The present work concentrates on the mixed convective flow of Casson-Williamson nanoliquid over a rough, vertical, slender cylinder. The impact of bioconvection by the movement of oxytactic microbes and an inclined magnetic field is discussed. The diffusion of liquid oxygen in conjunction with nanoparticles is taken into account. A set of coupled, highly nonlinear partial differential equations is used to model the proposed problem. The nonsimilar transformations are used to reduce them into a set of dimensionless equations. The quasilinearization technique and an implicit finite difference scheme are used for mathematical simplification. The effects of many important parameters namely, mixed convection parameter \(\lambda \left( { - 2 \le \lambda \le 10} \right),\) Williamson parameter \(W\left( {0 \le W \le 2} \right),\) Brownian motion parameter \(Nb\left( {0.1 \le Nb \le 0.6} \right),\) Magnetic field \(M\left( {0 \le M \le 0.6} \right),\) roughness parameter \(\alpha \left( {0.01 \le \varepsilon \le 0.1} \right),\) frequency parameter \(n\left( {10 \le n \le 50} \right),\) bio-convection’s Lewis number \(Lb\left( {0.1 \le Lb \le 0.5} \right),\) Rayleigh number \(Rb\left( {0.1 \le Rb \le 1.0} \right),\) Lewis number \(Le\left( {1 \le Le \le 10} \right),\) Schmidt number \(Sc\left( {1 \le Sc \le 3} \right),\) and Peclet number \(Pe\left( {1 \le Pe \le 5} \right)\) on the flow, heat, and mass transfer characteristics are investigated and illustrated through graphs. The skin friction coefficient increases by approximately 14% and 75% when the roughness parameter escalates from 0.01 to 0.05 and the magnetic parameter changes from 0 to 0.3, respectively. The coefficient of heat transfer upsurges for the roughness parameter and possesses lower values against the magnetic parameter. The microbial density increases by about 40% as the Peclet number rises from 1 to 2. The mass transfer of liquid oxygen is significantly greater with elevated values of the Peclet number and the bioconvection Lewis number. The fluid velocity for the Casson nanofluid is greater than that of the Casson-Williamson nanofluid, while the liquid temperature exhibits the opposite behaviour. The results show a high degree of concordance compared to previously published works. The present model may solve various biological, bioengineering, biomedical, geophysical activities, architectural thermal insulation, and ecological problems.
