In this research work, we introduce the influences of the shape of nanoparticles and joule heating in the hydromagnetic flow of hybrid nanofluids between non-parallel plates. A mixture base fluid (H2O (50%)-C2H6O2 (50%)), a hybrid nanofluid containing hybrid nanoparticles (graphene oxide-molybdenum disulfide, GO-MoS2) as nanoparticles, is considered. The non-linear governing equations are reduced into ordinary-differential equations (ODEs) by similarity transformations. The non-linear ordinary-differential equation (ODE) is solved numerically utilizing 4th–5th order Runge-Kutta-Fehlberg (RKF-45) with a shooting method and analytically using the homotopy analysis method (HAM). The effect of the rarefaction parameter, Reynolds number, channel angle, Hartmann number, electric field parameter, and the shape of nanoparticles on fluid velocity and skin friction are discussed and presented in a graphical form. Also, the theoretical results and the effectiveness of the homotopy analysis method (HAM) are confirmed by numerical tests and presented graphically coupled with detailed discussions.

In the field of drilling there is increasing interest in topics such as degradation of drilling tools and estimation of penetration speed, as well as efforts to optimize geometrical parameters and drilling processes. The current study was based on an original experimental setup that estimates the actual operating conditions of drilling tools and proposed mathematical models with and without interactions. These models characterize the penetration speed of a widely used compact polycrystalline diamond (PDC) oil-drilling bit. The special focus of this study was on the cutter penetration bit, with the aim of investigating the influence of four operating variables weight on bit (WOB), bit rotational speed (RPM), cutting angle , and compressive strength Cs on yield maximum penetration rate, using Taguchi's design-of-experiment concepts. In the study, 27 experimental runs based on Taguchi's L27 orthogonal array were performed with signal to-noise (S/N) ratio, analysis of variance (ANOVA), and regression analysis being used, with penetration rate as response variables. From the optimization and experimental analyses conducted, it was observed that WOB3 (160 kgf), RPM3 (152 rpm), 3 (45), and Cs1 (640 kgf/cm 2) had significant influence on penetration rate. The optimal values obtained during the study optimization using the Taguchi approach were validated by confirmation experiments.