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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 G 1 = G 2 , K 1 = K 2 , and E 1 = E 2 , and the values of the electric parts are α 1 = α 2 = 0 and λ 1 = λ 2 = 0 ; (ii) the general solutions of HSIEs reduce to half-plane materials if G 2 = 0 , and the values of α 1 = α 2 = 0 , λ 1 = λ 2 = 0 and κ 2 = 0 . These substitutions also partially validate the general solution derived from this study. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]