The flow of fluids over the boundaries of a rotating disc has many practical uses, including boundary-layer control and separation. Therefore, the aim of this study is to discuss the impact of unsteady magnetohydrodynamics (MHD) hybrid ferrofluid flow over a stretching/shrinking rotating disk. The time-dependent mathematical model is transformed into a set of ordinary differential equations (ODE’s) by using similarity variables. The bvp4c method in the MATLAB platform is utilised in order to solve the present model. Since the occurrence of more than one solution is presentable, an analysis of solution stabilities is conducted. Both solutions were surprisingly found to be stable. Meanwhile, the skin friction coefficient, heat transfer rate—in cooperation with velocity—and temperature profile distributions are examined for the progressing parameters. The findings reveal that the unsteadiness parameter causes the boundary layer thickness of the velocity and temperature distribution profile to decrease. A higher value of magnetic and mass flux parameter lowers the skin friction coefficient. In contrast, the addition of the unsteadiness parameter yields a supportive effect on the heat transfer rate. An increment of the magnetic parameter up to 30% reduces the skin friction coefficient by 15.98% and enhances the heat transfer rate approximately up to 1.88%, significantly. In contrast, the heat transfer is rapidly enhanced by improving the mass flux parameter by almost 20%.

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 $0.5\%\le {\varphi }_1,{\varphi }_2\le 1.5\%$ (Al${}_2$O${}_3$ and Cu volumetric concentration), $0\le K\le 0.2$ (curvature parameter), $2.6<S\le 3.2$ (suction parameter) and $-2.5<\lambda \le 0.5$ (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 ${\varphi }_1=0.5\%$ (alumina) and ${\varphi }_2=1.5\%$ (copper) has the highest skin friction coefficient and heat transfer rate, followed by the hybrid nanofluids with volumetric concentrations $({\varphi }_1=1\%,{\varphi }_2=1\%)$ and $({\varphi }_1=1.5\%,{\varphi }_2=0.5\%)$, respectively. Surprisingly, the flat plate surface abates the separation of boundary layer while it enhances the heat transfer process.

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.

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.

A new mathematical model is developed for the analytical study of two cracks in the upper plane of dissimilar materials under various mechanical loadings, i.e., shear, normal, tearing and mixed stresses with different geometry conditions. This problem is developed into a new mathematical model of hypersingular integral equations (HSIEs) by using the modified complex potentials (MCPs) function and the continuity conditions of the resultant force and displacement with the crack opening displacement (COD) function as the unknown. The newly obtained mathematical model of HSIEs are solved numerically by utilizing the appropriate quadrature formulas. Numerical computations and graphical demonstrations are carried out to observe the profound effect of the elastic constants ratio, mode of stresses and geometry conditions on the dimensionless stress intensity factors (SIFs) at the crack tips.