A hybrid numerical–ridge regression study of Carreau fluid flow with thermal radiation and the stability of dual solutions

This work introduces a novel hybrid approach for analyzing the boundary layer flow, heat, and mass transfer of a Carreau fluid over a shrinking surface under the influence of thermal radiation. Employing Lie group analysis and appropriate similarity transformations, we reduce the governing nonlinear partial differential equations to a system of coupled ordinary differential equations. The resulting equations are numerically solved with MATLAB's built-in boundary value solver. The results reveal the existence of dual solutions within particular ranges of the shrinking and suction parameters. A subsequent stability analysis, based on the sign of the least eigenvalue, confirms that only the upper branch of the solution is physically stable and realizable. Furthermore, to overcome the computational cost of repeated numerical simulations and to enable rapid estimation of heat transfer characteristics across a specific parameter space, a quadratic ridge regression framework is developed to predict the Nusselt number. By integrating polynomial expansion with regularization, this approach ensures reliable predictions while preserving physically meaningful interactions among governing parameters. By combining traditional numerical methods with this data-driven predictive model, this study provides a powerful tool for modeling complex non-Newtonian fluid behavior. This methodology offers both analytical insight and the computational speed required for real-time simulations and system optimization in fields, such as polymer manufacturing, thermal management, and biomedical engineering.