Fluid and Thermodynamics pp 197-269 | Cite as

# An Almanac of Simple Flow Problems of Ideal Fluids

## 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. 3. 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. 6.

## 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## References

- 1.Batchelor, G. K. (1967).
*An introduction to Fluid Mechanics*. Cambridge, UK: Cambridge University Press.Google Scholar - 2.Duncan, W. J., Thom, A. S., & Young, A. D. (1970).
*Mechanics of Fluids*. London: Hodder Arnold.Google Scholar - 3.Fröbe, S. und Wassermann, A.:
*Die Bedeutendsten Mathematiker*, p 112. Marix Verlag, Wiesbaden (2007)Google Scholar - 4.Green, G.: An essay on the application of mathematical analysis to the theories of electricity and magnetism. Nottingham (1828). [Printed for the Author by T. Wheelhouse, Nottingham. (Quarto, vii + 72 pages.)]Google Scholar
- 5.Green, G.: Mathematical Investigations concerning the Laws of the Equilibrium of Fluids analogous to the Electric Fluid, with other similar Researches. Cambridge Philosophical Society, 12 Nov 1832, printed in the Transactions 1833. Quatro, 64 pages.) Vol. III, Part I Communicated by Sir Edward French Bromhead, Bart., M.A., F.R.S.L. and EGoogle Scholar
- 6.Kreyszig, E. (2006).
*Advanced Engineering Mathematics*(9th ed.). Hoboken, NJ, USA: Wiley.Google Scholar - 7.Kundu, P. K., Cohen, I. M., & Dowling, D. R. (2016).
*Fluid Mechanics*(6th ed.). Oxford: Academic Press.Google Scholar - 8.Neumann, C.G.:
*Vorlesungen über Riemann’s Theorie der Abelschen Integrale*(1865)Google Scholar - 9.Neumann, C.G.:
*Hydrodynamische Untersuchungen: nebst einem Anhang über die Probleme der Eelektrostatik und die magnetische Induktion*. Teubner Leipzig (1883)Google Scholar - 10.Popula, L.:
*Mathematik für Ingenieure und Naturwissenschaftler*. Band 3 (3. Auflage) Vieweg, Braunschweig, Wiesbaden (1999)Google Scholar - 11.Popula, L.:
*Mathematik für Ingenieure und Naturwissenschaftler*. Bände 3 (9. Auflage) Vieweg, Braunschweig, Wiesbaden (2000)Google Scholar - 12.Sokolnikoff, I. S., & Redheffer, R. M. (1966).
*Mathematics of Physics and modern Engineering, XI+721 p*(2nd ed.). New York: McGraw-Hill.Google Scholar