Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/150

§ 100 matter more than passing attention and to discuss the subject from its controversial aspect.

In brief, Kelvin's objections appear to consist in the following: (1) Any system of discontinuous flow is inconsistent with his (Kelvin's) theorem of least energy, and therefore cannot exist. (2) That a surface of discontinuity in an inviscid fluid (whose physical continuity is unbroken) is essentially unstable and, if formed, will break up. (3) That in a real fluid possessed of viscosity a surface of discontinuity is impossible.

§ 101. Kelvin's Objections Discussed.—It is certainly true that the discontinuous system of flow violates Lord Kelvin's theorem; it is evident, however, that this theorem rests definitely upon the hypothesis of continuity, and it is precisely this hypothesis that Helmholtz has deliberately set aside. Consequently the objection is without weight.

In considering the behaviour of an inviscid fluid a certain ambiguity exists. Since rotation cannot be imparted to or abstracted from the fluid, there may be an infinite variety of possible forms of flow under given boundary conditions which are ordinarily excluded by hypothesis since they cannot be generated from rest. The Kelvin theorem of least energy is proved only for motions that can be generated from rest, and does not of necessity apply to motions that cannot be so produced.

It is conceivable that if a fluid possessed viscosity in a very small degree only, its motions, if generated and continued for a short period of time, would not sensibly depart from the Eulerian form, but if continued for a long time an entirely different system might eventually be evolved. On this basis, which supposes a cumulative change in the form of flow, the inviscid fluid may, after an infinite lapse of time, develop forms of flow quite foreign to the Eulerian theory, and such forms of flow will obviously be independent of Kelvin's theorem. The supposition of an infinite lapse of time merely constitutes an extension of the hypothesis of the perfect fluid, to simulate as far as possible the conditions