Mathematical solutions for coupled nonlinear equations based on bioconvection in MHD Casson nanofluid flow

The mathematical formulation of fluid flow problems often results in coupled nonlinear partial differential equations (PDEs); hence, their solutions remain a challenging task for researchers. The present study offers a solution for the flow differential equations describing a bio-inspired flow field of non-Newtonian fluid with gyrotactic microorganisms. A methanol-based nanofluid with ferrous ferric oxide, copper, and silver nanoparticles was considered in a stretching permeable cylinder. The chemical reaction, activation energy, viscous dissipation, and convective boundary conditions were considered. The Casson fluid, a non-Newtonian fluid model, was used as flowing over a cylinder. The fundamental PDEs were established using boundary layer theory in a cylindrical coordinate system for concentration, mass, momentum, and microorganisms' field. These PDEs were then transformed into nonlinear ODEs by applying transforming variables. ODEs were then numerically solved in MATLAB software using the built-in solver bvp4c algorithm. We established an artificial neural network (ANN) model, incorporating Tan-Sig and Purelin transfer functions, to enhance the accuracy of predicting skin friction coefficient (SFC) values along the surface. The networks were trained using the Levenberg–Marquardt method. Quantitative results show that the ferrous ferric oxide nanofluid is superior in increasing Nusselt number, Sherwood number, velocity, and microorganism density number; silver nanofluid is superior in increasing skin friction coefficient, temperature, and concentration. Interestingly, heat transfer rate decreases with the magnetic and curvature parameters and Eckert number, whereas the skin friction coefficient increases with the magnetic parameter and Darcy–Forchheimer number. The present results are validated with the previous existing studies.