Mathematical modeling of magneto-bioconvective Eyring–Powell nanofluid with activation energy and gyrotactic effects across various geometries

This work formulates a mathematical framework to analyze the magneto-bioconvective transport of an Eyring–Powell nanofluid containing gyrotactic microorganisms over flat plate, wedge, and stagnation-point geometries embedded in a porous medium. The Eyring–Powell constitutive relation is represented through two material constants, namely the viscoelastic parameter ( λ 1 ) and the nonlinear stretching parameter ( λ 2 ), which determine the fluid's shear response. The model accounts for nonlinear thermal radiation, activation energy, variable thermal conductivity, internal heat generation, Brownian diffusion, thermophoretic motion, and chemical reaction effects. By applying similarity transformations, the governing nonlinear PDE system is converted into a set of coupled ODEs, which are solved numerically using a fourth-order Runge–Kutta method combined with the shooting technique. Numerical results highlight the strong sensitivity of transport characteristics to governing parameters. For instance, in the flat-plate case, increasing λ 1 from 0.1 to 0.7 enhances the skin-friction coefficient from 1.128 to 1.387, whereas the same variation in λ 2 reduces it slightly from 1.129 to 1.121. For wedge geometry, the local Nusselt number ( N u x ) decreases from 0.6164 to 0.6112 as λ 1 increases, while in stagnation-point flow, the Sherwood number ( S h x ) falls from 0.4838 to 0.4741. A rise in the magnetic parameter ( M ) from 0.2 to 0.6 lowers N u x by about 0.002 and simultaneously raises the skin-friction coefficient by approximately 0.21 in the plate configuration. Activation energy ( E ) significantly increases solutal concentration and microorganism density by more than 6%, whereas the reaction-rate constant ( K r ) produces a nearly equivalent reduction. Brownian motion ( Nb ) and thermophoresis ( Nt ) elevate fluid temperature while decreasing S h x by over 0.01, and higher Peclet ( Pe ) and Schmidt ( Sc ) numbers reduce microorganism density and solute concentration by 5–10%. Overall, the developed formulation provides a predictive tool for applications such as bio-convective heat exchangers, microfluidic systems, and porous-medium transport under magnetic environments. • A unified model is formulated for magneto-bioconvective Eyring–Powell nanofluid flow with microorganisms across flat plate, wedge, and stagnation-point geometries, capturing the influence of activation energy, nonlinear thermal radiation, chemical reactions, and porous media effects. • The use of three geometries enables direct comparison of transport performance: plate flow enhances wall shear but reduces heat flux under magnetic effects, wedge flow shows modest heat transfer reduction, while stagnation-point flow exhibits the most significant decline in solutal transport, making plate geometry comparatively more efficient. • Numerical analysis using Runge–Kutta with shooting demonstrates opposite rheological influences of Eyring–Powell parameters: elasticity ( λ 1 ) strengthens drag while nonlinear shear ( λ 2 ) weakens it. Magnetic fields consistently increase wall shear but suppress heat transport due to Lorentz force effects. • Activation energy enhances solutal concentration and microorganism density by over 6%, while the chemical reaction rate produces a comparable reduction, highlighting the competing roles of molecular resistance and reaction kinetics in nanoparticle and microorganism transport. • The results provide engineering insights for optimizing bio-convective heat exchangers, porous media reactors, and microfluidic systems, where careful selection of geometry, magnetic field strength, and kinetic parameters is crucial for controlling coupled momentum, heat, and mass transport.