Advanced study of wavy dynamical behavior of suspended living organisms in generalized nanomaterials with magnetic and buoyancy effects.

This study presents a novel mathematical framework to investigate the undulating flow of a Cross fluid containing gyrotactic microorganisms under bioconvection. The problem is important because microorganism-induced density gradients can significantly influence momentum, heat, and mass transport in biological and engineering systems, such as microfluidic devices and nanofluid-based thermal management. The novelty of this work lies in simultaneously analyzing the effects of key parameters Peclet number, Rayleigh number, and magnetic field strength on drag force, heat and solute transport rates, and microorganism density, extending beyond previous studies that considered only partial effects. The transformed nonlinear boundary value problem is solved numerically using an improved bvp4c algorithm, which enhances computational stability and accuracy compared to conventional approaches. The results indicate that increasing the Péclet and Rayleigh numbers leads to a notable reduction in drag force and in the rates of heat and mass transfer. Moreover, higher values of the magnetic field parameter and Rayleigh number cause a significant decrease in the Sherwood number, demonstrating their strong influence on solute transport. The outcomes are presented through detailed graphical analyses and validated by comparison with previously published studies, confirming the reliability of the improved numerical method and the consistency of the physical model.