期刊
PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES
卷 460, 期 2045, 页码 1433-1452出版社
ROYAL SOC
DOI: 10.1098/rspa.2003.1218
关键词
conformal mapping; non-harmonic functions; nonlinear diffusion; advection-diffusion; electrochemical transport
Conformal mapping has been applied mostly to harmonic functions, i.e. solutions of Laplace's equation. In this paper, it is noted that some other equations are also conformally invariant and thus equally well suited for conformal mapping in two dimensions. In physics, these include steady states of various nonlinear diffusion equations, the advection-diffusion equations for potential flows, and the Nernst-Planck equations for bulk electrochemical transport. Exact solutions for complicated geometries are obtained by conformal mapping to simple geometries in the usual way. Novel examples include nonlinear advection-diffusion layers around absorbing objects and concentration polarizations in electrochemical cells. Although some of these results could be obtained by other methods, such as Boussinesq's streamline coordinates, the present approach is based on a simple unifying principle of more general applicability. It reveals a basic geometrical equivalence of similarity solutions for a broad class of transport processes and paves the way for new applications of conformal mapping, e.g. to non-Laplacian fractal growth.
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