Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/126

§ 79 case, the cyclic motion being supposed to take place about a filament of negligible diameter; the resultant motion is again found to consist of two distinct systems of flow, one internal and the other external to the surface $$a\ a\ a\ a$$; the field is plotted in full for equal increments of both $$\psi$$ and $$\phi$$.

It may be pointed out here that any systems that individually possess velocity potential must of necessity possess velocity potential in their resultant, for otherwise two irrotational systems would, in combination, possess rotation, which is manifestly impossible.

'''§ 80. Consequences of Inverting $$\boldsymbol{\psi}$$, $$\boldsymbol{\phi}$$ Functions in Special Case. Force at Eight Angles to Motion.'''—In Fig. 48 the curves of $$\psi$$ and $$\phi$$ if interchanged would obviously give the case of a source or sink, the flow being vertical instead of horizontal. In this inverted reading of the diagram we again find two systems of flow; the surface of separation $$e\ e\ e\ e$$ passes away to infinity, and has parallel asymptotes. It is frequently convenient when reading any flow diagram to temporarily suppose the functions inverted in this way.

A remarkable and important fact in connection with a cyclic system with superposed translation is the existence of a reaction or force at right angles to the direction of motion, such force in the case represented in Fig. 48 being an upicard force acting on the filament, that is to say, a downward force must be applied to the filament in order that the motion as a steady state should be stable. Where the fluid is bounded externally the force must be supposed to act between the external boundary and the filament or such other body as constitutes the inner boundary.

The necessity for this applied force may be demonstrated in several ways, but it is in the first place necessary to consider the distribution of kinetic energy and pressure in the region occupied by the field of flow.

§ 81. Kinetic Energy.—The expression for the kinetic energy of any dynamic system is $$\tfrac{1}{2}mr^2$$, where $$m$$ is the mass and $$r =$$