Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/328

§ 210 permissible in a propeller we require to somewhat extend our knowledge of the dynamics of the periptery.

At first sight it might be supposed that the blades of a propeller in their helical paths are related in a similar manner to a number of superposed aeroplanes, and the law of maximum proximity will be the same in both cases. Such is not the case. Where we have to deal with a battery of superposed planes or aerofoils, whether vertically over one another, as in Fig. 131 (α), or (in order to better simulate the conditions) like a flight of steps (Fig. 131 (b)), the supporting reaction is continually derived from the virgin fluid, and the line of pressure reaction of any plane, or any component of it, only cuts the path of that plane once. In the case of the propeller, on the contrary, the component of the pressure reaction of any blade in the line of motion cuts the paths of that blade an in- definite number of times. We have here to deal with a fact new to our theory.

Let us suppose that we substitute for our propeller blade some device that acts directly on the fluid without involving the complexity of the cyclic or peripteral motion, and let us stipulate that this hypothetical device produce the same total reaction with the same expenditure of energy as the original aerofoil or propeller blade.

On the Newtonian basis we know (§ 3) that if $$W$$ be the total reaction, $$m$$ the mass per second of the fluid dealt with, and $$v$$ be the velocity imparted in the direction of the reaction—

$$W = m\ v$$ or  $$v = \frac{W}{m} .$$

But the energy per second $$= \frac{m\ v^2}{2}$$

Energy per second $= \frac{W^2}{2\ m}$ or if $W$ is constant, energy per second $\propto \frac{1}{m}$