Invariant solutions for thermal energy transfer in tangent hyperbolic nanofluids via lie groups

The purpose of this research is to comprehend and examine intricate fluid flow processes involving chemically reactive mixed bioconvection, nonlinear thermal radiation, and magnetohydrodynamics (MHD). The significance of these flows is crucial for numerous applications in industrial operations, environmental science, and biomedical engineering. This study aims to characterize the behavior of such systems, optimize their performance, and provide important insights for both theoretical and practical advancements by focusing on tangent hyperbolic nanofluid (THNF) with gyrotactic microorganisms while considering the effects of partial slips. We analyze various types of fluid systems with distinct dynamics influenced by nonlinear thermal radiation, heat generation/absorption effects, Joule heating, thermophoresis, Brownian motion, Soret and Dufour effects, chemical reactions, and the presence of microorganisms. Evaluating these effects not only contributes to scientific knowledge but also yields practical outcomes in areas such as bioengineering and environmental remediation. Our approach involves transforming the highly nonlinear partial differential equations (PDEs) into ordinary differential equations (ODEs) to facilitate numerical computation. This conversion streamlines the analysis and simplifies the process of obtaining numerical solutions. We employ a one-parameter Lie scaling method as a powerful mathematical framework to simplify the equations while retaining their essential characteristics. Through numerical methods and graphical representations, we gain a deeper understanding of system behavior under different slip conditions and parameter variations. The principal results of this research contribute to the advancement of knowledge in fluid mechanics. The major findings show that the power-law index decreases the skin friction by 0.035719 in the no-slip case, while it increases by 0.040138 under momentum slip. Across variations in inclination, mixed convection, bioconvection Rayleigh number, buoyancy parameter, and power-law index, the density of microorganisms exhibits consistently low numerical impact in both motile slip and no-slip cases.