Free Spiral Vortex.—As in a radiating and circular current the equations of motion are the same, they will also apply to a vortex in which the motion is compounded of these motions in any proportions, provided the radial component of the motion varies inversely as the radius as in a radial current, and the tangential component varies inversely as the radius as in a free vortex. Then the whole velocity at any point will be inversely proportional to the radius of the point, and the fluid will describe stream lines having a constant inclination to the radius drawn to the axis of the current. That is, the stream lines will be logarithmic spirals. When water is delivered from the circumference of a centrifugal pump or turbine into a chamber, it forms a free vortex of this kind. The water flows spirally outwards, its velocity diminishing and its pressure increasing according to the law stated above, and the head along each spiral stream line is constant.
§ 35. Forced Vortex.—If the law of motion in a rotating current is different from that in a free vortex, some force must be applied to cause the variation of velocity. The simplest case is that of a rotating current in which all the particles have equal angular velocity, as for instance when they are driven round by radiating paddles revolving uniformly. Then in equation (2), § 33, considering two circular stream lines of radii r and r + dr (fig. 37), we have ρ = r, ds = dr. If the angular velocity is α, then v = αr and dv = αdr. Hence
dH = α2r dr/g + α2r dr/g = 2α2r dr/g.
Comparing this with (1), § 33, and putting dz = 0, because the motion is horizontal,
dp/G + α2r dr/g = 2α2r dr/g,
dp/G = α2r dr/g,
p/G = α2/2g + constant.
(9)
Let p1, r1, v1 be the pressure, radius and velocity of one cylindrical section, p2, r2, v2 those of another; then
p1/G − α2r12 / 2g = p2/G − α2r22/2g;