Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/287

we have traced the action and reaction merely from one part of the live stream to another part. It remains to be shown in what manner the motion of the fluid in the region $a\ a\ a$ (Fig. 114) is counterpart to some component resistance in the line of flight not otherwise essential.

Let us suppose a limited stratum of fluid to be dealt with, firstly, by a system of superposed aeroplanes; and secondly, by a kind of honeycomb of curved tubes whose leading orifices point in the direction of flight, and whose trailing or discharge orifices make the same angle with the line of flight as the angle $\beta$ of the aeroplane system. Then, if $W$ be the total weight supported in either case, the resistance in the case of the aeroplanes will be (for small values of $\beta$) $=W\ \beta ,$ but in the case of the curved tubes it will only amount to half this quantity, or $=W\ {\frac {\beta }{2}},$ the operating surfaces in either case being supposed frictionless.

It is therefore evident that the aeroplane involves twice as great a resistance to traction as that aerodynamically necessary,^[1]and from what we know of the Kirchhofl' form of flow we can see that this added traction is employed in generating and maintaining the spurting forward of the fluid round the leading edge, indicated by the lines $a\ a\ a\ a$ in the figure. When the work expended in traction is entirely devoted to diverting the stream, as in the theoretical case of the curved pipe system, then there is no spurting forward of the fluid, and no discontinuity in the system of flow; and on the other hand, the operating surfaces are fully exposed to frictional resistance. "When the stream is brusquely diverted by an aeroplane there is an aerodynamic resistance involved in excess of that necessary to divert the flow, and this, by giving rise to a form of flow of the discontinuous type, diminishes the frictional resistance.

Owing in part to the return current in the dead water region, and in part to the forward motion of the fluid on the front face

1. This is on the basis of ignoring the cyclic reaction; if this be taken into account the aeroplane is at a still greater disadvantage.

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