Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/208

§ 136 founded on the Principle of Work, shows that a given pressure is capable of generating a certain definite velocity in the fluid; thus, representing the pressure of the fluid by its hydrostatic head, the latter gives the height through which a body would require to fall in order to acquire the corresponding velocity. If we arrogate that the converse is true, i.e., that a given velocity is capable of generating the corresponding pressure, and that the conditions present in the case of the Normal Plane are such that this corresponding pressure will be generated, then we obtain the result: $$P = .5\ \rho\ V^2 ,$$ for, if $$s =$$ “head,” we have: $$s = \frac{V^2}{2\ g}$$ and mass whose weight constitutes pressure $$= p\ s ,$$ or pressure $$= g\ \rho\ s = g\ \rho\ \frac{V^2}{2\ g} = .5\ \rho\ V^2 .$$

4. The Helmholtz Kirchhof Method.—This method is based on the theory of Discontinuous Motion (Chap. III., § 97); the solution is only known in the case of a lamina bounded by parallel lines of infinite length; in this case the expression is: $$P = 44\ \rho\ V^2 .$$

The Helmholtz theory of discontinuous motion is in all probability the correct theory of the fluid whose viscosity is vanishingly small; and the above result may therefore be taken as rigidly accurate. It is unfortunate that the mathematical difficulties of this method have only been overcome in a few isolated cases.

To summarise, we have:—

And experimental determinations in ordinary viscous fluids as follows:—