Conformal Mapping and Bipolar Coordinate for Eccentric Laplace Problems

Abstract

Boundary value problems on the eccentric annulus are quite complex and cannot directly be solved analytically using cartesian or polar coordinates. Many mathematical techniques have been used to solve such a problem by using conformal mapping and bipolar coordinate. In the literature, Carrier and Pearson [Partial differential equation‐theory and technique. New York, Academic Press, 1976, pp 68–71], Muskhelishvili [Some basic problems of the mathematical theory of elasticity. Noordhoff, Groningen; 1953, pp 175–179], Ling [Torsion of an eccentric circular tube, Technical Report, No. 1, Chinese Bureau of Aeronautical Research, 1940], Timoshenko and Goordier [Theory of Elasticity. New York, McGraw‐Hill; 1972, pp 196–202], Shen [Null‐field approach for Laplace problems with circular boundaries using degenerate kernels, Master thesis, National Taiwan Ocean University, Keelung, Taiwan, 2005], Lebedev et al. [Worked Problems in Applied Mathematics. New York, Dover; 1965] have solved this kind of problems using similar techniques. By using transformation in a transformed plane in the complex variable theory, we can obtain the analytical solution easily. We focus on the connection between conformal mapping and curvilinear coordinates, and figure out the relation to take integration by way of mapping in the complex plane. All the transformations and curvilinear coordinates can be unified using the viewpoint of conformal mapping. Their relationship among available methods can be constructed by translation, stretching, rotation and inversion. Finally, an example of eccentric domain is solved by using various mappings and curvilinear coordinates and their relations are linked. Not only geometry transformation is addressed but also the solution of the Laplace equation is obtained.