Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/151

Rh obtaining in the case of the nearly inviscid fluid, discussed further in § 104.

On the second objection, i.e., the supposed instability of the surface of discontinuity in a perfect fluid, we are treading on very different ground, and reference should be made to Kelvin's article. There is certainly nothing to prevent the supposition of the momentary existence of a surface of discontinuity in an inviscid fluid, and it is difficult to see how it can be destroyed, in view of the fact that it contains rotation which by the theorem of Lagrange can never leave the infinitesimal film of fluid that initially constitutes the surface. It is certain that such a system of flow cannot break up into finite vortex rings, for if the rotation be distributed over a finite quantity of fluid in the core of such vortex rings, the theorem of Lagrange has been violated, and if the rotation be confined to a core that is vanishingly small the energy required to create one such ring is infinite.

§ 102. Discussion on Controversy (continued).—On the third objection, as to discontinuity in the case of the real fluid, it is unnecessary to dwell at length. Neither Helmholtz nor his followers could ever have supposed that the discontinuity exists as a surface under actual conditions, but rather as a stratum containing rotation. It has been elsewhere pointed out (§ 20), that in the case of the real fluid the conception of a surface of discontinuity must be looked upon as an abstraction of that which is essential in a somewhat complex phenomenon, and it is this fact that Kelvin appears to overlook; he points out that the surface will, if formed, break up at once into a series of vortex filaments, or vortex rings, and this view is in all probability correct; it may also be found practicable to assess the pressure reduction on the back of a plate on the basis of vortex theory, as suggested in Kelvin's article. It appears, however, to the author that all this may be considered in the light of an extension rather than a controversion of the Helmholtz theory.

In the course of his criticism Kelvin suggests certain cases of Rh