On the solution of a conformal mapping problem by means of Weierstrass functions

TL;DR

This paper presents a novel method for solving a conformal mapping problem involving elliptic functions, with applications to fluid flow in hydraulic structures, using Weierstrass functions and recursive series calculations.

Contribution

The paper introduces a new conformal mapping formula based on Weierstrass functions, including a refined proof of recursive coefficients for the sigma function.

Findings

Derived a simple conformal mapping formula with four parameters

Demonstrated the method's effectiveness through a numerical experiment

Analyzed the limit case as dam width tends to zero

Abstract

The conformal mapping problem for the section of a channel filled with porous material under a rectangular dam onto the upper half-plane is considered. Similar problems arise in computing of fluid flow in hydraulic structures. As a solution method, the representation of Christoffel-Schwartz elliptic integral in terms of Weierstrass functions is used. The calculation is based on Taylor series for the sigma function, the coefficients of which are determined recursively. A simple formula for a conformal mapping is obtained, which depends on four parameters and uses the sigma function. A numerical experiment was carried out for a specific area. The degeneration of the region, which consists in the dam width tending to zero, is considered, and it is shown that the resulting formula has a limit that implements the solution of the limiting problem. A refined proof of Weierstrass recursive formula for the coefficients of Taylor series of the sigma function is presented.