# Conformal Mappings with SymPy: Towards Python-driven Analytical Modeling in Physics

## Abstract¶

This contribution shows how the symbolic computing Python library SymPy can be used to improve flow force modeling due to a Couette-type flow, i.e. a flow of viscous fluid in the region between two bodies, where one body is in tangential motion relative to the other. This motion imposes shear stresses on the fluid and leads to a corresponding fluid flow. The flow forces exerted on the moving component are of interest in many applications, for example in system simulations of electrohydraulic valves. There, an eccentrically mounted cylindrical core (the armature) moves within an oil-filled tube (the polecap), experiencing fluid forces due to the viscous oil. SymPy can help to understand the range of validity as well as the limitations of analytical relations that are commonly used as standard approximations for these type of forces in many leading system simulation tools. In order to motivate these approaches, this contribution elucidates how the velocity of the flow is determined analytically by solving the Stokes equation in an eccentric annulus with a conformal mapping-approach. Afterwards analytical postprocessing leads to the corresponding flow force. The results obtained with SymPy are then checked against full 3D computational fluid dynamics (CFD) simulations. This work concludes with the combination of new Couette flow force approximations and similar results for the known Poiseuille flow (i.e. fluid flow induced by a pressure difference) to derive new relations for a combined Couette-Poiseuille flow force. This article is addressed to natural scientists and engineers that are interested in the application of conformal mappings and Taylor-expansions with the help of SymPy when solving partial differential equations analytically.