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 $\lambda$ . We also observed that the bifurcation of the solutions occurs in the region of $\lambda <0$ , 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.

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 ( ${\mathit{Re}}_x^{1/2}{C}_f$ ) and the heat transfer rate at the surface ( ${\mathit{Re}}_x^{-1/2}N{u}_x$ ) enhance with the added Cu nanoparticle ( ${\phi }_2$ ) and for larger magnetic strength ( $M$ ). Moreover, the values of ${\mathit{Re}}_x^{1/2}{C}_f$ decrease, whereas the values of ${\mathit{Re}}_x^{-1/2}N{u}_x$ 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.

The steady flow and heat transfer past a permeable stretching/shrinking surface in a hybrid nanofluid with a convective boundary condition is studied. The governing equations of the problem are transformed to the similarity equations by using similarity transformation technique. The problem is solved numerically using the boundary value problem solver (bvp4c) in Matlab software. The plots of the skin friction coefficient and the local Nusselt number as well as the velocity and temperature profiles for selected parameters are presented. Results show that dual solutions exist for a certain range of the stretching/shrinking and suction parameters. The critical values of these parameters decrease with the increasing of the copper (Cu) nanoparticle volume fractions. It is found that the heat transfer rate for hybrid nanofluid is higher than that for nanofluid for the impermeable stretching surface. It is also found that the increasing of the copper (Cu) nanoparticle volume fractions enhances the skin friction coefficient and reduces the local Nusselt number for the shrinking surface. The rise in Biot number leads to the increment of the temperature at the surface and widen the thermal boundary layer for both branches. A temporal stability analysis is performed to determine the stability of the dual solutions in a long run, and it is revealed that only one of them is stable while the other is unstable.