The flow of a fluid across a revolving disc has several technical and industrial uses. Examples of rotating disc flows include centrifugal pumps, viscometers, rotors, fans, turbines, and spinning discs. An important technology with implications for numerous treatments utilized in numerous sectors is the use of hybrid nanofluids (HNFs) to accelerate current advancements. Through investigation of ternary nanoparticle impacts on heat transfer (HT) and liquid movement, the thermal properties of tri-HNFs were to be ascertained in this study. Hall current, thermal radiation, and heat dissipation have all been studied in relation to the use of flow-describing equations. The ternary HNFs under research are composed of the nanomolecules aluminum oxide (Al2O3), copper oxide (CuO), silver (Ag), and water (H2O). For a number of significant physical characteristics, the physical situation is represented utilizing the boundary layer investigation, which produces partial differential equations (PDEs). The rheology of the movement is extended and computed in a revolving setting under the assumption that the movement is caused by a rotatingfloppy. Before the solution was found using the finite difference method, complicated generated PDEs were transformed into corresponding ODEs (Keller Box method). A rise in the implicated influencing factors has numerous notable physical impacts that have been seen and recorded. The Keller Box method (KBM) approach is also delivered for simulating the determination of nonlinear system problems faced in developing liquid and supplementary algebraic dynamics domains. The rate of entropy formation rises as the magnetic field parameter and radiation parameter increase. Entropy production rate decreases as the Brinkman number and Hall current parameter become more enriched. The thermal efficiency of ternary HNFs compared to conventional HNFs losses to a low of 4.8% and peaks to 5.2%.

The formation of entropy in a mixed convection Casson nanofluid model with Arhenius activation energy is examined in this paper using magnetohydrodynamics (MHD). The expanding sheet, whose function of sheet velocity is nonlinear, confines the Casson nanofluid. The final equations, which are obtained from the first mathematical formulations, are solved using the MATLAB built-in solver bvp4c. Utilizing similarity conversion, ODEs are converted in their ultimate form. A number of graphs and tabulations are also provided to show the effects of important flow parameters on the results distribution. Slip parameter was shown to increase fluid temperature and decrease entropy formation. On the production of entropy, the Brinkman number and concentration gradient have opposing effects. In the presence of nanoparticles, the Eckert number effect's augmentation of fluid temperature is more significant. Furthermore, a satisfactory agreement is reached when the findings of the current study are compared to those of studies that have been published in the past.

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In this work, a weighted H lder function that approximates a Jacobi polynomial which solves the second order singular Sturm-Liouville equation is discussed. This is generally equivalent to the Jacobean translations and the moduli of smoothness. This paper aims to focus on improving methods of approximation and finding the upper and lower estimates for the degree of approximation in weighted H lder spaces by modifying the modulus of continuity and smoothness. Moreover, some properties for the moduli of smoothness with direct and inverse results are considered.