Nonlinear dynamics of triple-stratified bioconvective maxwell nanofluid flow over a porous rotating disk with a darcy–forchheimer effect: Residual error and convergence perspective

Abstract

This study investigates the nonlinear dynamics of a triple-stratified bioconvective Maxwell nanofluid flow over a porous rotating disk subject to Darcy–Forchheimer resistance, accounting for thermal, solutal, and microorganism stratification effects. The governing nonlinear partial differential equations are transformed into dimensionless ordinary differential equations using similarity transformations and solved using both the homotopy analysis method (HAM) and the ND-Solve numerical technique, enabling a robust cross-validation of graphical and tabulated results. The influences of key physical parameters such as Deborah number, Forchheimer inertial drag, porosity, thermal radiation, nanoparticle transport, chemical reactions, and bioconvection parameters on velocities (radial, axial, and tangential), temperature, concentration, microorganism density, entropy generation (EG), Bejan number profiles and also on the crucial engineering parameters skin friction, Nusselt, Sherwood, and motile density numbers are comprehensively analyzed. Particular emphasis is placed on HAM convergence control and residual error analysis, ensuring solution accuracy and stability. The combined analytical–numerical framework provides reliable insights into complex nanobioconvective transport phenomena relevant to advanced thermal, biomedical, and energy systems. This study is novel in integrating triple stratification (thermal, solutal, and microorganism) with bioconvective Maxwell nanofluid flow over a porous rotating disk under Darcy–Forchheimer resistance, a coupling not previously reported in the literature, while also ensuring solution reliability through residual error and convergence analysis using HAM and ND-Solve. It fills key research gaps by providing a validated nonlinear framework that simultaneously captures viscoelastic effects, nanoparticle transport, and microorganism dynamics within a rotating porous medium. The findings have strong real-life relevance to thermal management systems, biomedical transport processes, bioreactors, geothermal and porous energy systems, offering reliable guidance for optimizing complex heat, mass, and bioconvective transport in advanced engineering applications.”