d[zeta] [Omega]1 [Omega]2
------- = 0, -------- = --------, (27)
dt [xi] [eta]
[phi] = [psi] = nt, suppose, (28)
[Omega]1[xi] + [Omega]2 [eta] = [Omega][omega], (29)
d[phi] a² + c² [omega]
------ = [zeta]- ------- -------[zeta], (30)
dt a² - c² [Omega]
d[psi] 2a² [Omega]
------ = - ------- -------[zeta], (31)
dt a² + c² [omega]
a² + c² [omega] 2a² [Omega]
1 - ------- ------- = - ------- -------, (32)
a² - c² [Omega] a² + c² [omega]
/ [omega] a² + c² \² (a² - c²)(9a² - c²)
( ------- - ½ ------- ) = -------------------, (33)
\ [Omega] a² - c² / 4(a² + c²)
and a state of steady motion is impossible when 3a > c > a.
An experiment was devised by Lord Kelvin for demonstrating this, in
which the difference of steadiness was shown of a copper shell filled
with liquid and spun gyroscopically, according as the shell was slightly
oblate or prolate. According to the theory above the stability is
regained when the length is more than three diameters, so that a modern
projectile with a cavity more than three diameters long should fly
steadily when filled with water; while the old-fashioned type, not so
elongated, would be highly unsteady; and for the same reason the gas
bags of a dirigible balloon should be over rather than under three
diameters long.
40. _A Liquid Jet._--By the use of the complex variable and its
conjugate functions, an attempt can be made to give a mathematical
interpretation of problems such as the efflux of water in a jet or of
smoke from a chimney, the discharge through a weir, the flow of water
through the piers of a bridge, or past the side of a ship, the wind
blowing on a sail or aeroplane, or against a wall, or impinging jets of
gas or water; cases where a surface of discontinuity is observable, more
or less distinct, which separates the running stream from the dead water
or air.
Uniplanar motion alone is so far amenable to analysis; the velocity
function [phi] and stream function [psi] are given as conjugate
functions of the coordinates x, y by
w = [f](z) where z = x + yi, w = [phi] + [psi]i, (1)
and then
dw d[phi] d[psi]
-- = ------ + i------ = -u + vi; (2)
dz dx dx
so that, with u = q cos [theta], v = q sin [theta], the function
dz Q Q Q
[zeta] = -Q -- = -------- = ---(u + vi) = --- (cos [theta] + i sin [theta]), (3)
dw (u - vi) q² q
gives [zeta] as a vector representing the reciprocal of the velocity q
in direction and magnitude, in terms of some standard velocity Q.
To determine the motion of a jet which issues from a vessel with plane
walls, the vector [zeta] must be Constructed so as to have a constant
direction [theta] along a plane boundary, and to give a constant skin
velocity over the surface of a jet, where the pressure is constant.