One-parameter lie scaling for tri-hybrid maxwell nanofluid in electro-magnetohydrodynamics field with bio-convection and blowing effects

The growing demand for enhanced heat transfer efficiency in industrial applications has driven significant research into advanced nano-fluids, with tri-hybrid formulations representing the latest frontier in thermal management technology. While conventional fluids exhibit limited thermal conductivity that restricts their cooling capabilities, the incorporation of multiple nanoparticle types offers unprecedented opportunities for performance optimization. Despite extensive research on mono and hybrid nano-fluids, the complex interactions within tri-hybrid systems under combined electromagnetic fields and bio-convective conditions remain inadequately understood, particularly for non-Newtonian Maxwell fluids exhibiting both viscous and elastic properties. The current research introduces a mathematical model aimed at examining the flow of a non-Newtonian electro-magneto-hydrodynamic (EMHD) tri-hybrid Maxwell nano-fluid (THMNF) across a stretching porous surface. This model takes into account various factors, porous medium, including heat transport rate influenced by blowing, radiation, heat source and sink effects, convective boundary conditions, and the presence of microorganisms causing bio-convection. One-parameter Lie group scaling analysis is employed to identify the symmetries and establish similarity transformations that are essential for transforming partial differential equations (PDEs) into coupled ordinary differential equations (ODEs), which are then solved numerically by the bvp4c method in MATLAB. An inclusive review is performed on essential factors, which encompass the Biot coefficient, bio-convection Rayleigh coefficient and non-linear convection factor. This study examines the effects of these factors on different flow distributions, including velocity, thermal, and microorganisms, thereby enhancing the depth and practical significance of the results. The validity and robustness of the findings are established through a comparative analysis of the numerical results reported in the existing literature. Results demonstrate that tri-hybrid nano-fluid achieves superior energy transfer capabilities compared to mono and hybrid nanofluids as 12.2%(Φ1=Φ2=Φ3=0.02), 25.4%(Φ1=Φ2=Φ3=0.04), and 41.3%(Φ1=Φ2=Φ3=0.06). The magnetic parameter Mtfadversely impacts velocity profiles while positively enhancing thermal and microbial concentration distributions. The bioconvection parameter Ωtf effectively control microbial distribution patterns. Biot numbers bi1 and bi3 show substantial enhancement in thermal and microorganism profiles, respectively.