# An Almanac of Simple Flow Problems of Ideal Fluids

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

## Abstract

A primer on vector analysis lays the mathematical foundation used later in the chapter. Gauss’, Stokes’ laws and the Green identities provide the mathematical prerequisites to determine a vector field from its sources and vortices. These concepts are then applied to potential flows due to a concentrated isotropic source, a doublet, parallel flows around a sphere and flows far away from compact three-dimensional source distributions. It is proved that (1) harmonic velocity fields of potential fields in a three-dimensional simply connected region subject to Neumann boundary condition delivers a unique velocity field; (2) that Kelvin’s energy theorem holds, according to which the rotational motion in a simply connected region has the smaller kinetic energy than any other motion with the same Neumann boundary condition, and (3) the potential obeys the maximum-minimum principle, i.e. its maximum and minimum values are assumed on the boundary of the simply connected domain. Moreover, for steady potential flows around stationary three-dimensional rigid bodies, the force exerted by the fluid on the body vanishes. By contrast, the motion induced force on a body in general time dependent potential flow is given by the virtual added mass. These results change in plane potential flow around a two dimensional finite rigid region because the exterior of such a finite region is not simply connected, as already seen in Chap. . The interior two-dimensional problem, or the flow within the whole plane is a potential flow in a simply connected region. It is best treated with complex variable theory. Except for a few first examples, this technique is reserved to Chap. .

## Keywords

Gauss Stokes laws Green’s identities Vortex free flow fields Potential fields Motion induced force on a body in potential flow Plane flow configurations

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