Novel Approach for Solving Conformal Mapping Problems
Speaker: Dave Giri (Pro-Tech, CA & Dept. of ECE, University of New Mexico, USA)
Title: Novel Approach for Solving Conformal Mapping Problems
Abstract: Conformal (- angle preserving) transformations map a problem in an inconvenient geometry (z plane) into a new plane with convenient geometry. The problem is solved in the mapped plane (w) and the solution is transformed back to the original plane. One of the most celebrated conformal transformations is the Joukowski mapping which transforms a circular cylinder into a family of airfoil shapes. This is a problem in fluid mechanics where one needs to solve for the flow field. Knowing the velocity and pressure of air flow in the circular cylinder case, it can be transformed into the solution around the airfoil. If we know the flow field (velocity and pressure) around the airfoil of interest, the “lift” can then be calculated to complete the problem. The computations are generally performed using numerical methods. Conformal transformation which maps one complex plane into another has been used in many scientific disciplines for solving complex problems. In the context of problems in electromagnetics, a complex physical plane z is mapped into a complex potential plane w via a transformation z = f (w). f (w) is a transcendental, but analytic function of the complex variable w. In this presentation, we describe a way of obtaining the desired inversion w = f -1 (z) with the use of Cauchy Integrals. Although our method can apply to any analytical conformal transformation problem, as an illustrative example, we consider a two dimensional problem of two parallel plates of finite width. If the two plates are conical in shape, ( a 3 D problem), a stereographic projection is possible that converts the 3D problem into a 2 D problem for which conformal transformation becomes useful.