§ 13. Volume of Flow.—Let A (fig. 6) be any ideal plane surface, of area ω, in a stream, normal to the direction of motion, and let V be the velocity of the fluid. Then the volume flowing through the surface A in unit time is
Q = ωV.
(1)
Thus, if the motion is rectilinear, all the particles at any instant in the surface A will be found after one second in a similar surface A′, at a distance V, and as each particle is followed by a continuous thread of other particles, the volume of flow is the right prism AA′ having a base ω and length V.
If the direction of motion makes an angle θ with the normal to the surface, the volume of flow is represented by an oblique prism AA′ (fig. 7), and in that case
Q = ωV cos θ.
| Fig. 7. |
If the velocity varies at different points of the surface, let the surface be divided into very small portions, for each of which the velocity may be regarded as constant. If dω is the area and v, or v cos θ, the normal velocity for this element of the surface, the volume of flow is
Q = ∫ v dω, or ∫ v cos θ dω,
as the case may be.