dorsal/arxiv
View SchemaConformal mapping of some non-harmonic functions in transport theory
| Authors | Martin Z. Bazant |
|---|---|
| Categories | |
| ArXiv ID | physics/0302086 |
| URL | https://arxiv.org/abs/physics/0302086 |
Abstract
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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"abstract": "Conformal mapping has been applied mostly to harmonic functions, i.e.\nsolutions of Laplace\u0027s equation. In this paper, it is noted that some other\nequations are also conformally invariant and thus equally well suited for\nconformal mapping in two dimensions. In physics, these include steady states of\nvarious nonlinear diffusion equations, the advection-diffusion equations for\npotential flows, and the Nernst-Planck equations for bulk electrochemical\ntransport. Exact solutions for complicated geometries are obtained by conformal\nmapping to simple geometries in the usual way. Novel examples include nonlinear\nadvection-diffusion layers around absorbing objects and concentration\npolarizations in electrochemical cells. Although some of these results could be\nobtained by other methods, such as Boussinesq\u0027s streamline coordinates, the\npresent approach is based on a simple unifying principle of more general\napplicability. It reveals a basic geometrical equivalence of similarity\nsolutions for a broad class of transport processes and paves the way for new\napplications of conformal mapping, e.g. to non-Laplacian fractal growth.",
"arxiv_id": "physics/0302086",
"authors": [
"Martin Z. Bazant"
],
"categories": [
"physics.chem-ph",
"cond-mat"
],
"title": "Conformal mapping of some non-harmonic functions in transport theory",
"url": "https://arxiv.org/abs/physics/0302086"
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