A general solution of Oseen equations

Abstract

Lamb [1945. Hydrodynamics, sixth ed. Cambridge University Press, Cambridge, NewYork.] suggested a solution of Oseen [1927. Hydrodynamic. Akademische Verlag, Leipzig.] equations. However, this solution is not a general solution. We propose a new form of solution of Oseen equations and show that it is indeed a general solution. An example of an Oseenlet is discussed. We derive a necessary and sufficient condition for a divergence-free vector to represent the velocity in an Oseen flow.

A further note on the force discrepancy for wing theory in Euler flow

“…This is supported by the far-field Oseen flow description, for which the general Oseen flow representation contains singular and coupled inviscid-viscous terms. Even for large Reynolds number, the viscous terms cannot be assumed to be bounded and small [7,24]. This view is supported in this paper, where we find that these viscous terms are necessary not only for the correct evaluation of the normal force as was shown in the author's previous papers [9,6], but also for the correct evaluation of the axial force.…”

“…Chadwick [7] gives a far-field expansion for the Oseen velocity and demonstrates that in general the velocity decomposition into an inviscid part and a viscous part, the LambGoldstein velocity decomposition, does not hold, and instead inviscid and viscous terms are coupled together. This is also found by Venkatalaxmi et al [24] who demonstrated that the Lamb-Goldstein velocity decompostion does not produce a general solution, and propose a general solution such that the viscous and potential velocity parts are coupled in a similar way as given by the expressions for the oseenlets [19]. The coupled viscous terms provide a significant lift contribution as demonstrated in the case of lift on a slender body [8] and on a wing [9].…”

Application of a vortex tracking method to the piston-like behaviour in a semi-entrained vertical gap

“…This requires a mathematical model for the decay, and one such model is due to Oseen, see e.g. [18]. However, any substantial decay of single Oseen vortices that we investigated takes in the order of ten or more periods, while the double vortices seem from informal physical experiments to dissipate basically within a period.…”

A Novel Application of the Classical Banach Fixed Point Theorem

“…It is also today one of the richly enhanced field of research in Mathematics. The applications of elliptic partial differential equations are almost unrestricted spreading across fields like Fluid Mechanics, Electro-magnetics, Biological systems and finance ( [3][4][5][6][7][8] and the references therein). Though many a times, one would attempt analytical solutions, the real life problems may strictly demand numerical treatment.…”