输运理论中某些非调和函数的共形映射
Conformal mapping of some non-harmonic functions in transport theory
摘要：
保角映射主要应用于调和函数，即拉普拉斯方程的解。在本文中，它指出，其他一些方程也是保形不变的，因此同样适合于保形映射在二维。在物理学中，这些包括各种非线性扩散方程的稳态，势流的对流扩散方程，以及体电化学输运的能斯特-普朗克方程。复杂几何的精确解通常是通过保角变换到简单几何得到的。新的例子包括非线性对流扩散层周围的吸收对象和浓度极化的电化学电池。虽然这些结果中的一些可以通过其他方法，如Boussinesq的流线坐标，本方法是基于一个简单的统一的原则，更普遍的适用性。它揭示了一个基本的几何等价的相似性解决方案的广泛的一类运输过程，并铺平了道路，为新的应用保角映射，例如非拉普拉斯分形增长。
英文摘要：
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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