Python implementation of the hybrid analytical and numerical method for analytical conversion of numerical thermofluidic boundary layer solutions

The primary objective of this study is to introduce a novel framework for numerically solving the governing equations of boundary layer theory. In this approach, the equations are first solved using an arbitrary numerical method, and the resulting data are subsequently transformed into polynomial-based analytical expressions. This hybrid analytical and numerical method (HAN method) offers a significant advancement by eliminating the reliance on traditional analytical or semi-analytical techniques, which often involve substantial complexity and limitations. The HAN method enables researchers to leverage numerical solvers while simultaneously obtaining analytical representations of the solution. This study employs the HAN method to solve novel governing equations for Blasius-type flow with heat transfer, mass transfer, and entropy generation because current governing equations are re-scaled equations, derived for uniform free-stream flow, are distinct from previous stretching-sheet models. Although the re-scaled governing equations represent a significant mathematical innovation, the HAN methodology stands out as the most prominent contribution of this study. The HAN method first obtains high-accuracy numerical solutions, then converts them into compact analytical expressions, enabling rapid parametric analysis of key dimensionless groups—Prandtl, Schmidt, Eckert, and Brinkman numbers. This work provides four key innovations: a dual literature review, the derivation of novel governing equations, a step-by-step HAN solution guide with Python code, and presentation of novel results. The findings show that increasing the Prandtl number thins the thermal boundary layer and increases the Nusselt number, while the Schmidt number thins the concentration boundary layer and increases the Sherwood number. Entropy generation rises significantly with higher Brinkman numbers and diffusive parameters, and the Bejan number increases with the temperature ratio, indicating a shift toward thermal irreversibility. The velocity field remains unaffected by these parameters. Analytical solutions from HAN show excellent agreement with finite-difference numerical results, validating the method's accuracy.