[psi] = m log r, [phi] = m[theta],
[psi] + [phi]i = m log re^(i[theta]) (8)
gives a state of vortex motion, circulating round Oz, called a
straight or columnar vortex.
A single vortex will remain at rest, and cause a velocity at any point
inversely as the distance from the axis and perpendicular to its
direction; analogous to the magnetic field of a straight electric
current.
If other vortices are present, any one may be supposed to move with
the velocity due to the others, the resultant stream-function being
[psi] = [Sigma]m log r = log [Pi]r^m; (9)
the path of a vortex is obtained by equating the value of [psi] at the
vortex to a constant, omitting the r^m of the vortex itself.
When the liquid is bounded by a cylindrical surface, the motion of a
vortex inside may be determined as due to a series of vortex-images,
so arranged as to make the flow zero across the boundary.
For a plane boundary the image is the optical reflection of the
vortex. For example, a pair of equal opposite vortices, moving on a
line parallel to a plane boundary, will have a corresponding pair of
images, forming a rectangle of vortices, and the path of a vortex will
be the Cotes' spiral
r sin 2[theta] = 2a, or x^(-2) + y^(-2) = a^(-2); (10)
this is therefore the path of a single vortex in a right-angled
corner; and generally, if the angle of the corner is [pi]/n, the path
is the Cotes' spiral
r sin n[theta] = na. (11)
A single vortex in a circular cylinder of radius a at a distance c
from the centre will move with the velocity due to an equal opposite
image at a distance a²/c, and so describe a circle with velocity
mc/(a² - c²) in the periodic time 2[pi](a² - c²)/m. (12)
Conjugate functions can be employed also for the motion of liquid in a
thin sheet between two concentric spherical surfaces; the components
of velocity along the meridian and parallel in colatitude [theta] and
longitude [lambda] can be written
d[phi] 1 d[psi] 1 d[psi] d[psi]
-------- = ----------- ---------, ----------- --------- = - --------, (13)
d[theta] sin [theta] d[lambda] sin [theta] d[lambda] d[theta]
and then
[phi] + [psi]i = F(tan ½[theta]·e^([lambda]i)). (14)
28. _Uniplanar Motion of a Liquid due to the Passage of a Cylinder
through it._--A stream-function [psi] must be determined to satisfy
the conditions
[Nabla]²[psi] = 0, throughout the liquid; (1)
[psi] = constant, over any fixed boundary; (2)
d[psi]/ds = normal velocity reversed over a solid boundary, (3)
so that, if the solid is moving with velocity U in the direction Ox,
d[psi]/ds = -Udy/ds, or [psi] + Uy = constant over the moving
cylinder; and [psi] + Uy = [psi]´ is the stream function of the
relative motion of the liquid past the cylinder, and similarly [psi] -
Vx for the component velocity V along Oy; and generally
[psi]´ = [psi] + Uy - Vx (4)
is the relative stream-function, constant over a solid boundary moving
with components U and V of velocity.
If the liquid is stirred up by the rotation R of a cylindrical body,
d[psi]/ds = normal velocity reversed
dx dy
= -Rx-- - Ry--, (5)
ds ds
[psi] + ½R(x² + y²) = [psi]´, (6)
a constant over the boundary; and [psi]´ is the current-function of
the relative motion past the cylinder, but now