Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/42

§ 20 and developed by Kirchhoff, Helmboltz, and others, to account for the phenomenon of resistance in fluid motion. The analytical theory, based on the hypothesis of continuity, does not in general lead to results in harmony with experience. All bodies, according to the Eulerian theory, are of streamline form, provided that the hydrostatic pressure of the fluid is sufficient to prevent cavitation; we know that in practice this is not the case.

According to the teaching of Helmboltz and Kirchhoff, a kinetic discontinuity can be treated as if it were a physical discontinuity; that is to say, the contents of the dead-water region can be ignored; and this method of treatment is now generally recognised, although not universally so. The controversial aspect of the subject is discussed at length at the conclusion of Chap. III.

The principal objection to the theory of discontinuity is that in an inviscid fluid a surface of discontinuity involves rotation, and therefore, by a certain theorem of Lagrange, it is a condition that cannot be generated. A further objection sometimes raised is that such a condition as that contemplated would be unstable, and that the surface of discontinuity, even if formed, would break up into a multitude of eddies. Whether this is the case or not in an inviscid fluid, it is certain that in a fluid possessed of viscosity a surface of discontinuity does commence to break up from the instant of its formation; but as this breaking up does not affect the problem in any important degree, the objection in the case of the inviscid fluid is probably also without weight.

In a real fluid a finite difference of velocity on opposite sides of any surface would betoken an infinite tangential force. Consequently the discontinuity becomes a stratum rather than a surface, and the stratum will either be a region in which a velocity gradient exists (§ 31), or it will become the seat of turbulent motion (§ 37), the latter in all probability.

The conception of the discontinuity as a surface and the method involving this conception are in no way affected by these