Magneto-bioconvective Oldroyd-B nanofluid flow over a porous stretching surface with motile microorganisms

• A new multiphysics model integrates viscoelasticity, MHD, bioconvection, and entropy generation in nanofluid flow. • The Oldroyd-B fluid with gyrotactic microorganisms flows over a porous stretching surface simulating reactor walls. • The Keller box method is applied to solve the nonlinear system with thermal, solutal, and motile species interactions. • Entropy generation increases with stronger magnetic field and Brownian motion but reduces with fluid relaxation. • Findings provide guidance for optimizing non-Newtonian nanofluid transport in chemical and biological heat transfer devices. This study numerically analyzes MHD mixed convection and bioconvection in an Oldroyd-B nanofluid with motile microorganisms over a porous stretching surface, incorporating entropy generation to assess transport behavior. The governing momentum, energy, solute concentration, and microorganism conservation partial differential equations (PDEs) are transformed into Ordinary differential equations (ODEs) via similarity variables and solved using a finite-difference Keller-box method with Newton–Raphson iteration approach. Key dimensionless parameters, including magnetic field strength (M), Brownian motion factor (Nb), thermophoresis (Nt), bioconvection Péclet number (Pe), bio-convection Lewis number (Lb), Entropy optimization (Ec), the fluid’s relaxation (λ₁) and retardation (λ₂) times, are varied to assess their influence on the velocity (f′ ( η ) ) , temperature ( θ ( η ) ) , nanoparticle concentration ( φ ( η ) ) and microorganism distribution ( χ ( η ) ) profiles. The results show that imposing a stronger magnetic field (M) significantly slows the flow and thickens the thermal boundary layer, raising fluid temperatures, while enhancing nanoparticle and microorganism concentrations near the surface due to suppressed convection. The microorganism distribution ( χ ( η ) ) and temperature rate achieved maximum at Ec=0.20 ( 0 . 01 ≤ Ec ≥ 0 . 20 ), and Pr=10.0 ( 1 ≤ Pr ≥ 10 ), respectively. Larger values of Pe (stronger bioconvection) and higher microorganism concentrations enhance mixing and heat transfer, producing cooler fluid near the wall and steeper microorganism gradients. The Oldroyd-B parameters exhibit significant effects: increasing relaxation time λ₁ reduces velocity and nanoparticle distribution but raises temperature due to elastic energy storage, while increasing retardation time λ₂ further suppresses flow and diffusion. These findings provide important insights into optimizing thermal performance and species transport in bioreactors by tuning magnetic and bioconvection parameters in viscoelastic nanofluid systems.