An integral equation method in conformal mapping of regions with circular slit

In traditional integral equation methods, the calculation of singular integrals often leads to numerical difficulties, especially when dealing with complex regions containing slits. To address the problems of mapping distortion and integration difficulties, this paper proposes a novel method that combines a premap function with the generalized Neumann kernel integral equation method, aimed at simulating irrotational planar flow with arc-shaped obstacles. Using a premap function based on the Joukowski transformation, a complex region is mapped to a regular region with smooth boundaries, significantly improving numerical stability and solution accuracy. An iterative algorithm is developed in conjunction with the integral equation method to simulate the flow characteristics in complex regions. Numerical simulations show that the method efficiently and stably handles flow fields in multi-connected regions, providing a reliable tool for applications in engineering and physical sciences.