Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/132

§ 85 great, may be circumscribed by a circle of finite diameter; that is to say, no finite quantity of energy, however great, will cover the whole field. When the diameter of the filament becomes zero the energy internal to any line of flow also becomes infinite, but so long as we regard the motion as cyclic, we are not entitled to regard the filament as of zero diameter; it is legitimate to suppose the filament of very small diameter, so small as not, by its size, to affect the superposed motion of translation.

Now since a pure cyclic motion round a fixed filament does not result in any displacement of the fluid in translation, its mean velocity in each of the co-ordinate directions of space is zero. Consequently, if such a motion be superposed on one of pure translation, the energy of the combined system is the sum of the separate energies and is infinite. Moreover, this result is independent of the energy of the motion of translation (which is never available except to an external system), for if we take the fluid at rest, at infinity (in the $$x$$ and $$y$$ directions), and the filament to undergo the translation, the problem is unaffected, and we have proved that to generate a cyclic motion about a filament in motion (Fig. 48) requires the same quantity of energy as to generate the same cyclic motion about a filament at rest, and in both cases where the expanse of fluid is infinite the total energy required is infinite also.

§ 86. Two Opposite Cyclic Motions on Translation.—In the case of the superposition of a system consisting of two cyclic motions of equal value and opposite sign, such as that obtained by the interchange of the functions $$\psi$$ and $$\phi$$ in the source and sink system (Fig. 40), the energy is finite, for the system consists of a limited number of squares, and consequently the energy required to generate such a system about two filaments moving uniformly in space is also finite; the resultant stream lines of such a superposition are given in Fig. 50. Such a system possesses a plane of symmetry $$\mathit{A}\;\mathit{A}$$, and the motion of the fluid on either side of this plane will be in nowise affected