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Numerous researchers studied on the properties of hybrid copper-alumina/water nanofluid in order to have better thermal efficiency that could be used in industrial applications. Therefore, the present study accentuates the effect of MHD on Marangoni boundary layer of hybrid copper-alumina/water nanofluid flow over a stretching surface. Furthermore, the governing boundary layer equations which is partial differential equations (PDEs) are transformed into a set of ordinary differential equations (ODEs) using similarity transformations. Then, these problems are solved numerically using shooting method through Maple software. It is shows that the rate of heat transfer decreases with the increase in magnetic parameter. The velocity and temperature profiles, as well as local Nusselt number are observed with the specific parameter namely, magnetic parameter, nanoparticles volume fraction and suction. By increasing the volume fraction for copper and magnetic parameter, the velocity profiles decrease while the temperature profiles increase.

A comprehensive study of magnetohydrodynamics (MHD) hybrid copper-alumina nanofluid flow towards a permeable stretching sheet embedded in porous medium is analyzed analytically. The flow problem is mathematical modeled into nonlinear partial differential equations, which are reduced into ordinary differential equations via similarity transformation. The analytical solutions for momentum and energy equations are found. The effects of the porosity on velocity and temperature profiles are analyzed graphically, while the skin friction coefficient and the local Nusselt number are displayed in data tabulation. The results show that increasing the porosity parameter decrease the skin friction coefficient and heat transfer rate.

The purpose of this study is to investigate the boundary layer of Carreau fluid and heat transfer over an exponentially stretching plate derived in a vertical porous with variable surface thermal flux. The partial differential equations that represent the momentum equation and heat equation are commuted into nonlinear ODEs by applying similarity transformations and results found numerically. The impact of several emerging dimensionless parameters labelled the Weissenberg number (We), the power-law index ( ), Velocity slip ( ), Thermal jump ( ), and Prandtl number ( ) on the velocity profile and heat transfer on the boundary layer are showed in detail. In more detail, also the influence of physical parameters on local skin friction and Sherwood number are studied. The shooting method with the explicit technique is used to find the solution and all results are illustrated graphically and numerically. We noted that by increasing power index, radiation parameter and velocity slip, the velocity profile increases, and the temperature profile decreases. Furthermore, it is deduced that rising the thermal radiation parameter reduces the local Nusselt number.

The 2D steady flow of CNTs nanofluids and heat transfer over the moving plate through a uniform free stream under the effect of magnetohydrodynamics (MHD) is studied. The movement of plate is presumed in the opposite or same direction. A mathematical modelling that is governed by a system of partial differential equations (PDEs) subjected to boundary conditions is transformed into a system of nonlinear ordinary equations (ODEs). An attempt at finding the expected outcomes is successfully executed by solving ODEs system using MATLAB bvp4c solver. The effect of various parameters such as magnetic field, CNTs volume friction and moving parameter on velocity and temperature profile, skin friction, Nusselt number and heat transfer rate are investigated numerically. The results are presented using tabular and graphical illustration. From the results, the increment of magnetic field into the flow will increase in both the skin friction and the heat transfer coefficient. Besides that, non-unique solutions are obtained when the plate moves in the range of and . For comparing the performance of base fluid, kerosene works better than water as a base liquid. We are also showing that SWCNTs is more effective both in the skin friction and the heat transfer coefficient compare to MWCNTs.

This study highlights the hybrid Fe3O4-CoFe2O4/H2O ferrofluid flow and heat transfer with the effects of linear heat generation, magnetic field and suction on a rotating disk. Using the similarity transformation, the mathematical model is simplified and reduced to a similarity set of equations. The bvp4c solver is used for computational analysis as well as the stability analysis procedure. The present model is successfully validated with previous results, and also verified with the fulfillment of the asymptote profiles. Triple solutions are observed within a limited range of testing parameters. The flow progress of Fe3O4-CoFe2O4/H2O is reduced when some changes made by varying the magnetic and suction parameters while a reverse result is obtained for the third solution. Only the suction parameter boosts the thermal progress of Fe3O4-CoFe2O4/H2O. The stability analysis surprisingly shows that two of the solutions have positive smallest eigenvalues and align with the physical results.

The flow between bounded surfaces is known as internal flow. The internal flow between disks has many significant applications, such as gas turbine rotors, rotating machinery, food processing technology, and air cleaning machines. In the current study, the nanofluid flow between two disks, nonpermeable and stationary, and the other permeable, rotating and shrinking, is analysed. The governing partial differential equations and boundary conditions are proposed with the inclusion of radiation and heat generation effects. Then, similarity transformations are utilised in deriving the nonlinear ordinary differential equations and boundary conditions for computation using the bvp4c solver. Multiple solutions are obtained, and only the first solution is stable. The combination Mn-ZnFe2O4/C2H6O2 nanofluid is found to produce the lowest magnitude of skin friction coefficient and the highest heat transfer rate.

The Blasius flow over a movable and permeable plate is envisaged in this study. Water-based hybrid nanofluid is incorporated with the insertion of thermal radiation, suction, and a convectively heated plate. The governing partial differential equations that simulate the fluid model are modified to ordinary differential equations through the implementation of self-similar transformation. A numerical solver known as bvp4c in Matlab is adopted to solve the problem numerically through the finite difference code with the Lobatto IIIa formula. Non-unique solutions are acquirable when the plate and the flow move in a dissimilar direction. As conducting the stability analysis, it is validated that the first solution is stable and reliable. The findings reveal that the imposition of stronger thermal radiation and greater Biot number for convection can lead to a better heat transfer performance. The 2% volume fraction of copper in the 1% volume fraction of alumina nanofluid composition would lead to greater skin friction when the plate is moving oppositely from the flow direction compared to the lesser volume fraction of copper. The boundary layer separation also can be efficiently prevented by composing a 2% copper volume fraction in the 1% alumina-water nanofluid compared to the lesser copper volume fraction.

A computational analysis has been performed for Casson hybrid nanofluid natural convection heat transfer in trapezoidal cavity containing a partially heated at bottom wall, and the incline walls are set to be at the cold temperature while the top wall is supposed to be adiabatic. The hybrid nanoparticles, alumina and copper are dispersed in ethylene glycol and water mixture has been used as a working fluid. The finite element method is used to solve the nondimensional governing partial differentials equations. The numerical results are carried out for different governing parameters such as Casson parameter Rayleigh number and hybrid nanoparticles volume fraction and heated zone length. The influence of these parameters on streamlines, isotherms, velocities and average Nusselt number will be the focus in this study.

Due to the significance of magnetic nanofluids in environmental and biomedical sectors, this study is designed to analyze the available solutions alongside with the flow and thermal behaviours of radiative hybrid ferrofluid flow in a three-dimensional system subjected to the shrinking surface. The case of Fe3O4-CoFe2O4/water is considered in this work. The initial procedure is conducted by reducing the complex model into a system of nonlinear differential equations using similarity transformation technique. The results are generated using the bvp4c package in the Matlab software and graphically presented. The existence of dual solutions leads to the treatment of stability analysis where the first solution is affirmed as the physical solution. Meanwhile, the impact of thermal radiation, magnetic field and suction are also observed for the distributions of thermal rate and skin friction coefficients. These distributions boost with the imposition of magnetic field and suction while a deterioration in thermal rate is observed with the rise of thermal radiation.

Recent developments in the field of fluid dynamics have led to a new interest in stability analysis. The numerical solution obtained from the problems of the flow at the stagnation point, as well as the heat transfer with MHD and thermal radiation effects over a shrinking sheet, is used to carry out a stability analysis. The flow of this problem is considered in nanofluids and Buongiorno's model is employed. The boundary layer equation is obtained by reducing the governing equations to an ordinary differential equation. Partial differential equations are converted to ordinary differential equations using a suitable similarity transformation. The bvp4c simulation on Matlab is then used to solve ordinary differential equations. According to the numerical data, the dual solutions occur in a specific range of α. The parameter α refers to the stretching/shrinking where shrinking (less than 0) is the main reason the dual solution exists. The stability analysis is presented graphically and in tabular form to prove that there are two solutions to the problem and only one of them is stable. As a result, our research shows that the solution is only stable in the first solution, but not in the second.

The intention of this article is to investigate the impact of slip of MHD stagnation point flow over a permeable shrinking/stretching sheet with double stratification in a porous medium. Employing the appropriate similarity transformations and non-dimensional variables, the governing partial differential equations were reduced into a set of nonlinear ordinary differential equations. These equations were solved using shooting method and influence of pertinent variables on velocity, temperature and concentration are computed and analyzed. It was found that the slips have the propensity to control boundary layer flow and as the velocity slip increases, the momentum, thermal, and concentration boundary layer thickness become thinner for velocity slip. Therefore, velocity slip acts as a boost for enhancement of the velocity profile in the boundary layer region, whereas temperature and concentration profiles decelerate with the velocity slip. It is also shown that the skin friction coefficient has decreased as the values of velocity slip increase while the It was found that the slips have the propensity to control boundary layer flow and as the velocity slip increases, the momentum, thermal, and concentration boundary layer thickness become thinner for velocity slip. Therefore, velocity slip acts as a boost for enhancement of the velocity profile in the boundary layer region, whereas temperature and concentration profiles decelerate with the velocity slip. It is also shown that the skin friction coefficient has decreased as the values of velocity slip increase while the local Nusselt number and the local Sherwood number are increasing. A comparison with previous studies available in the literature has been done and found an excellent agreement by comparing the numerical results in two decimal places which supports the validity of the present analysis.

The Tiwari-Das model is used to investigate magnetohydrodynamic stagnation point flow and heat transfer past a nonlinear stretching or shrinking cylinder in nanofluid with viscous dissipation and heat generation using. The partial differential equations, also known as governing equations, were reduced to nonlinear ordinary differential equations using similarity transformation. MATLAB with the bvp4c solver is used for numerical computing. The controlling parameter, such as nanoparticle volume fraction, magnetic, curvature, nonlinear, radiation, and heat generation parameters, as well as Eckert and Grashof numbers, influence the skin friction coefficient, heat transfer rate, velocity, and temperature profiles. The results are presented as graphs to show the influence of the variables studied. In some circumstances of stretching and shrinking cases, dual solutions can be obtained.

Thermal radiation enhances heat transfer, and it is used widely in manufacturing and materials processing applications. Thus, steady two-dimensional boundary layer flow over an exponentially porous shrinking sheet of nanofluids was considered in the influence of thermal radiation related to partial slip boundary conditions and suction. This paper aims to study the nanofluid penetrable flow over an exponentially shrinking sheet with thermal radiation and partial slip. The effects of silver (Ag) nanoparticles with two different types of base fluids named water and kerosene oil are investigated in this study. First, the governing equations and boundary conditions are transformed to a non-linear ordinary differential equation and then solved using bvp4c solver. Using Matlab software, it is found that the dual solution exists in some values from the suction parameter. Furthermore, we identified both nanoparticle volume fraction and suction parameter increase, leading to the rise in velocity profile. Moreover, the suction parameter increases both skin friction coefficient and Nusselt number increase.