Abstract
The determination of the flow of a viscous fluid along a body of given contour is one of the most important problems of hydrodynamics. When we consider the stationary flow along a body at rest and start from the equations of motion for an incompressible viscous fluid, the solution is required of a system of four partial differential equations with the velocity components and the pressure as variables. This solution has to satisfy the boundary conditions, which for the case are that the velocity of the fluid along the surface of the body is zero, and that at infinity the flow asymptotically approaches to a parallel motion with given constant velocity. In the following we will suppose that this latter velocity has the value V and is directed along the negative x-axis. Until now the rigorous solution of these equations presents unsurpassable difficulties. However, many applications of hydrodynamics do not require a solution of general validity, but a solution for the case in which the internal friction of the fluid is very small and may be put nearly equal to zero. We shall confine ourselves to this case.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Compare f. i. H. Lamb. Hydrodynamics (Cambridge), art. 76, 77, 78; Ph. Frank u. R. VON Mises, Die Differential~ und Integralgleichungen der Mechanik und Physik (Braunschweig 1927) II, p. 775 and seq.
The most important of these papers are: C. W. Oseen, Zur Theorie des Flüssig-keitswiderstandes, Nova Acta Reg. Soc. Scient. Upsaliensis (IV. 4) 1914; Beiträge zur Hydrodynamik I, Ann. d. Physik 46, p. 231, 1915; and in particular: Hydrodynamik (Bd. I der Sammlung Mathematik in Monographien und Lehrbüchern, Leipzig 1927), p. 211 and seq.; N. Zeilon, On potential problems in the theory of fluid resistance, Kungl. Svenska Vetenskapsakademiens Handlingar, III Ser. I: l, 1924; Beiträge zur Theorie des asymp~ totischen Flüssigkeitswiderstandes, Nova Acta Reg. Soc. Scient. Upsaliensis 1927.
In treating this example, and also in the following one (the oblique plate) ZeilON uses a theorem given by Hilbert in order to solve the boundary problem for Ф. This leads to a rather tedious calculation.
In a paper by the present author published in these Proceedings, Vol. 23, p. 1082, 1921, the flow around a cylinder has been calculated by means of a Fourier expansion for the potential. This method does not converge quickly. Limiting the expansion to 4 terms, for the coefficient in the expression (15) for Q was found: 2,36, with 11 terms: 2,30, the rigorous value being 2(π—2) = 2,283. In the paper0020mentioned a diagram of the streamlines etc. has been given.
Compare N. Zeilon, On potential problems, etc., p. 34, table I.
At small values of α the circulation C becomes equal to πaVα. As Zeilon remarks, this is half the value given by the theory of KUTTA and Joukowsky.
Zeilon, Beiträge zur Theorie,… p. 35.
In the following terms of this series also a logarithm occurs.
When the compensational flow u′, v′ is not introduced, we find: K + I = — ρ VQ. This result has been obtained by Lagally; compare M. lagally, Zeitschr. f. angew. Mathematik u. Mechanik, 2, p. 409, 1922.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1995 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Burgers, J.M. (1995). Physics — On Oseen’s theory for the approximate determination of the flow of a fluid with very small friction along a body . In: Nieuwstadt, F.T.M., Steketee, J.A. (eds) Selected Papers of J. M. Burgers. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-0195-0_3
Download citation
DOI: https://doi.org/10.1007/978-94-011-0195-0_3
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-4088-4
Online ISBN: 978-94-011-0195-0
eBook Packages: Springer Book Archive