as well as the velocities and the temperature profiles, are presented. It is interesting to note that the friction factor for both x and y directions exhibits the symmetries pattern. Furthermore, the outcomes show that the friction factor increase with the rising of the hybrid nanoparticles and the porous medium parameter. Meanwhile, hybrid nanoparticles led to enhance the heat transfer rate but the porous medium parameter shows the opposite effect. Moreover, the value of the shear-to-strain-rate ratio where the occurrence of zero friction on the wall is also determined. • The hybrid nanofluid flow on a non-axisymmetric stagnation point region of a flat plate in a porous medium is studied. • The governing equations are transformed into the similarity equations and then solved using the bvp4c function. • The outcomes show that the friction factor increase with the rising of the hybrid nanoparticles and the porous medium parameter. • Hybrid nanoparticles led to enhance the heat transfer rate but the porous medium parameter shows the opposite effect. [ABSTRACT FROM AUTHOR]

This paper examines the hybrid nanoparticles and the magnetic field impacts on the mixed convection boundary layer flow and heat transfer caused by an inclined shrinking–stretching surface in a hybrid nanofluid. Silver (Ag) is added into a MgO–water nanofluid to form Ag-MgO–water hybrid nanofluid. By making use of proper similarity transformations, the governing equations are transformed to ordinary differential equations. The problem is numerically solved with the help of the MATLAB function bvp4c. The influences of the chosen parameters on the temperature, velocity, heat transfer rate and the skin friction coefficient are addressed and graphically illustrated. The results show that increasing the magnetic parameter substantially improves the heat transfer rate and increases the skin friction coefficient. The findings also suggest that increasing the nanoparticle volume fraction φ 2 (Ag) improves the skin friction coefficient while decreasing the heat transfer rate. For both stretching and shrinking instances, non-unique (dual) solutions are discovered. Only the first solution is stable, according to the temporal stability analysis of the dual solutions. [ABSTRACT FROM AUTHOR]

This paper examines the impact of hybrid nanoparticles on the stagnation point flow towards a curved surface. Silica (SiO2) and alumina (Al2O3) nanoparticles are added into water to form SiO2-Al2O3/water hybrid nanofluid. Both buoyancy-opposing and -assisting flows are considered. The governing partial differential equations are reduced to a set of ordinary differential equations, before being coded in MATLAB software to obtain the numerical solutions. Findings show that the solutions are not unique, where two solutions are obtained, for both buoyancy-assisting and -opposing flow cases. The local Nusselt number increases in the presence of the hybrid nanoparticles. The temporal stability analysis shows that only one of the solutions is stable over time. [ABSTRACT FROM AUTHOR]

The time-dependent stagnation point flow (SPF) and heat transfer of a water-based hybrid nanofluid (Cu − Al 2 O 3 / Water) from a radially permeable shrinking or stretching surface is examined. The similarity technique is employed to transform the governing equations of hybrid nanofluid (Cu − Al 2 O 3 / Water) into similarity equations. These similarity equations are solved numerically using bvp4c function in MATLAB software. The numerical outcomes are acquired for particular values of the selected parameters. The results notice that dual solutions exist, up to a definite amount of the suction, unsteady strengths, and nanoparticle volume fraction. The critical amount declines due to nanoparticle volume fraction and augments due to suction and unsteady parameters. Also, it is seen that hybrid nanofluid (Cu − Al 2 O 3 / Water) augments the rate of heat transfer relative to the regular fluid. The temporal stability analysis is implemented to determine the stability of the dual solutions, and it is found that only one of them is stable and thus physically reliable as time passes. • We simulate the hybrid nanofluid flow over a radially stretching/shrinking sheet. • The solutions were obtained numerically using bvp4c solver in Matlab software. • Two (dual) solutions are possible for a certain range of parameter. • Only one of the two solutions are stable as time evolves. • The heat transfer rate of Al 2 O 3 -Cu/water hybrid nanofluid is greater compared to the regular fluid. [ABSTRACT FROM AUTHOR]

In this paper, we examine the influence of hybrid nanoparticles on flow and heat transfer over a permeable non-isothermal shrinking surface and we also consider the radiation and the magnetohydrodynamic (MHD) effects. A hybrid nanofluid consists of copper (Cu) and alumina (Al2O3) nanoparticles which are added into water to form Cu-Al2O3/water. The similarity equations are obtained using a similarity transformation and numerical results are obtained via bvp4c in MATLAB. The results show that dual solutions are dependent on the suction strength of the shrinking surface; in addition, the heat transfer rate is intensified with an increase in the magnetic parameter and the hybrid nanoparticles volume fractions for higher values of the radiation parameter. Furthermore, the heat transfer rate is higher for isothermal surfaces as compared with non-isothermal surfaces. Further analysis proves that the first solution is physically reliable and stable. [ABSTRACT FROM AUTHOR]

This study investigates a hybrid nanofluid flow towards a stagnation region of a vertical plate with radiation effects. The hybrid nanofluid consists of copper (Cu) and alumina (Al2O3) nanoparticles which are added into water to form Cu-Al2O3/water nanofluid. The stagnation point flow describes the fluid motion in the stagnation region of a solid surface. In this study, both buoyancy assisting and opposing flows are considered. The similarity equations are obtained using a similarity transformation and numerical results are obtained via the boundary value problem solver (bvp4c) in MATLAB software. Findings discovered that dual solutions exist for both opposing and assisting flows. The heat transfer rate is intensified with the thermal radiation (49.63%) and the hybrid nanoparticles (32.37%). [ABSTRACT FROM AUTHOR]

• We simulate the stagnation-point flow and heat transfer over a shrinking sheet. • The alumina (Al 2 O 3) and copper (Cu) nanoparticles are suspended in water to form Al 2 O 3 -Cu/water hybrid nanofluid. • The solutions were obtained numerically using bvp4c in Matlab software. • Dual solutions exist for a certain range of the shrinking strength. • A temporal stability analysis is performed to determine the stability of the solutions as time evolves. This study investigates the Hiemenz flow of hybrid nanofluid over a shrinking sheet. The similarity equations are obtained using similarity variables and then solved using the bvp4c solver. The outcomes showed that dual solutions occur for the shrinking case, in the range of - 1.24657 < λ ≤ - 1.1 with λ c = - 1.24657 is the point of bifurcation between the solutions. Meanwhile, the solution is unique for λ > - 1.1. Besides, the heat transfer rate is intensified with the rise of hybrid nanoparticles. Moreover, as hybrid nanoparticles increases, the friction on the surface is increased for λ < 1 , while it is decreased for λ > 1 , and no friction occurs when λ = 1. Finally, these solutions are tested using the stability analysis where the outcomes found that the first solution is stable and acceptable. [ABSTRACT FROM AUTHOR]

This numerical study was devoted to examining the occurrence of non-unique solutions in boundary layer flow due to deformable surfaces (cylinder and flat plate) with the imposition of prescribed surface heat flux. The hybrid Al 2 O 3 -Cu/water nanofluid was formulated using the single phase model with respective correlations of hybrid nanofluids. The governing model was simplified by adopting a similarity transformation. The transformed differential equations were then numerically computed using the efficient bvp4c solver with the ranges of the control parameters 0.5 % ≤ ϕ 1 , ϕ 2 ≤ 1.5 % (Al 2 O 3 and Cu volumetric concentration), 0 ≤ K ≤ 0.2 (curvature parameter), 2.6 < S ≤ 3.2 (suction parameter) and − 2.5 < λ ≤ 0.5 (stretching/shrinking parameter). Dual steady solutions are presentable for both a cylinder (K > 0) and a flat plate (K = 0) with the inclusion of only the suction (transpiration) parameter. The real and stable solutions were mathematically validated through the stability analysis. The Al 2 O 3 -Cu/water nanofluid with ϕ 1 = 0.5 % (alumina) and ϕ 2 = 1.5 % (copper) has the highest skin friction coefficient and heat transfer rate, followed by the hybrid nanofluids with volumetric concentrations (ϕ 1 = 1 % , ϕ 2 = 1 %) and (ϕ 1 = 1.5 % , ϕ 2 = 0.5 %) , respectively. Surprisingly, the flat plate surface abates the separation of boundary layer while it enhances the heat transfer process. [ABSTRACT FROM AUTHOR]

This paper examines the squeezed hybrid nanofluid flow over a permeable sensor surface with magnetohydrodynamics (MHD) and radiation effects. The alumina (Al2O3) and copper (Cu) are considered as the hybrid nanoparticles, while water is the base fluid. The governing equations are reduced to the similarity equations, using the similarity transformation. The resulting equations are programmed in Matlab software through the bvp4c solver to obtain the numerical solutions. It was found that the heat transfer rate was greater for the hybrid nanofluid, compared to the regular nanofluid. It was observed that dual solutions exist for some values of the permeable parameter S. The upper branch solutions of the skin friction coefficient ( Re x 1 / 2 C f ) and the heat transfer rate at the surface ( Re x − 1 / 2 N u x ) enhance with the added Cu nanoparticle ( φ 2 ) and for larger magnetic strength (M). Moreover, the values of Re x 1 / 2 C f decrease, whereas the values of Re x − 1 / 2 N u x increase for both branches, with the rise of the squeeze flow index (b). Besides, an increment of the heat transfer rate at the sensor surface for both branches was observed in the presence of radiation (R). Temporal stability analysis was employed to determine the stability of the dual solutions, and it was discovered that only one of them was stable and physically reliable as time evolves. [ABSTRACT FROM AUTHOR]

The steady mixed convection flow along a vertical surface embedded in a porous medium with hybrid nanoparticles are examined. Both assisting and opposing flows are considered. We consider alumina (Al 2 O 3) and copper (Cu) as the hybrid nanoparticles which are suspended in water to form Al 2 O 3 -Cu/water hybrid nanofluid. The similarity equations are gained from the governing equations using the similarity variables, and their solutions are obtained by the aid of the bvp4c solver in Matlab software. We found that there exist dual solutions for the case of opposing flow. Besides, it is noticed that the added hybrid nanoparticles delay the separation of the boundary layer. In addition, the velocity of the hybrid nanofluid is increased for the upper branch solutions, but the observation is reversed for the lower branch solutions as Cu nanoparticle increases. [ABSTRACT FROM AUTHOR]

The problem of a steady flow and heat transfer past a permeable moving thin needle in a hybrid nanofluid is examined in this study. Here, we consider copper (Cu) and alumina (Al2O3) as hybrid nanoparticles, and water as a base fluid. In addition, the effects of thermophoresis and Brownian motion are taken into consideration. A similarity transformation is used to obtain similarity equations, which are then solved numerically using the boundary value problem solver, bvp4c available in Matlab software (Matlab_R2014b, MathWorks, Singapore). It is shown that heat transfer rate is higher in the presence of hybrid nanoparticles. It is discovered that the non-uniqueness of the solutions is observed for a certain range of the moving parameter λ. We also observed that the bifurcation of the solutions occurs in the region of λ < 0 , i.e., when the needle moved toward the origin. Furthermore, we found that the skin friction coefficient and the heat transfer rate at the surface are higher for smaller needle sizes. A reduction in the temperature and nanoparticle concentration was observed with the increasing of the thermophoresis parameter. It was also found that the increase of the Brownian motion parameter leads to an increase in the nanoparticle concentration. Temporal stability analysis shows that only one of the solutions was stable and physically reliable as time evolved. [ABSTRACT FROM AUTHOR]

Modern electronic equipment frequently encounters thermal critical difficulties as a result of increased heat production or a reduction in effective surface area for heat exclusion. This most intriguing problem can be solved by either improving an optimal structure for the cooling system or improving heat transfer performance. In this case, nanofluid performs well in addressing all of these issues. In addition, the structure of boilers necessitates a thorough understanding of the two-phase flow behavior of heat transfer and pressure drop, which varies substantially from that of single-phase flow. The present works address the features of Agrawal axisymmetric flow induced by hybrid nanofluid with motile microorganisms past a porous moving disk with zero mass flux. Through the use of similarity variables, the partial differential equations that represent the two-phase flow problems are eased to ordinary differential equations and then used a bvp4c technique to find the numerical dual solutions. The influences of pertaining control parameters on the dimensionless friction factor, the heat transfer, and the motile microorganisms are investigated and portrayed in both the form of graphical illustrations as well as in the form of tables. The obtained results show that an upsurge in the solid volume fraction of nanoparticles leads to substantial magnification of the shear stress and heat transfer for both branches while the local motile microorganism flux reduces. In addition, the friction factor, motile microorganism flux, and heat transfer are enhanced due to the presence of suction. [ABSTRACT FROM AUTHOR]