In the present study, the magnetohydrodynamics (MHD) bio-convective flow and heat transfer of nanofluid, due to the swimming of the gyrotactic micro-organisms over a curved stretched sheet, is examined. In addition, thermophoresis and Brownian motion behaviors are also investigated by assuming slip conditions at the boundary. A non-linear system of partial differential equations (PDEs) is reduced to a system of ordinary differential equations (ODEs). For convergent solutions, the obtained ODE system is solved by the use of the BVP4C routine integrated MATLAB package. In addition, the impacts of different influential parameters on motile micro-organisms, temperature, velocity, and concentration profiles are deliberated. The velocity field is observed to be reduced when the slip parameter increases. As the main results, it is demonstrated that the distribution of motile microorganisms against the curvature parameter decreases significantly. Similarly, it is found that the nanofluid parameters (i.e., Brownian motion and thermophoresis parameters) and the Peclet number reduce the motile micro-organisms’ number. On the other hand, it is evidenced that the motile micro-organisms’ distribution can be improved with an increase in bio-convective Schmidt number.

The effective applications of Casson fluid in drilling processes, biological treatments, food processing, and bio‐engineering activities have caught the interest of a wide range of researchers. The suitable knowledge of heat transfer via non‐Newtonian fluid is essential for the achievement of best quality products in industry. Thus, the three‐dimensional Casson nanofluid flow over a stretching sheet with Arrhenius activation energy and exponential heat source effects is investigated in this paper using a computational process based on iterative power series (IPS) method. To provide useful insights into the physical and dynamic examinations of this topic, convective heat and convective mass boundary conditions are used. The developed model of nonlinear partial differential equations (PDEs) has been transformed into ordinary differential equations (ODEs) using similarity transformations. The numerical solution of the transformed ODEs is obtained by employing the IPS technique combined with shooting iteration approach. The results of this study are validated with the previous studies, and excellent agreements have been obtained. The behavior of various capable physical parameters is analyzed. It is observed that the thermal and concentration fields show an enhancement with respect to the exponential heat source parameter and thermal and concentration Biot numbers. Further, the Arrhenius activation energy parameter has shown a significant effect on the concentration field.

The primary objective of this study is to quantify the rate of entropy generation within the Magnetohydrodynamic (MHD) slip flow system over the inclined needle. Entropy generation is a measure of the irreversibility and inefficiency in the flow process. The slip flow condition at the fluid interface can significantly impact the flow characteristics and heat transfer rates. In the hybrid nanofluid flow, which consists of non-magnetic and magnetic (Al₂O₃ and Fe₃O₄) are nanoparticles,  are considered as the base fluid. Furthermore, the effects of inclined magnetic fields are taken into interpretation. The PDE’s governing equations are converted into ODE’s using similarity transformations and solved by a numerical technique based on BVP4C. The results illustrate that crucial parameter such as the magnetic parameter, mixed convection parameter, nanoparticles of solid volume fractions, and Prandtl numbers are pointedly impacted by momentum and thermal profiles. The entropy and Bejan number also consider being various relationship combined parameters. These analyses protest that raising the magnetic parameter estates an increase in the hybrid nanofluid thermal profile under slip circumstances. Examined magnetic field impact on flow and entropy generation in MHD flows, revealing significant changes in entropy generation due to interaction between magnetic field and nanoparticles. This analysis understands the impact of MHD and slip effects on entropy generation, particularly in the context of the newly emerging 50:50 fluid mixture. Hybrid nanofluids have been shown to have improved thermal conductivity compared to traditional fluids, which can enhance the cooling or heating capabilities of the inclined needle.

The study presents a mathematical model to predict the impact of interfacial nanolayers on mass and heat transfer phenomena in ternary hybrid nanofluid (THNF) flow. It also investigates the effect of TiO ₂ , SiO ₂ and Al ₂ O ₃ in non-Newtonian biological fluids (i.e. blood) using porous surfaces. The study includes uniform transverse magnetic flux (MF) and reduces the nonlinear partial differential equations (PDEs) to ordinary differential equations (ODEs) using similarity transformations then solved analytically and numerically, the analytical method has been established using the Adomian Decomposition Method (ADM) and by the numerical procedure (Explicit Runge Kutta Method), the present findings in specified cases are compared to findings obtained by the Homotopy Analysis Method (HAM)-based Mathematica package and by the previous literature for validation. The research focused on the influence of active parameters on various parameters, numerically and graphically examining their impact. Results show that the velocity profile experienced a 20% enhancement in the channel centerline region when using a 0.5 unsteadiness parameter (α) together with M = 1 as the magnetic parameter value. This enhancement was possible through the 15% reduction in momentum boundary layer thickness. The heat transfer capabilities of ternary hybrid nanofluids containing 2% nanoparticle volume fractions boost the performance by 25% better than pure base fluids.