Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/129

Rh tension whatever, and it leads to some simplification from a physical point of view to make this assumption and deal with the tension system that results. The consequences are the same whether the superposed pressure be taken account of in the ideal fluid, or whether it be regarded merely as a mechanical detail necessary to carrying the theory into the realms of reality.

We already know that the kinetic energy varies everywhere as $$v^2$$, and we now have it that the tension also varies everywhere as $$v^2$$ (pressure and tension being the same quantity but of reversed sign), consequently the tension on the fluid is everywhere proportional to the kinetic energy, that is the total tension on each element of the $$\psi$$, $$\phi$$ plotting is constant.

In the interpretation of this and the corresponding result as to energy the two-dimensional diagram must be regarded as consisting of a slice of unit thickness, the energy increment being that contained in the element consisting of the volume cut off by adjacent surfaces, the corresponding tension being measured over the surface of the element.

§ 83. Application of the Theorem of Energy.—A simple example of the application of the energy theorem is found in the case of a circular cylinder of infinite length in steady motion in an infinite region containing fluid.

Let Fig. 49 represent the cylinder in cross-section with the external field plotted for $$\psi$$ and $$\phi$$ with respect to space. Let the cylinder be supposed to consist of a thin shell filled with fluid having the necessary motion of translation only; then let the $$\psi$$, $$\phi$$ lines be plotted for the fluid within the cylinder as shown. Now if we count the complete squares within a quadrant, internal and external to the cylinder, the number is equal; further, for every part of a square internal to the sin-face there is a corresponding part external to the surface, and these fractional squares may be made as unimportant as we please by choosing increments of $$\psi$$ and $$\phi$$ small enough, consequently the energy external to the