Solution of a two-dimensional bubble shape in potential flow by the method of fundamental solutions

This paper describes an application of the method of fundamental solutions to steady-state free boundary problems arising in potential flow around deformable bodies. The solution in two-dimensional Cartesian coordinates is represented in terms of the fundamental solution of the Laplace equation together with the first-order polynomial augmentation. The collocation is used for determination of the expansion coefficients. The shape of the free boundary is interpolated in the global sense by parameterisation of its length and use of the cubic radial basis functions with the second-order polynomial augmentation. The components of the normal and curvature are calculated in an analytical way. A special algorithm, based on Bernoulli equation, is used for the iterative reshaping of the free boundary towards its equilibrium position. The algorithm is divided into pressure equilibrium, incompressibility, node relocation, and smoothing steps. A numerical example of a two-dimensional deformed incompressible bubble in potential flow is shown.