A three-dimensional hybrid nanofluid flow over a stretching/shrinking sheet is numerically studied. The hybrid nanofluid being considered in this study used water as the base fluid and mixed with two types of solid nanoparticles, namely alumina (Al2O3) and copper (Cu). The main focus of the current study is to examine the effect of magnetic field, Joule heating, and rotating sheet on the velocity, and temperature profiles. In addition, the impact of suction and stretching sheet on the variations of reduced skin friction, , and reduced heat transfer are studied as well. The fluid flow and heat transfer problem presented in this study is governed by a system of nonlinear partial differential equations (PDEs), which is then transformed into the corresponding system of high order nonlinear ordinary differential equations (ODEs) using similarity variables. The resulting system of higher order nonlinear ODEs is solved numerically using a boundary value solver known as bvp4c, which operates on the MATLAB computational platform. Results revealed that dual solutions exist for shrinking sheet while unique solutions are observed for stretching sheet with various values of Cu nanoparticles volume fraction and magnetic parameter. Dual solutions also exist for the value of the suction parameter greater than its critical point with various values of Cu nanoparticles volume fraction. Velocity profile of the hybrid nanofluid increases alongside with the value of magnetic parameter but declination was observed in the profile of and temperature, for both solutions as the value of Cu nanoparticles volume fraction increases. When the value of rotational parameter increases, both velocity and profiles increase for both solutions. This indicates that the momentum boundary layer thickness increases with increasing values of for both solutions, but thermal boundary layer thickness decreases for the first solution and increases for the second solution. Finally, an increment in the value of Eckert number causes the temperature of the hybrid nanofluid to rise as well for both first and second solutions.

The effect of thermal radiation on the three-dimensional magnetized rotating flow of a hybrid nanofluid has been numerically investigated. Enhancing heat transmission is a contemporary engineering challenge in a range of sectors, including heat exchangers, electronics, chemical and biological reactors, and medical detectors. The main goal of the current study is to investigate the effect of magnetic parameter, solid volume fraction of copper, Eckert number, and radiation parameter on velocity and temperature distributions, and the consequence of solid volume fraction on declined skin friction and heat transfer against suction and a stretching/shrinking surface. A hybrid nanofluid is a contemporary type of nanofluid that is used to increase heat transfer performance. A linear similarity variable is–applied to convert the governing partial differential equations (PDEs) into corresponding ordinary differential equations (ODEs). Using the three-stage Labatto III-A method included in the MATLAB software’s bvp4c solver, the ODE system is solved numerically. In certain ranges of involved parameters, two solutions are received. The temperature profile $\theta \left(\eta \right)$ upsurges in both solutions with growing values of $EC$ and $Rd$. Moreover, the conclusion is that solution duality exists when the suction parameter $S\ge S_{ci}$, while no flow of fluid is possible when $S<S_{ci}$. Finally, stability analysis has been performed and it has been found that only the first solution is the stable one between both solutions.

Hybrid nanofluid has an extensive range of real-world applications. Hybrid nanofluid is a new and advanced nanofluid modification extensively used to increase thermal efficiency in fluid flow systems. The main objective of this research is to study magnetohydrodynamics hybrid nanofluid flow numerically in two dimensional over a vertical exponentially shrinking sheet, considering the effects of Joule heating and thermal slip condition. Furthermore, using the Tiwari-Das model, the influence of the suction parameter on variations of reduced skin friction and reduced heat transfer is also explored. The hybrid nanofluid in this research is an Al2O3+Cu/water hybrid nanofluid, in which water is the base fluid, and two types of solid nanoparticles, namely Alumina (Al2O3) and copper (Cu), are combined together. The governing partial differential (PDEs) equations are transformed into the ordinary differential equations (ODEs) using exponential similarity variables. The resulting ordinary differential equations (ODEs) are numerically solved using the three-stage Labatto III-A technique in the "MATLAB software's" bvp4c solver. Hybrid nanofluids have greater thermal efficiency than nanofluids and base fluid. Dual solutions are obtained in specified ranges of suitable parameters. The temperature profile rises in both solutions as the Eckert value enhances. Besides, In the first and second solutions, the thermal boundary layer thickness decreased gradually as the thermal slip parameter increased. Finally, the conclusions presented that solution duality exists when the suction parameter , while no flow of fluid is possible when .

Hybrid nanofluid is considered a modern and improvised form of nanofluid which usually used to enhance the performance of heat transfer in fluid flow systems. Previous studies found hybrid nanofluid offered a wide range of applications and this opened up numerous new opportunities to further explore the unknown behaviour of hybrid nanofluid under different body geometries and physical parameters. This paper numerically studied a two-dimensional mixed convection and radiative Al2O3-Cu/H2O hybrid nanofluid flow over a vertical exponentially shrinking sheet with partial slip conditions. The main objective is to investigate the effect of mixed convection and radiation on the velocity and temperature profiles, as well as the effect of suction on reduced skin friction and reduced heat transfer with respect to solid volume fraction of copper, velocity, and thermal slips. Exponential similarity variables transformed the governing system of partial differential equations into a system of ordinary differential equations which is solved via MATLAB’s bvp4c solver. Outcomes showed that the value of the reduced heat transfer upsurges in the first solution but declines in the second solution when the velocity slip rises. The reduced heat transfer decreases in both dual solutions when thermal slip is enhanced. As the intensity of thermal slip increases, the reduced skin friction rises in the first solution and decreases in the second. As the mixed convection parameter increases, no obvious variation is noticed in the temperature distribution within the first solution, but increasing trend is observed within the second solution. An increment in the temperature distribution also observed within the dual solutions as the thermal radiation parameter increases. In summary, findings from this study are particularly useful to understand various behaviour of Al2O3-Cu/H2O hybrid nanofluid under the influence of mixed convection, radiation, and partial slip conditions when it flows over a vertical exponential shrinking sheet.

The application of hybrid nanofluid is now being employed to augment the efficiency of heat transfer rates. A numerical study was conducted to investigate the flow characteristics of water-based-alumina copper hybrid nanofluids towards a power-law form stretching/shrinking sheet. This study also considered the influence of magnetic, Joule heating, and thermal slip parameters. This study is significant because it advances our understanding of hybrid nanofluids in the presence of magnetic fields, power-law form stretching/shrinking sheet, and heat transfer mechanisms, providing valuable insights for optimizing and innovating thermal management systems in various industrial applications such as polymers, biological fluids, and manufacturing processes like extrusion, plastic and metal forming, and coating processes. The main objective of this study is to examine the impact of specific attributes, including suction and thermal slip parameters on temperature and velocity profiles. In addition, this exploration examined the reduced skin friction and reduced heat transfer in relation to the solid volume fraction copper and magnetic effects on shrinkage sheet and thermal slip parameter on suction effect. To facilitate the conversion of a nonlinear partial differential equation into a collection of ordinary differential equations, it is necessary to incorporate suitable similarity variables into the transformation procedure. The MATLAB bvp4c solver application is utilized in the conclusion process to solve ordinary differential equations. No solution was found in the sort of when , and . As the intensity of the Eckert number increases, the temperature profile and boundary layer thickness also increase. The reduced heat transfer rate upsurged in both solutions for solid volume fraction copper for shrinking sheet, while the opposite actions can be noticed in both solutions for thermal slip parameter for suction effect. Finally, the study conducted an analysis to identify two distinct solutions for shrinking sheet and suction zone, while considering different parameter values for the copper volume fractions, magnetic and thermal slip condition effect.

In this study, the dual solutions of two-dimensional hybrid nanofluid flow and heat transfer past a porous medium permeable shrinking sheet with influence of heat generation/absorption and convective boundary condition were examined using the bvp4c solver with the MATLAB computational framework. The utilization of two-dimensional hybrid nanofluid flow and heat transfer in the presence of a porous medium permeable shrinking sheet, considering the effects of heat generation/absorption and convective boundary conditions, has wide-ranging applications in industries such as cooling systems, aerospace, chemical engineering, biomedical applications, energy systems, microfluidics, automotive thermal management, industrial drying, and nuclear reactor cooling, etc. These applications employ the improved thermal characteristics of hybrid nanofluids and the efficient heat control offered by porous media and convective boundary conditions. The governing partial differential equations (PDEs) are transformed into a collection of higher-order ordinary differential equations (ODEs) together with their corresponding boundary conditions. These equations are subsequently shown both numerically and graphically. The main objective of this inquiry is to examine the relationship between the solid volume fraction of copper and the permeability parameter of the porous medium, specifically focusing on the values of f''(0) and - (0) according to the suction effect. The current research has also integrated the temperature and velocity profile of a hybrid nanofluid flow, which corresponds to the influence of porous medium permeability, heat generation/absorption, and convective boundary condition. Dual solutions are acquired by certain combinations of parameters. Non unique solutions are obtained when the critical point reaches , for suction effects. Increasing the intensity of heat generation absorption and convective boundary condition leads to a more pronounced temperature profile and thicker boundary layer.