Background: The study of non-Newtonian fluids in thin channels is crucial for advancing technologies in microfluidic systems and targeted industrial coating processes. Nanofluids, which exhibit enhanced thermal properties, are of particular interest. This paper investigates the complex flow and heat transfer characteristics of a Sutterby nanofluid (SNF) within a thin channel, considering the combined effects of magnetohydrodynamics (MHD), Brownian motion, and bioconvection of microorganisms. Analyzing such systems is essential for optimizing design and performance in relevant engineering applications. Method: The governing non-linear partial differential equations (PDEs) for the flow, heat, concentration, and bioconvection are derived. Using lubrication theory and appropriate dimensionless variables, this system of PDEs is simplified into a more simplified system of ordinary differential equations (ODEs). The resulting nonlinear ODEs are solved numerically using the boundary value problem (BVP) Midrich method in Maple software to ensure accuracy. Furthermore, data for the Nusselt number, extracted from the numerical solutions, are used to train an artificial neural network (ANN) model based on the Levenberg–Marquardt algorithm. The performance and predictive capability of this ANN model are rigorously evaluated to confirm its robustness for capturing the system’s non-linear behavior. Results: The numerical solutions are analyzed to understand the variations in velocity, temperature, concentration, and microorganism profiles under the influence of various physical parameters. The results demonstrate that the non-Newtonian rheology of the Sutterby nanofluid is significantly influenced by Brownian motion, thermophoresis, bioconvection parameters, and magnetic field effects. The developed ANN model demonstrates strong predictive capability for the Nusselt number, validating its use for this complex system. These findings provide valuable insights for the design and optimization of microfluidic devices and specialized coating applications in industrial engineering.

This study investigates the flow of non-Newtonian fluids for the fixed constraining boundary in order to construct a mathematical model for an incompressible, non-Newtonian fluid that passes through a little gap between a roll and substrate. By considering suitable dimensionless parameters, the dimensionless form of the governing equations is produced. The fluid flow governing equations have been made simpler with the application of lubrication approximation theory (LAT). The expression for velocity profile, coating thickness, separation force, pressure distribution, roll power input, pressure gradient, and separation point are obtained with the implementation of the perturbation technique. To investigate the impacts of different physical parameters such as Wiesenberger number, force and the dependence of the response factors coating thickness, power input, and roll separation force, sensitivity analysis and an experimental design based on response surface methodology (RSM) are employed. The correlation equation derived from the RSM demonstrates the relationship between the output and input parameters. The high values of coefficient of determination for output responses coating thickness, power input and roll separation force are 99.90, 99.91 and 97.16%, respectively, which demonstrates the validity of our analysis. Moreover, the impacts of involved parameters on the quantities of interest are displayed through various graphs and in tabular form. Moreover, it has been established that an increase in the Weissenberg number correlates with a drop in velocity; conversely, an increase in the embedding factor F is associated with an increase in velocity. Both the factors leads to decreasing the pressure gradient but an increasing for pressure distribution. The coating thickness and separation point diminish as power input and separation force escalate with increasing F. On the other hand, a larger Weissenberg number lowers the separation force while raising the coating thickness, power input, and separation point. Finally, we assure our results by comparing to previously published research. This study addresses a gap in the existing literature by exploring the rheological properties of the Williamson fluid model and integrating numerical methods along with response surface techniques in the coating process.