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The aim of this paper is to highlight the output of the investigation on the MHD and radiative flow and thermal characteristics of a non-Newtonian Reiner–Philippoff nanofluid with Brownian and thermophoresis diffusion effects. The model studied is embedded in the Buongiorno theory. This unique model is designed to observe both shear thickening and shear thinning properties on that particular fluid with embedded Brownian and thermophoresis diffusion implications. The proposed model consists of continuity, momentum, energy, and concentration equations constructed using the theoretical assumptions and are reduced to a set of ordinary differential equations (ODEs) before solving it using the bvp4c function in MATLAB software. Two solutions are observed, and their physical significance is justified using the temporal stability analysis. From the standpoint of the Reiner–Philippoff fluid parameter, the skin friction coefficient as well as the heat and mass transfer rates are at maximum for the shear-thickening fluid followed by the Newtonian and shear-thinning fluids. The thermophoresis parameter is noticed to decline the heat and mass transfer rate whereas the Brownian motion parameter boosts the mass transfer rate but decreases the heat transfer rate.

New formulations are produced for problems associated with multiple cracks in the upper part of thermoelectric-bonded materials subjected to remote stress using hypersingular integral equations (HSIEs). The modified complex stress potential function method with the continuity conditions of the resultant electric force and displacement electric function, and temperature and resultant heat flux being continuous across the bonded materials’ interface, is used to develop these HSIEs. The unknown crack opening displacement function, electric current density, and energy flux load are mapped into the square root singularity function using the curved length coordinate method. The new HSIEs for multiple cracks in the upper part of thermoelectric-bonded materials can be obtained by applying the superposition principle. The appropriate quadrature formulas are then used to find stress intensity factors, with the traction along the crack as the right-hand term with the help of the curved length coordinate method. The general solutions of HSIEs for crack problems in thermoelectric-bonded materials are demonstrated with two substitutions and it is strictly confirmed with rigorous proof that: (i) the general solutions of HSIEs reduce to infinite materials if G1=G2, K1=K2, and E1=E2, and the values of the electric parts are α1=α2=0 and λ1=λ2=0; (ii) the general solutions of HSIEs reduce to half-plane materials if G2=0, and the values of α1=α2=0, λ1=λ2=0 and κ2=0. These substitutions also partially validate the general solution derived from this study.

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.

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 paper examines the unsteady separated stagnation point (USSP) flow and thermal progress of Fe3O4–CoFe2O4/H2O on a moving plate subject to the heat generation and MHD effects. The model of the flow includes the boundary layer and energy equations. These equations are then simplified with the aid of similarity variables. The numerical results are generated by the bvp4c function and then presented in graphs and tables. The magnetic and acceleration (strength of the stagnation point flow) parameters are the contributing factors in the augmentation of the skin friction and heat transfer coefficients. However, the enhancement of heat generation parameter up to 10% shows a reduction trend in the thermal rate distribution of Fe3O4–CoFe2O4/H2O. This finding reveals the effectiveness of heat absorption as compared to the heat generation in the thermal flow process. From the stability analysis, the first solution is the physical solution. The streamline for the first solution acts as a normal stagnation point flow, whereas the second solution splits into two regions, proving the occurrence of reverse flow.

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%.

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.