r1v1 = r2v2
r1/r2 = v2/v1. (4)
The velocity would be infinite at radius 0, if the current could be
conceived to extend to the axis. Now, if the motion is steady,
H = p1/G + v1²/2g = p2/G + v2²/2g;
= p2/G + r1² + v1²/r2²2g;
(p2- p1)/G = v1²(1 - r1²/r2²)/2g; (5)
p2/G = H - r1²v1²/r2²2g. (6)
Hence the pressure increases from the interior outwards, in a way
indicated by the pressure columns in fig. 36, the curve through the
free surfaces of the pressure columns being, in a radial section, the
quasi-hyperbola of the form xy² = c³. This curve is asymptotic to a
horizontal line, H ft. above the line from which the pressures are
measured, and to the axis of the current.
[Illustration: FIG. 36.]
_Free Circular Vortex._--A free circular vortex is a revolving mass of
water, in which the stream lines are concentric circles, and in which
the total head for each stream line is the same. Hence, if by any slow
radial motion portions of the water strayed from one stream line to
another, they would take freely the velocities proper to their new
positions under the action of the existing fluid pressures only.
For such a current, the motion being horizontal, we have for all the
circular elementary streams
H = p/G + v²/2g = constant;
.: dH = dp/G + v dv/g = 0. (7)
Consider two stream lines at radii r and r + dr (fig. 36). Then in
(2), § 33, [rho] = r and ds = dr,
v² dr/gr + v dv/g = 0,
dv/v = -dr/r,
v [oo] 1/r, (8)
precisely as in a radiating current; and hence the distribution of
pressure is the same, and formulae 5 and 6 are applicable to this
case.
_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
[rho] = r, ds = dr. If the angular velocity is [alpha], then v =
[alpha]r and dv = [alpha]dr. Hence
dH = [alpha]²r dr/g + [alpha]²r dr/g = 2[alpha]²r dr/g.
Comparing this with (1), § 33, and putting dz = 0, because the motion
is horizontal,
dp/G + [alpha]²r dr/g = 2[alpha]²r dr/g,
dp/G = [alpha]²rdr/g,
p/G = [alpha]²/2g + 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 - [alpha]²r1²/2g = p2/G - [alpha]²r2²/2g;
(p2 - p1)/G = [alpha]²(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).
[Illustration: FIG. 37.]