This study presents a comprehensive analytical approach to address the complexities of flow and heat transfer in planar Taylor–Couette systems. Utilizing innovative simplifying assumptions and conversion variables, we analyze the fluid dynamics between two cylinders, where the outer cylinder is hotter, and the inner cylinder rotates at a higher velocity. Employing a cylindrical coordinate system, the research derives the governing equations for continuity, momentum, and energy in two dimensions under steady-state conditions. These nonlinear partial differential equations are transformed into a set of ordinary differential equations (ODEs) using specific assumptions and conversion variables, facilitating a more practical analysis of Taylor–Couette flow. The study leverages two distinct mathematical methods to solve the ODEs, introducing a novel application of a penalty function to replace the pressure term, which is traditionally used in numerical studies. Our findings indicate that with a Reynolds number (Re) of 900 and a Prandtl number (Pr) of 6.9, the dimensionless radial velocity approaches zero, validating the Taylor–Couette flow model. The analysis reveals a significant tangential velocity gradient between the inner and outer cylinders and an efficient heat transfer from the outer to the inner cylinder, with entropy values decreasing radially outward. Quantitative results include calculated Nusselt numbers of 1.58 for the inner cylinder and −0.58 for the outer cylinder, while skin friction coefficients are −0.0049 and −0.0012, respectively. The non-negative entropy values corroborate compliance with the second law of thermodynamics, ensuring the robustness of the results. Additionally, the study delves into the entropy generation, Nusselt number, and skin friction coefficient, offering a holistic view of the Taylor–Couette flow dynamics.
