Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/144

Let the area of the efflux jet be $A$; let s be the “head” of liquid whose density is $\rho$; let $r$ be the efflux velocity.

Let it be assumed that the pressure within the vessel is everywhere due to the hydrostatic head,—that is to say, let us suppose that the motions of the fluid within the vessel do not affect the pressure on its surfaces.

Then $r={\sqrt {2\ gs}}$, and mass of fluid passing out per second $\rho Av=\rho A{\sqrt {2\ gs}}$, or,

Momentum per second $=r\times \rho A{\sqrt {2\ gs}}=2\ \rho Ags,$

which is the reaction on vessel due to the “recoil” of efflux.

But pressure per unit area on wall of vessel at level of aperture $=\rho gs$, or, if $\alpha =$ area of wall of vessel on which pressure is relieved,

$\alpha ={\frac {2\ \rho Ags}{\rho gs}}=2\ A.$

That is to say, on the above assumption the aperture in the wall of the vessel is twice the area of the resulting jet.

When the aperture is a simple hole in the wall of the vessel (Fig. 53, A), the assumption is not strictly accurate, for the pressure in the region surrounding the hole is less than that due to the hydrostatic pressure owing to the converging of the lines of flow, and consequently the actual hole is of less area than that over which the pressure is effectively relieved, and the jet contracts less than the simple theory would indicate.

§ 96. The Borda Nozzle.—The conditions of hypothesis are most nearly conformed to by the Borda re-entrant nozzle (Fig. 53, B), in which the aperture is furnished with a short tube extending inward. Such an arrangement ensures, as closely as is possible in practice, that the pressure on the walls of the vessel is unaffected by the motion of the fluid. Experimenting with a circular cylindrical nozzle, Borda (1766) obtained the result $\alpha =1.94\ A,$ which is in sufficiently close agreement with theory. It is more usual to invert this expression, writing $A=.515\alpha .$

The complete solution of the path of flow at the free surface

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