Gyrotactic and oxytactic bioconvection in Reiner–Rivlin fluid flow past a spherical aerobic granular sludge particle

The conventional wastewater treatment is energy intensive, chemical-based and cannot efficiently eliminate specific contaminants from the waste water, leading to the growing popularity of the biologically-induced processes of aeration and degradation; yet, the bioconvection models proposed so far assume that the host fluid is Newtonian and that there is only one type of microorganism involved, although research indicates that activated sludge shows non-Newtonian behavior and that gyrotactic and oxytactic microorganisms usually exist simultaneously within aerobic reactors—an aspect taken care of by the current model using Reiner–Rivlin non-Newtonian hybrid nanofluid flow near a spherical granule. A model is developed considering variable viscosity of the fluid and buoyancy effects due to nanoparticles, microorganisms. Heat and mass transfer are coupled in the model to predict convective flow applicable for advanced wastewater treatments. Using similarity transformation technique, the non-linear partial differential equations were transformed into a set of ordinary differential equations using a group theory approach. Numerical solutions of the ODEs were obtained and tested for convergence to ensure validity of solutions. Parametric studies have been performed for various combinations of non-dimensional numbers such as Sc, Gr, Re, Le, Pr, and Ra. The local Nusselt number (Nu), skin friction coefficient, and Sherwood number are calculated as the output parameters. The aim of the present investigation is to mathematically determine the impact of non-Newtonian fluid flow behavior, double microbial swimming ability, and mixed nanoparticle suspension on the flow field adjacent to a spherical bio-carrier. It is noted that increasing values of Schmidt and Lewis numbers reduce the concentration boundary layer thickness, whereas increasing Grashof and Rayleigh numbers increase the buoyancy effect on velocity and heat transfer; increasing Reynolds and Prandtl numbers reduce the boundary layers and increase thermal gradients and local Nusselt number. What makes the present investigation novel is the modeling of both gyrotaxis and oxytaxis of microorganisms in the Reiner–Rivlin hybrid nanofluids (Fe3O4–Cu–TiO2). To verify stability and asymptotic accuracy of the computed solutions, step size variation experiments were conducted at η = 0.5 for $$ h = 0.1,0.05, $$ and $$0.025$$ against the reference solution computed for h_ref = 0.001 . This study contributes theoretically to the optimization of wastewater treatments based on biological approaches.