Cylindrical and rotational coordinate systems

✍ Scribed by Parry Moon; Domina Eberle Spencer

• Publisher: Elsevier Science
• Year: 1951
• Tongue: English
• Weight: 723 KB
• Volume: 252
• Category: Article
• ISSN: 0016-0032
• DOI: 10.1016/0016-0032(51)91054-x

✦ Synopsis

Conformal transformation of the complex plane has been employed in potential theory since the time of Cauchy and Riemann. A rectangular map of uniformly spaced lines in the w-plane may be regarded as the representation of a uniform field, with equipotentials as one family of lines and flow lines as the orthogonal family. A functional transformation yields a z-plane map, which is conformal to the original map and which again (I) a represents a solution of Laplace's equation.

Note, however, that this well-known method gives a solution of Laplace's equation only for a plane distribution or for a cylindrical distribution where the potential is independent of the distance along the generators of the cylinders. It does not even apply in the symmetrical rotational case, though it is often assumed to be an approximation in this case. Moreover, it gives the solution only for the Dirichlet problem with constant potentials on the boundaries, and is notdirectly applicable to arbitrary potential distributions, to Neumann problems, or to mixed boundary conditions.

These limitations are removed to a great extent if one regards the z-plane map as a new coordinate system. This plane system is made into a cylindrical or a rotational system by translation or rotation. Solutions of the Laplace, Helmholtz, wave, and diffusion equations are then obtained for any boundary conditions if the coordinate system allows separation of variables. The idea is an obvious one but it seems never to have been properly exploited. The present paper outlines this method of finding new coordinate systems and indicates how it can be employed in solving field problems.

PROCEDURE