(p2 − p1) / G = α2 (r22 − r12) / 2g = (v22 − v12) / 2g.
(10)
That is, the pressure increases from within outwards in a curve which in radial sections is a parabola, and surfaces of equal pressure are paraboloids of revolution (fig. 37).
| Fig. 37. |
Dissipation of Head in Shock
§ 36. Relation of Pressure and Velocity in a Stream in Steady Motion when the Changes of Section of the Stream are Abrupt.—When a stream changes section abruptly, rotating eddies are formed which dissipate energy. The energy absorbed in producing rotation is at once abstracted from that effective in causing the flow, and sooner or later it is wasted by frictional resistances due to the rapid relative motion of the eddying parts of the fluid. In such cases the work thus expended internally in the fluid is too important to be neglected, and the energy thus lost is commonly termed energy lost in shock. Suppose fig. 38 to represent a stream having such an abrupt change of section. Let AB, CD be normal sections at points where ordinary stream line motion has not been disturbed and where it has been re-established. Let ω, p, v be the area of section, pressure and velocity at AB, and ω1, p1, v1 corresponding quantities at CD. Then if no work were expended internally, and assuming the stream horizontal, we should have
p/G + v2/2g = p1/G + v12/2g.
(1)
But if work is expended in producing irregular eddying motion, the head at the section CD will be diminished.