# Function-Theoretical Methods Applied to Plane Potential Flows

Chapter
Part of the Advances in Geophysical and Environmental Mechanics and Mathematics book series (AGEM)

## Abstract

Plane two-dimensional potential flows can be treated by methods of complex variable theory. A complex valued function of a complex variable consists of a real and imaginary part, which represent, respectively, the real velocity potential and stream function of a velocity field, which satisfy the Cauchy-Riemann differential equations. Such a function may be viewed as a conformal (angle preserving) map from the $$z = x + \iota y$$ plane to the $$\zeta = \xi + \iota \eta$$ plane. More generally, if $$\phi \!\!\!\!\phi (z)$$ represents a potential flow in the z-plane around a given body and $$z = h(\zeta )$$ is a conformal map from the z- to the $$\zeta$$-plane, then $$\phi \!\!\!\!\phi (h(\zeta )) = \tilde{\phi \!\!\!\!\phi }(\zeta )$$ describes the potential flow in the $$\zeta$$-plane around the image of the given body. Of immense constructive use is the following obvious extension of Riemann’s theorem that ‘every simply connected domain can conformally be mapped onto any other simply connected domain. The above facts are illustrated by means of a large number of examples: (generalized) stagnation point flows in and around wedges; source, circulation and dipole flows; flows with circulation around cylinders and plates; flows around circular segments and Joukowski and von Kármán-Treffts profiles, etc. The circle theorem of Milne-Thomson achieves the reflection or mirror picture of a source at the periphery of the circle. This property can be used to construct potential jet flows through an orifice (as first solved by Kirchhoff in 1860), or a periodic arrangement of such slits. The chapter ends with an introduction into the theory of Schwarz-Christoffel transformations and the demonstration of its usefulness in a number of hydraulically significant applications as potential flows through elliptical and polygonal pipes, as well as flows over a topographic step or from a bottom channel into a half space, etc.

## Keywords

Complex variable methods Cauchy-Riemann equations Riemann’s theorem Potential flows around Joukowski and von Kármán-Treffts profiles Schwarz-Christoffel transformations and their use in potential flows

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