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CONFORMAL MAPPING

DOI: 10.1615/AtoZ.c.conformal_mapping

Conformal mapping or conformal transformation describes a mapping on a complex plane that preserves the angles between the oriented curves in magnitude and in sense. That is, the images of any two intersecting curves, taken with their corresponding orientation, make the same angle of intersection as the curves, both in magnitude and direction. A mapping that preserves the magnitude of each angle but not necessarily the sense is described as isogonal.

The engineering usefulness of conformal mapping is both in grid generation and in solving certain boundary value problems in two-dimensional potential theory, heat conduction and electrostatic potential by mapping a given complicated region (problem plane) onto a much simpler one (solution plane). An important role in the mapping task is performed by linear transformations, bilinear transformations, the Schwarz Christoffel transformation and other special functions [see Churchill, Brown & Verhey (1974)].

For instance, the problem of calculating the free surfaces of a two-dimensional inviscid fluid jet is extremely complicated analytically because the dynamic and nonlinear boundary conditions depend on the free surface, whose shape must be determined as part of the solution. However, by conformally mapping the physical flow region onto a unit circle, a solution can be found numerically with relative ease (Dias, Elcrat & Trefethen, 1987). Analytic conformal mapping functions between the problem plane and the solution plane only exist for a limited range of problems. For most practical problems, the conformal map is represented by an integral equation that has to be solved numerically. Some numerical solution schemes and the resolution of practical boundary conditions are given in a monograph by Trefethen (1986).

REFERENCES

Churchill, R. V., Brown, J. W., and Verhey, R. F. (1974) Complex Variables and Applications, McGraw Hill, New York.

Dias, F., Elcrat, A. R., and Trefethen, L. (1987) Ideal Jet in Two Dimensions, Journal of Fluid Mechanics, 185, 275.

Trefethen, L. (1986) Numerical Conformal Mapping, North-Holland, Amsterdam.

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