Bioconvective MHD flow of Williamson nanofluid with swimming microorganisms and cross-diffusion effects induced by nonlinear stretching surface in porous media

• This investigation discovers the mutual encouragement of Nn. • Nar stretching, bioconvection, and cross-diffusion on non-Newtonian flow due to porous medium, a gap often overlooked in prior research. • The character of motile microbes is emphasized in enhancing mass and heat transfer while diminishing nanoparticle sedimentation, enhancing nanofluid stability. • Outcomes show that growing magnetic field strength and Williamson parameter overpower fluid velocity, while Brownian motion and thermal radiation significantly raise temperature distribution. • The converted ODE system is efficiently solved via MATLAB’s bvp4c function and confirmed against benchmark results, donation consistent insight into multiphysics transport mechanisms. Developing heat and mass transfer effectiveness in fluid mechanics is precarious for current engineering, industrial and technological applications. But, nanoparticle sedimentation considerably challenges nanofluid enactment, posturing a major contest to constancy and proficiency. Previous investigations have often unnoticed inclusive analyses of magnetohydrodynamic flow integrating nonlinear stretching, living microorganisms, and cross-diffusion within spongy media. Exploring this gap, the current study examines magnetohydrodynamic flow in Williamson nanofluid with Soret and Dufour due to nonlinear stretched sheet entrenched in a Darcy permeable medium, integrating motile microorganisms and mass transfer. A novel aspect of this work lies in exploring bioconvection driven by motile microorganisms, which alleviates nanoparticle sedimentation and develops thermal and mass transfer enactment. The governing partial differential equations of Williamson nanofluid involving Soret and Dufour are simplified using similarity alterations into ordinary differential equations, resolved mathematically via bvp4c function endorsed against established benchmarks. Fundamental discoveries disclose that enlarged magnetic field strength and Williamson parameter reduce fluid velocity field. Conversely, Brownian motion and thermal radiation raise temperature profiles. These perceptions improvement the understanding of nanofluid dynamics in multifactorial environs, proposing potential uses in systems involving concurrent thermal, magnetic, and hydrodynamic resistor.