Heat transfer analysis of magnetohydrodynamic sutterby nanofluid using deep neural networks with Bayesian regularization in thin channel formed by counter-rotating rolls

Background The study of non-Newtonian fluids in tiny channels is critical for developing innovations in microfluidic devices and specialized industrial coating processes. Nanofluids, which have improved thermal characteristics, have become especially interesting. Objective This study investigates heat transfer in non-Newtonian nanofluids (N-N-Fs) using a deep neural network with Bayesian regularization (DNN-BRA). The model predicts the Nusselt number behavior of magnetohydrodynamic Sutterby nanofluid (MHD-SNF) in a thin channel formed by counter-rotating rolls. The proposed framework addresses complex fluid dynamics relevant to coating processes, electronic cooling, magnetic separation, and biomedical applications. Methodology The governing partial differential equations (PDEs) are reduced to ordinary differential equations (ODEs) through appropriate dimensionless parameters. Numerical solutions for temperature, velocity, concentration, and microorganism distributions are obtained using a stream function formulation and the Richardson extrapolation-based finite difference method (FDM). Nusselt number datasets are generated across various fluid parameters to train and test the DNN-BRA model. Regression analysis, error histograms, and statistical metrics including the loss function, Theil’s inequality coefficient (TIC), mean absolute deviation (MAD), relative error (RE), mean squared error (MSE), mean absolute error (MAE), root mean squared error (RMSE), and coefficient of determination (R²), are used to evaluate model performance. Key Findings The optimal MSE values of 7.0796 × 10 − 10 , 1.5671 × 10 − 8 , 3.9629 × 10 − 8 , are achieved within 7, 3, 3, and 4 epochs for different cases. Statistical metrics confirm strong convergence and validation, with T I C = 1.3207 × 10 − 5 , M A D = 1.3012 × 10 − 5 , and R 2 =0.99998, demonstrating the robustness and predictive reliability of the DNN-BRA model. The results show that the microorganism profile decreases with Brownian motion and Schmidt number, while velocity, temperature, and microorganism concentration increase with Brownian motion, thermophoresis, Grashof number, bioconvection, and Peclet number. Originality/Value This work presents the Bayesian-regularized AI-numerical framework for modeling of non-Newtonian nanofluids in roll coating.