Page:Elementary Text-book of Physics (Anthony, 1897).djvu/149

§ 119] If we consider any stream of incompressible fluid, of which the cross-sections at two points where the velocities of the elements are $$v_{1}$$ and $$v_{2}$$ have respectively the areas $$A_{1}$$ and $$A_{2}$$, we can deduce at once from the condition of continuity

119. Velocity of Efflux.—We shall now apply this principle to discover the velocity of efflux of a liquid from an orifice in the walls of a vessel.

Consider any small portion of the liquid, bounded by stream lines, which we may call a filament. Represent the velocity of the filament at $$B$$ (Fig. 39) by $$v_{1}$$, and at $$C$$ by $$v$$, and the areas of the cross-sections of the filament at the same points by $$A_{1}$$ and $$A$$. We have then, as above, $$A_{1}v_{1} = Av.$$. We assume that the flow has been established for a time sufficiently long for the motion to become steady. The energy of the mass contained in the filament between $$B$$ and $$C$$ is, therefore, constant. Let $$V_{1}$$ represent the potential or the potential energy of unit mass at $$B$$ due to gravity, $$V$$ the potential at $$C$$, and $$d$$ the density of the liquid. The mass that enters at $$B$$ in a unit of time or the rate at which mass enters at $$B$$ is $$dA_{1}v_{1}$$. The rate at which mass goes out at is the equal quantity $$dAv$$. The energy entering at $$B$$ is $$dA_{1}v_{1}(\tfrac{1}{2}v_{1}^2 + V_{1})$$, the energy passing out at $$C$$ is $$dAv (\tfrac{1}{2}v^2 + V).$$

If the pressures at $$B$$ and $$C$$ on unit areas be expressed by $$p_{1}$$ and $$p$$, the rate at which work is done at $$B$$ on the entering mass by the pressure $$p_{1}$$, is $$p_{1}A_{1}v_{1}$$, and at $$C$$ on the outgoing mass is $$pAv.$$ This may be seen by considering the cross-section of the filament at $$C$$. The pressure $$p$$ acting on each unit of area of that cross-section is equivalent to a force $$pA$$, and $$v$$ is the rate at which the cross-section moves forward, so that $$pAv$$ is the rate at which the pressure does work. The energy within the filament remaining constant, the incoming must equal the outgoing energy; therefore