Eccentric cylindrical problems in polymer processing

In manufacturing, the gap between cylinders is rarely perfectly concentric. Moreover, many manufacturers deliberately use cylindrical gap eccentricity to improve processing performance. In polymer processing, fluids are commonly transported through the annular gap between eccentrically arranged cylinders, as observed in pipe extrusion, wire coating, and calendering processes, introducing complex flow behavior and asymmetric transport phenomena. The shape of such an eccentric cylindrical gap cannot be represented exactly using Cartesian or cylindrical coordinates. Instead, bipolar or eccentric cylindrical coordinates can be adopted to describe eccentric cylindrical geometries. These coordinate systems provide an appropriate mathematical framework for modeling eccentric-cylinder problems and analyzing transport phenomena in complex geometries. By using bipolar or eccentric cylindrical coordinates, a more realistic representation of the actual eccentric cylindrical gap can be obtained. In this paper, we review bipolar and eccentric cylindrical coordinates, describe how they are defined and related, and summarize their analytical applications. Representative analytical solutions for velocity profiles, pressure distributions, stress distributions, forces, heat transfer, and other relevant transport characteristics under different eccentricities are discussed. Applications in polymer processing, petroleum engineering, lubrication, and electromagnetics are also reviewed to demonstrate the usefulness of these coordinate systems in different engineering problems . We also describe the limitations of these two coordinate systems and the challenges associated with their implementation. Finally, we identify areas in which analytical solutions remain limited and discuss transient and hybrid analytical–numerical approaches as potential directions for future development.