Chapter II: Front Matter (2)

The naval architect distinguishes between the _stability of form_,
represented by the righting couple W.BM, and the _stability of
ballasting_, represented by W.BG. Ballasted with G at B, the righting
couple when the ship is heeled through [theta] is given by W.BM.
tan[theta]; but if weights inside the ship are raised to bring G above
B, the righting couple is diminished by W.BG.tan[theta], so that the
resultant righting couple is W·GM·tan[theta]. Provided the ship is
designed to float upright at the smallest draft with no load on board,
the stability at any other draft of water can be arranged by the
stowage of the weight, high or low.

19. Proceeding as in § 16 for the determination of the C.P. of an
area, the same argument will show that an inclining couple due to the
movement of a weight P through a distance c will cause the ship to
heel through an angle [theta] about an axis FF´ through F, which is
conjugate to the direction of the movement of P with respect to an
ellipse, not the momental ellipse of the water-line area A, but a
confocal to it, of squared semi-axes

a² - hV/A, b² - hV/A, (1)

h denoting the vertical height BG between C.G. and centre of buoyancy.
The varying direction of the inclining couple Pc may be realized by
swinging the weight P from a crane on the ship, in a circle of radius
c. But if the weight P was lowered on the ship from a crane on shore,
the vessel would sink bodily a distance P/wA if P was deposited over
F; but deposited anywhere else, say over Q on the water-line area, the
ship would turn about a line the antipolar of Q with respect to the
confocal ellipse, parallel to FF´, at a distance FK from F

FK = (k² - hV/A)/FQ sin QFF´ (2)

through an angle [theta] or a slope of one in m, given by

1 P P V
sin [theta] = --- = ----- = --- · -------- FQ sin QFF´, (3)
m wA·FK W Ak² - hV

where k denotes the radius of gyration about FF´ of the water-line
area. Burning the coal on a voyage has the reverse effect on a
steamer.

HYDRODYNAMICS

20. In considering the motion of a fluid we shall suppose it non-viscous, so that whatever the state of motion the stress across any section is normal, and the principle of the normality and thence of the equality of fluid pressure can be employed, as in hydrostatics. The practical problems of fluid motion, which are amenable to mathematical analysis when viscosity is taken into account, are excluded from treatment here, as constituting a separate branch called "hydraulics" (q.v.). Two methods are employed in hydrodynamics, called the Eulerian and Lagrangian, although both are due originally to Leonhard Euler. In the Eulerian method the attention is fixed on a particular point of space, and the change is observed there of pressure, density and velocity, which takes place during the motion; but in the Lagrangian method we follow up a particle of fluid and observe how it changes. The first may be called the statistical method, and the second the historical, according to J. C. Maxwell. The Lagrangian method being employed rarely, we shall confine ourselves to the Eulerian treatment.

_The Eulerian Form of the Equations of Motion._

21. The first equation to be established is the _equation of continuity_, which expresses the fact that the increase of matter within a fixed surface is due to the flow of fluid across the surface into its interior.

In a straight uniform current of fluid of density [rho], flowing with
velocity q, the flow in units of mass per second across a plane area
A, placed in the current with the normal of the plane making an angle
[theta] with the velocity, is [rho]Aq cos [theta], the product of the
density [rho], the area A, and q cos [theta] the component velocity
normal to the plane.

Generally if S denotes any closed surface, fixed in the fluid, M the
mass of the fluid inside it at any time t, and [theta] the angle which
the outward-drawn normal makes with the velocity q at that point,

dM/dt = rate of increase of fluid inside the surface, (1)

= flux across the surface into the interior
_ _
/ /
= - | | [rho]q cos [theta] dS,
_/_/

the integral equation of continuity.

In the Eulerian notation u, v, w denote the components of the velocity
q parallel to the coordinate axes at any point (x, y, z) at the time
t; u, v, w are functions of x, y, z, t, the independent variables; and
d is used here to denote partial differentiation with respect to any
one of these four independent variables, all capable of varying one at
a time.

To transfer the integral equation into the differential equation of
continuity, Green's transformation is required again, namely,
_ _ _ _ _
/ / / /d[xi] d[eta] d[zeta] \ / /
| | | ( ----- + ------ + ------- )dx dy dz = | | (l[xi] + m[eta] + n[zeta]) dS, (2)
_/_/_/ \ dx dy dz / _/_/

or individually
_ _ _ _ _
/ / / d[xi] / /
| | | ----- dx dy dz = | | l[xi] dS,..., (3)
_/_/_/ dx _/_/

where the integrations extend throughout the volume and over the
surface of a closed space S; l, m, n denoting the direction cosines of
the outward-drawn normal at the surface element dS, and [xi], [eta],
[zeta] any continuous functions of x, y, z.

The integral equation of continuity (1) may now be written
_ _ _ _ _
/ / / d[rho] / /
| | | ----- dx dy dz = | | (l[rho]u + m[rho]v + n[rho]w) dS = 0, (4)
_/_/_/ dt _/_/

which becomes by Green's transformation
_ _ _
/ / / /d[rho] d([rho]u) d([rho]v) d([rho]w)\
| | | ( ------ + --------- + --------- + -------- ) dx dy dz = 0, (5)
_/_/_/ \ dt dx dy dz /

leading to the differential equation of continuity when the
integration is removed.

22. The equations of motion can be established in a similar way by considering the rate of increase of momentum in a fixed direction of the fluid inside the surface, and equating it to the momentum generated by the force acting throughout the space S, and by the pressure acting over the surface S.

Taking the fixed direction parallel to the axis of x, the time-rate of
increase of momentum, due to the fluid which crosses the surface, is
_ _ _ _
/ / / /
- | | [rho]uq cos [theta] dS = - | | (l[rho]u² + m[rho]uv + n[rho]uw) dS, (1)
_/_/ _/_/

which by Green's transformation is
_ _ _
/ / / /d([rho]u²) d([rho]uv) d([rho]uw)\
- | | | (---------- + ---------- + ---------- ) dx dy dz. (2)
_/_/_/ \ dx dy dz /

The rate of generation of momentum in the interior of S by the
component of force, X per unit mass, is
_ _ _
/ / /
| | | [rho]X dx dy dz, (3)
_/_/_/

and by the pressure at the surface S is
_ _ _ _ _
/ / / / / dp
- | | lp dS = - | | | -- dx dy dz, (4)
_/_/ _/_/_/ dx

by Green's transformation.

The time rate of increase of momentum of the fluid inside S is
_ _ _
/ / / d([rho]u)
| | | --------- dx dy dz; (5)
_/_/_/ dt

and (5) is the sum of (1), (2), (3), (4), so that
_ _ _
/ / / /d[rho]u d[rho]u² d[rho]uv d[rho]uw dp \
| | | ( ------- + -------- + -------- + -------- - [rho]X + -- ) dx dy dz = 0, (6)
_/_/_/ \ dt dx dy dz dx /

leading to the differential equation of motion

d[rho]u d[rho]u² d[rho]uv, d[rho]uw dp
------- + -------- + -------- + -------- = [rho]X - --, (7)
dt dx dy dz dx

with two similar equations.

The absolute unit of force is employed here, and not the gravitation
unit of hydrostatics; in a numerical application it is assumed that
C.G.S. units are intended.

These equations may be simplified slightly, using the equation of
continuity (5) § 21; for

d[rho]u d[rho]u² d[rho]uv d[rho]uw
------- + -------- + -------- + --------
dt dx dy dz

/du du du du \
= [rho] ( -- + u-- + v-- + w-- )
\dt dx dy dz /

/d[rho] d[rho]u d[rho]v d[rho]w \
+ u ( ------ + ------- + ------- + ------- ), (8)
\ dt dx dy dz /

reducing to the first line, the second line vanishing in consequence
of the equation of continuity; and so the equation of motion may be
written in the more usual form

du du du du 1 dp
-- + u-- + v-- + w-- = X - ----- --, (9)
dt dx dy dz [rho] dx

with the two others

dv dv dv dv 1 dp
-- + u-- + v-- + w-- = Y - ----- --, (10)
dt dx dy dz [rho] dy

dw dw dw dw 1 dp
-- + u-- + v-- + w-- = Z - ----- --. (11)
dt dx dy dz [rho] dz

23. As a rule these equations are established immediately by determining the component acceleration of the fluid particle which is passing through (x, y, z) at the instant t of time considered, and saying that the reversed acceleration or kinetic reaction, combined with the impressed force per unit of mass and pressure-gradient, will according to d'Alembert's principle form a system in equilibrium.

To determine the component acceleration of a particle, suppose F to
denote any function of x, y, z, t, and investigate the time rate of F
for a moving particle; denoting the change by DF/dt,

DF F(x + u[delta]t, y + v[delta]t, z + w[delta]t, t + [delta]t) - F(x, y, z, t)
-- = lt·----------------------------------------------------------------------------
dt [delta]t

dF dF dF dF
= -- + u-- + v-- + w--; (1)
dt dx dy dz

and D/dt is called particle differentiation, because it follows the
rate of change of a particle as it leaves the point x, y, z; but

dF/dt, dF/dx, dF/dy, dF/dz (2)

represent the rate of change of F at the time t, at the point, x, y,
z, fixed in space.

The components of acceleration of a particle of fluid are consequently

Du du du du du
-- = -- + u-- + v-- + w--, (3)
dt dt dx dy dz

Dv dv dv dv dv
-- = -- + u-- + v-- + w--, (4)
dt dt dx dy dz

Dw dw dw dw dw
-- = -- + u-- + v-- + w--, (5)
dt dt dx dy dz

leading to the equations of motion above.

If F (x, y, z, t) = 0 represents the equation of a surface containing
always the same particles of fluid,

DF dF dF dF dF
-- = 0, or -- + u-- + v-- + w-- = 0, (6)
dt dt dx dy dz

which is called the differential equation of the _bounding surface_. A
bounding surface is such that there is no flow of fluid across it, as
expressed by equation (6). The surface always contains the same fluid
inside it, and condition (6) is satisfied over the complete surface,
as well as any part of it.

But turbulence in the motion will vitiate the principle that a
bounding surface will always consist of the same fluid particles, as
we see on the surface of turbulent water.

24. To integrate the equations of motion, suppose the impressed force
is due to a potential V, such that the force in any direction is the
rate of diminution of V, or its downward gradient; and then

X = -dV/dx, Y = -dV/dy, Z = -dV/dz; (1)

and putting

dw dv du dw dv du
-- - -- = 2[xi], -- - -- = 2[eta], -- - -- = 2[zeta], (2)
dy dz dz dx dx dy

d[xi] d[eta] d[zeta]
----- + ------ + ------- = 0, (3)
dx dy dz

the equations of motion may be written

du dH
-- - 2v[zeta] + 2w[eta] + -- = 0, (4)
dt dx

dv dH
-- - 2w[xi] + 2u[zeta] + -- = 0, (5)
dt dy

dw dH
-- - 2u[eta] + 2w[xi] + -- = 0, (6)
dt dz

where
_
/
H = | dp/[rho] + V + ½q², (7)
_/

q² = u² + v² + w², (8)

and the three terms in H may be called the pressure head, potential
head, and head of velocity, when the gravitation unit is employed and
½q² is replaced by ½q²/g.

Eliminating H between (5) and (6)

D[xi] du dv dw / du dv dw \
----- - [xi]-- - [eta]-- - [zeta]-- + [xi]( -- + -- + -- ) = 0, (9)
dt dx dx dx \ dx dy dz /

and combining this with the equation of continuity

1 D[rho] du dv dw
----- ------ + -- + -- + -- = 0, (10)
[rho] dt dx dy dz

we have

D /[xi] \ [xi] du [eta] dv [zeta] dw
-- ( ----- ) - ----- -- - ----- -- - ------ -- = 0, (11)
dt \[rho]/ [rho] dx [rho] dx [rho] dx

with two similar equations.

Putting

[omega]² = [xi]² + [eta]² + [zeta]², (12)

a _vortex line_ is defined to be such that the tangent is in the
direction of [omega], the resultant of [xi], [eta], [zeta], called the
components of molecular rotation. A small sphere of the fluid, if
frozen suddenly, would retain this angular velocity.

If [omega] vanishes throughout the fluid at any instant, equation (11)
shows that it will always be zero, and the fluid motion is then called
_irrotational_; and a function [phi] exists, called the _velocity
function_, such that

udx + vdy + wdz = -d[phi], (13)

and then the velocity in any direction is the space-decrease or
downward gradient of [phi].

25. But in the most general case it is possible to have three
functions [phi], [psi], m of x, y, z, such that

udx + vdy + wdz = -d[phi] - md[psi], (1)

as A. Clebsch has shown, from purely analytical considerations
(_Crelle_, lvi.); and then

d([psi], m) d([psi], m) d([psi], m)
[xi] = ½ -----------, [eta] = ½ -----------, [zeta] = ½ -----------, (2)
d(y, z) d(z, x) d(x, y)

and

d[psi] d[psi] d[psi] dm dm dm
[xi]------ + [eta]------ + [zeta]------ = 0, [xi]-- + [eta]-- + [zeta]-- = 0, (3)
dx dy dz dx dy dz

so that, at any instant, the surfaces over which [psi] and m are
constant intersect in the vortex lines.

Putting

d[phi] d[psi]
H - ------ - m ------ = K, (4)
dt dt
the equations of motion (4), (5), (6) § 24 can be written

dK d([psi],m)
-- - 2u[zeta] + 2w[eta] - ---------- = 0, ..., ...; (5)
dx d(x,t)

and therefore

dK dK dK
[xi]-- + [eta]-- + [zeta]-- = 0. (6)
dx dy dz

Equation (5) becomes, by a rearrangement,

dK d[psi] /dm dm dm dm \
-- - ------ ( -- + u-- + v-- + w-- )
dx dx \dt dx dy dz /

dm / d[psi] d[psi] d[psi] d[psi] \
+ -- ( ------ + u------ + v------ + w------ ) = 0, ..., ..., (7)
dx \ dt dx dy dz /

dK d[psi] Dm dm D[psi]
-- - ------ -- + -- ------ = 0, ..., ..., (8)
dx dx dt dx dt

and as we prove subsequently (§ 37) that the vortex lines are composed
of the same fluid particles throughout the motion, the surface m and
[psi] satisfies the condition of (6) § 23; so that K is uniform
throughout the fluid at any instant, and changes with the time only,
and so may be replaced by F(t).

26. When the motion is _steady_, that is, when the velocity at any
point of space does not change with the time,

dK
-- - 2v[zeta] + 2w[eta] = 0, ..., ... (1)
dx

dK dK dK dK dK dK
[xi]-- + [eta]-- + [zeta]-- = 0, u-- + v-- + w-- = 0, (2)
dx dy dz dx dy dz
and
_
/
K = | dp/[rho] + V + ½q² = H (3)
_/

is constant along a vortex line, and a _stream line_, the path of a
fluid particle, so that the fluid is traversed by a series of H
surfaces, each covered by a network of stream lines and vortex lines;
and if the motion is irrotational H is a constant throughout the
fluid.

Taking the axis of x for an instant in the normal through a point on
the surface H = constant, this makes u = 0, [xi] = 0; and in steady
motion the equations reduce to

dH/d[nu] = 2v[zeta] - 2w[eta] = 2q[omega] sin [theta], (4)

where [theta] is the angle between the stream line and vortex line;
and this holds for their projection on any plane to which d[nu] is
drawn perpendicular.

In plane motion (4) reduces to

dH / dQ q \
----- = 2q[zeta] = q ( -- + --- ), (5)
d[nu] \ dv r /

if r denotes the radius of curvature of the stream line, so that

1 dp dV dH d½q² q²
----- ----- + ----- = ----- - ----- = ---, (6)
[rho] d[nu] d[nu] d[nu] d[nu] r

the normal acceleration.

The osculating plane of a stream line in steady motion contains the
resultant acceleration, the direction ratios of which are

du du du d½q² d½q² dH
u-- + v-- + w-- = ---- - 2v[zeta] + 2w[eta] = ---- - --, ..., (7)
dx dy dz dx dx dx

and when q is stationary, the acceleration is normal to the surface H
= constant, and the stream line is a geodesic.

Calling the sum of the pressure and potential head the statical head,
surfaces of constant statical and dynamical head intersect in lines on
H, and the three surfaces touch where the velocity is stationary.

Equation (3) is called Bernoulli's equation, and may be interpreted as
the balance-sheet of the energy which enters and leaves a given tube
of flow.

If homogeneous liquid is drawn off from a vessel so large that the
motion at the free surface at a distance may be neglected, then
Bernoulli's equation may be written

H = p/[rho] + z + q²/2g = P/[rho] + h, (8)

where P denotes the atmospheric pressure and h the height of the free
surface, a fundamental equation in hydraulics; a return has been made
here to the gravitation unit of hydrostatics, and Oz is taken
vertically upward.

In particular, for a jet issuing into the atmosphere, where p = P,

q²/2g = h - z, (9)

or the velocity of the jet is due to the head k - z of the still free
surface above the orifice; this is Torricelli's theorem (1643), the
foundation of the science of hydrodynamics.

27. _Uniplanar Motion._--In the uniplanar motion of a homogeneous
liquid the equation of continuity reduces to

du dv
-- + -- = 0, (1)
dx dy

so that we can put

u = -d[psi]/dy, v = d[psi]/dx, (2)

where [psi] is a function of x, y, called the stream- or
current-function; interpreted physically, [psi] - [psi]0, the
difference of the value of [psi] at a fixed point A and a variable
point P is the flow, in ft.³/second, across any curved line AP from A
to P, this being the same for all lines in accordance with the
continuity.

Thus if d[psi] is the increase of [psi] due to a displacement from P
to P´, and k is the component of velocity normal to PP´, the flow
across PP´ is d[psi] = k·PP´; and taking PP´ parallel to Ox, d[psi] =
vdx; and similarly d[psi]= -udy with PP´ parallel to Oy; and generally
d[psi]/ds is the velocity across ds, in a direction turned through a
right angle forward, against the clock.

In the equations of uniplanar motion

dv du d²[psi] d²[psi]
2[zeta] = -- - -- = ------ + ------ = -[Nabla]²[psi], suppose, (3)
dx dy dx² dy²

so that in steady motion

dH d[psi] dH d[psi] dH
-- + [Nabla]²[psi]------ = 0, -- + [Nabla]²[psi]------ = 0, ------ + [Nabla]²[psi] = 0, (4)
dx dx dy dy d[psi]

and [Nabla]²[psi] must be a function of [psi].

If the motion ia irrotational,

d[phi] d[psi] d[phi] d[psi]
u = - ------ = - ------, v = - ----- = ------, (5)
dx dy dy dx´

so that [psi] and [phi] are conjugate functions of x and y,

[phi] + [psi]i = [f](x + yi), [Nabla]²[psi] = 0, [Nabla]²[phi] = 0; (6)

or putting

[phi] + [psi]i = w, x + yi = z, w = [f](z).

The curves [phi] = constant and [psi] = constant form an orthogonal
system; and the interchange of [phi] and [psi] will give a new state
of uniplanar motion, in which the velocity at every point is turned
through a right angle without alteration of magnitude.

For instance, in a uniplanar flow, radially inward towards O, the flow
across any circle of radius r being the same and denoted by 2[pi]m,
the velocity must be m/r, and

[phi] = m log r, [psi] = m[theta],
[phi] + [psi]i = m log re^(i[theta]), w = m log z. (7)

Interchanging these values

[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

V²[psi]´ + 2R = 0, (7)

throughout the liquid.

Inside an equilateral triangle, for instance, of height h,

[psi]´ = -2R[alpha][beta][gamma]/h, (8)

where [alpha], [beta], [gamma] are the perpendiculars on the sides of
the triangle.

In the general case [psi]´ = [psi] + Uy - Vx + ½R(x² + y²) is the
relative stream function for velocity components, U, V, R.

29. _Example 1._--Liquid motion past a circular cylinder.

Consider the motion given by

[omega] = U(z + a²/z), (1)

so that

/ a²\ / a² \
[phi] = U ( r + -- ) cos [theta] = U ( 1 + -- )x, (2)
\ r / \ r² /

/ a²\ / a² \
[psi] = U ( r - -- ) sin [theta] = U ( 1 - -- )y.
\ r / \ r² /

Then [psi] = 0 over the cylinder r = a, which may be considered a
fixed post; and a stream line past it along which [psi] = Uc, a
constant, is the curve

/ a²\
( r - -- ) sin [theta] = c, (x² + y²)(y - c) - a²y = 0 (3)
\ r /

a cubic curve (C3).

Over a concentric cylinder, external or internal, of radius r = b,

/ a²\
[psi]´ = [psi] + U1y = [U ( 1 - --- ) + U1] y, (4)
\ b²/

and [psi]´ is zero if

U1/U = (a² - b²)/b²; (5)

so that the cylinder may swim for an instant in the liquid without
distortion, with this velocity U1, and [omega] in (1) will give the
liquid motion in the interspace between the fixed cylinder r = a and
the concentric cylinder r = b, moving with velocity U1.

When b = 0, U1 = [oo]; and when b = [oo], U1 = -U, so that at infinity
the liquid is streaming in the direction xO with velocity U.

If the liquid is reduced to rest at infinity by the superposition of
an opposite stream given by [omega] = -Uz, we are left with

[omega] = Ua²/z, (6)

[phi] = U(a²/r) cos [theta] = Ua²x/(x² + y²), (7)

[psi] = -U(a²/r) sin [theta] = -Ua²y/(x² + y²), (8)

giving the motion due to the passage of the Cylinder r = a with
velocity U through the origin O in the direction Ox.

If the direction of motion makes an angle [theta]´ with Ox,

d[phi] / d[phi] 2xy
tan[theta]´ = ----- / ----- = ------ = tan 2[theta], [theta] = ½[theta]´, (9)
dy / dx x² - y²

and the velocity is Ua²/r².

Along the path of a particle, defined by the C3 of (3),

y² y(y - c)
sin² ½[theta]´ = ------- = -------, (10)
x² + y² a²

d[theta]´ 2y - c dy
½ sin [theta]´ --------- = ------ --, (11)
ds a² ds

on the radius of curvature is ¼a²/(y - ½c), which shows that the curve
is an Elastica or Lintearia. (J. C. Maxwell, _Collected Works_, ii.
208.)

If [phi]1 denotes the velocity function of the liquid filling the
cylinder r = b, and moving bodily with it with velocity U1,

[phi]1 = -U1x, (12)

and over the separating surface r = b

[phi] U / a²\ a² + b²
--------- = - -- ( 1 + -- ) = -------, (13)
[phi]1 U1 \ b²/ a² - b²

and this, by § 36, is also the ratio of the kinetic energy in the
annular interspace between the two cylinders to the kinetic energy of
the liquid moving bodily inside r = b.

Consequently the inertia to overcome in moving the cylinder r = b,
solid or liquid, is its own inertia, increased by the inertia of
liquid (a² + b²)/(a² - b²) times the volume of the cylinder r = b;
this total inertia is called the effective inertia of the cylinder r =
b, at the instant the two cylinders are concentric.

With liquid of density [rho], this gives rise to a kinetic reaction to
acceleration dU/dt, given by

a² + b² dU a² + b² dU
[pi][rho]b² ------- -- = ------- M´--, (14)
a² - b² dt a² - b² dt

if M´ denotes the mass of liquid displaced by unit length of the
cylinder r = b. In particular, when a = [oo], the extra inertia is M´.

When the cylinder r = a is moved with velocity U and r = b with
velocity U1 along Ox,

a² / b² \ b² / a²\
[phi] = U ------- ( --- + r ) cos [theta] - U1------- ( r + --- ) cos [theta], (15)
b² - a² \ r / b² - a² \ r /

a² / b² \ b² / a²\
[psi] = -U ------- ( --- - r ) sin [theta] - U1------- ( r - --- ) sin [theta]; (16)
b² - a² \ r / b² - a² \ r /

and similarly, with velocity components V and V1 along Oy

a² / b² \ b² / a²\
[phi] = V ------- ( --- + r ) sin [theta] - V1------- ( r + --- ) sin [theta], (17)
b² - a² \ r / b² - a² \ r /

a² / b² \ b² / a²\
[psi] = V ------- ( --- - r ) cos [theta] + V1------- ( r - --- ) cos [theta], (18)
b² - a² \ r / b² - a² \ r /

and then for the resultant motion
a² z a²b² U + Vi
w = (U² + V²) ------- ------ + ------- ------
b² - a² U + Vi b² - a² z

a² z a²b² U1 + V1i
-(U1² + V1²) ------- -------- + ------- --------. (19)
b² - a² U1 + V1i b² - a² z

The resultant impulse of the liquid on the cylinder is given by the
component, over r = a (§ 36),
_
/ / b² + a² 2b² \
X = | [rho][phi] cos [theta]·ad[theta] = [pi][rho]a² ( U ------- - U1 ------- ); (20)
_/ \ b² - a² b² - a²/

and over r = b
_
/ / 2a² b² + a² \
X1 = | [rho][phi] cos [theta]·bd[theta] = [pi][rho]b² ( U ------ - U1------- ), (21)
_/ \ b² - a² b² - a² /

and the difference X - X1 is the component momentum of the liquid in
the interspace; with similar expressions for Y and Y1.

Then, if the outside cylinder is free to move

V1 2a² b² - a²
X1 = 0, -- = -------, X = [pi][rho]a²U -------. (22)
U b² + a² b² + a²

But if the outside cylinder is moved with velocity U1, and the inside
cylinder is solid or filled with liquid of density [sigma],

U1 2[rho]b²
X = -[pi][rho]a²U, -- = --------------------------------,
U [rho](b² + a²) + [sigma](b² - a²)

U - U1 ([rho] - [sigma])(b² - a²)
------ = ---------------------------------, (23)
U1 [rho](b² + a²) + [sigma](b² - a²)

and the inside cylinder starts forward or backward with respect to the
outside cylinder, according as [rho] > or < [sigma].

30. The expression for [omega] in (1) § 29 may be increased by the
addition of the term

im log z = -m[theta] + im log r, (1)

representing vortex motion circulating round the annulus of liquid.

Considered by itself, with the cylinders held fixed, the vortex sets
up a circumferential velocity m/r on a radius r, so that the angular
momentum of a circular filament of annular cross section dA is
[rho]mdA, and of the whole vortex is [rho]m[pi](b² - a²).

Any circular filament can be started from rest by the application of a
circumferential impulse [pi][rho]mdr at each end of a diameter; so
that a mechanism attached to the cylinders, which can set up a uniform
distributed impulse [pi][rho]m across the two parts of a diameter in
the liquid, will generate the vortex motion, and react on the cylinder
with an impulse couple -[rho]m[pi]a² and [rho]m[pi]b², having
resultant [rho]m[pi](b² - a²), and this couple is infinite when b =
[oo], as the angular momentum of the vortex is infinite. Round the
cylinder r = a held fixed in the U current the liquid streams past
with velocity

q´ = 2U sin [theta] + m/a; (2)

and the loss of head due to this increase of velocity from U to q´ is

q´² - U² (2U sin [theta] + m/a)² - U²
-------- = ----------------------------, (3)
2g 2g

so that cavitation will take place, unless the head at a great
distance exceeds this loss.

The resultant hydrostatic thrust across any diametral plane of the
cylinder will be modified, but the only term in the loss of head which
exerts a resultant thrust on the whole cylinder is 2mU sin[theta]/ga,
and its thrust is 2[pi][rho]mU absolute units in the direction Cy, to
be counteracted by a support at the centre C; the liquid is streaming
past r = a with velocity U reversed, and the cylinder is surrounded by
a vortex. Similarly, the streaming velocity V reversed will give rise
to a thrust 2[pi][rho]mV in the direction xC.

Now if the cylinder is released, and the components U and V are
reversed so as to become the velocity of the cylinder with respect to
space filled with liquid, and at rest at infinity, the cylinder will
experience components of force per unit length

(i.) - 2[pi][rho]mV, 2[pi][rho]mU, due to the vortex motion;

(ii.) - [pi][rho]a² dU/dt, -[pi][rho]a² dV/dt, due to the kinetic
reaction of the liquid;

(iii.) 0, -[pi]([sigma] - [rho])a²g, due to gravity,

taking Oy vertically upward, and denoting the density of the cylinder
by [sigma]; so that the equations of motion are

dU dU
[pi][rho]a²-- = - [pi][rho]a²-- - 2[pi][rho]mV, (4)
dt dt

dV dV
[pi][rho]a²-- = - [pi][rho]a²-- + 2[pi][rho]mV - [pi]([sigma] - [rho])a²g, (5)
dt dt

or, putting m = a²[omega], so that the vortex velocity is due to an
angular velocity [omega] at a radius a,

([sigma] + [rho])dU/dt + 2[rho][omega]V = 0, (6)

([sigma] + [rho])dV/dt - 2[rho][omega]U + ([sigma] - [rho])g = 0. (7)

Thus with g = 0, the cylinder will describe a circle with angular
velocity 2[rho][omega]/([sigma] + [rho]), so that the radius is
([sigma] + [rho])v/2[rho][omega], if the velocity is v. With [sigma] =
0, the angular velocity of the cylinder is 2[omega]; in this way the
velocity may be calculated of the propagation of ripples and waves on
the surface of a vertical whirlpool in a sink.

Restoring [sigma] will make the path of the cylinder a trochoid; and
so the swerve can be explained of the ball in tennis, cricket,
baseball, or golf.

Another explanation may be given of the sidelong force, arising from
the velocity of liquid past a cylinder, which is encircled by a
vortex. Taking two planes x = ± b, and considering the increase of
momentum in the liquid between them, due to the entry and exit of
liquid momentum, the increase across dy in the direction Oy, due to
elements at P and P´ at opposite ends of the diameter PP´, is

[rho]dy (U - Ua²r^(-2) cos 2[theta] + mr^(-1) sin [theta])(Ua²r^(-2) sin 2[theta] + mr^(-1) cos [theta])
+ [rho]dy (- U + Ua²r^(-2) cos 2[theta] + mr^(-1) sin [theta])(Ua²r^(-2) sin 2[theta] - mr^(-1) cos [theta])
= 2[rho]dymUr^(-1)(cos [theta] - a^2r^(-2)cos 3[theta]), (8)

and with y = b tan [theta], r = b sec [theta], this is

2[rho]mUd[theta] (1 - a²b^(-2) cos 3[theta] cos [theta]), (9)

and integrating between the limits [theta] = ±½[pi], the resultant, as
before, is 2[pi][rho]mU.

31. _Example 2.--Confocal Elliptic Cylinders._--Employ the elliptic
coordinates [eta], [xi], and [zeta] = [eta] + [xi]i, such that

z = c ch[zeta], x = c ch [eta] cos [xi], y = c sh [eta] sin [zeta]; (1)

then the curves for which [eta] and [xi] are constant are confocal
ellipses and hyperbolas, and

d(x, y)
J = -------, [xi]) = c²(ch²[eta] - cos² [xi])
d([eta]

= ½c²(ch 2[eta] - cos 2[xi]) = r1r2 = OD², (2)

if OD is the semi-diameter conjugate to OP, and r1, r2 the focal
distances,

r1, r2 = c(ch[eta] ± cos [xi]); (3)

r² = x² + y² = c²(ch²[eta] - sin² [xi])

= ½c²(ch 2[eta] + cos 2[xi]). (4)

Consider the streaming motion given by

w = m ch([zeta] - [gamma]), [gamma] = [alpha] + [beta]i, (5)

[phi] = m ch([eta] - [alpha]) cos ([xi] - [beta]),
[psi] = m sh([eta] - [alpha]) sin ([xi] - [beta]). (6)

Then [psi] = 0 over the ellipse [eta] = [alpha], and the hyperbola
[xi] = [beta], so that these may be taken as fixed boundaries; and
[psi] is a constant on a C4.

Over any ellipse [eta], moving with components U and V of velocity,

[psi]´ = [psi] + Uy - Vx = [m sh([eta] - [alpha]) cos [beta] + Uc sh[eta]] sin [xi]

-[m sh ([eta] - [alpha]) sin [beta] + Vc ch [eta] cos [xi]; (7)

so that [psi]´ = 0, if

m sh([eta] - [alpha]) m sh([eta] - [alpha])
U = - --- ------------------- cos [beta], V = - --- - ------------------- sin [beta], (8)
c sh[eta] c ch[eta]

having a resultant in the direction PO, where P is the intersection of
an ellipse [eta] with the hyperbola [beta]; and with this velocity the
ellipse [eta] can be swimming in the liquid, without distortion for an
instant.

At infinity

m m
U = - --- e^(-a) cos [beta] = - ----- cos [beta],
c a - b

m m
V = - --- e^(-a) sin [beta] = - ----- sin [beta], (9)
c a + b

a and b denoting the semi-axes of the ellipse [alpha]; so that the
liquid is streaming at infinity with velocity Q = m/(a + b) in the
direction of the asymptote of the hyperbola [beta].

An ellipse interior to [eta] = [alpha] will move in a direction
opposite to the exterior current; and when [eta] = 0, U = [oo], but V
= (m/c) sh [alpha] sin [beta].

Negative values of [eta] must be interpreted by a streaming motion on
a parallel plane at a level slightly different, as on a double Riemann
sheet, the stream passing from one sheet to the other across a cut SS´
joining the foci S, S´. A diagram has been drawn by Col. R. L.
Hippisley.

The components of the liquid velocity q, in the direction of the
normal of the ellipse [eta] and hyperbola [xi], are

-mJ^(-1)sh([eta] - [alpha]) cos([xi] - [beta]),
mJ^(-1)ch([eta] - [alpha]) sin ([xi] - [beta]). (10)

The velocity q is zero in a corner where the hyperbola [beta] cuts the
ellipse [alpha]; and round the ellipse [alpha] the velocity q reaches
a maximum when the tangent has turned through a right angle, and then

[root](ch 2[alpha] - cos 2[beta])
q = Qe^a ---------------------------------; (11)
sh 2[alpha]

and the condition can be inferred when cavitation begins.

With [beta] = 0, the stream is parallel to x0, and

[phi] = m ch([eta] - [alpha])cos [xi]

= -Uc ch([eta] - [alpha])sh [eta] cos [xi]/sh([eta] - [alpha]) (12)

over the cylinder [eta], and as in (12) § 29,

[phi]1 = -Ux = -Uc ch [eta] cos [xi], (13)

for liquid filling the cylinder; and

[phi] th [eta]
------ = --------------------, (14)
[phi]1 th ([eta] - [alpha])

over the surface of [eta]; so that parallel to Ox, the effective
inertia of the cylinder [eta], displacing M´ liquid, is increased by
M´th [eta]/th([eta]- [alpha]), reducing when [alpha] = [oo] to M´th
[eta] = M´(b/a).

Similarly, parallel to Oy, the increase of effective inertia is M´/th
[eta] th([eta] - [alpha]), reducing to M´/th [eta] = M´(a/b), when
[alpha] = [oo], and the liquid extends to infinity.

32. Next consider the motion given by

[phi] = m ch 2([eta] - [alpha]) sin 2[xi],
[psi] = -m sh 2([eta] - [alpha]) cos 2[xi]; (1)

in which [psi] = 0 over the ellipse [alpha], and

[psi]´ = [psi] + ½R(x² + y²)
= [-m sh 2([eta] - [alpha]) + ¼Rc²] cos 2[xi] + ¼Rc² ch 2[eta], (2)

which is constant over the ellipse [eta] if

¼Rc² = m sh 2([eta] - [alpha]); (3)

so that this ellipse can be rotating with this angular velocity R for
an instant without distortion, the ellipse [alpha] being fixed.

For the liquid filling the interior of a rotating elliptic cylinder of
cross section

x²/a² + y²/b² = 1, (4)

[psi]1´ = m1(x²/a² + y²/b²) (5)

with

[nabla]²[psi]1´ = -2R = -2m1(1/a² + 1/b²),

[psi]1 = m1(x²/a² + y²/b²) - ½R(x² + y²)
= -½R(x² - y²)(a² - b²)/(a² + b²), (6)

[phi]1 = Rxy(a² - b²)/(a² + b²),

w1 = [phi]1 + [psi]1i = -½iR(x + yi)²(a² - b²)/(a² + b²).

The velocity of a liquid particle is thus (a² - b²)/(a² + b²) of what
it would be if the liquid was frozen and rotating bodily with the
ellipse; and so the effective angular inertia of the liquid is (a² -
b²)²/(a² + b²)² of the solid; and the effective radius of gyration,
solid and liquid, is given by

k² = ¼(a² + b²), and ¼(a² - b²)²/(a² + b²). (7)

For the liquid in the interspace between [alpha] and [eta],

[phi] m ch 2([eta] - [alpha]) sin 2[xi]
------ = -------------------------------------------
[phi]1 ¼Rc² sh 2[eta] sin 2[xi](a² - b²)/(a² + b²)

= 1/th 2([eta] - [alpha])th 2[eta]; (8)

and the effective k² of the liquid is reduced to

¼c²/th 2([eta] - [alpha]) sh 2[eta], (9)

which becomes ¼c²/sh 2[eta] = 1/8 (a² - b²)/ab, when [alpha] = [oo],
and the liquid surrounds the ellipse [eta] to infinity.

An angular velocity R, which gives components -Ry, Rx of velocity to a
body, can be resolved into two shearing velocities, -R parallel to Ox,
and R parallel to Oy; and then [psi] is resolved into [psi]1 + [psi]2,
such that [psi]1 + ½Rx² and [psi]2 + ½Ry² is constant over the
boundary.

Inside a cylinder

[phi]1 + [psi]1i = -½iR(x + yi)²a²/(a² + b²), (10)

[phi]2 + [psi]2i = ½iR(x + yi)²b²/(a² + b²), (11)

and for the interspace, the ellipse [alpha] being fixed, and [alpha]1
revolving with angular velocity R

[phi]1 + [psi]1i = -1/8 iRc²sh 2([eta] - [alpha]
+ [xi]i)(ch 2[alpha] + 1)/sh 2([alpha]1 - [alpha]), (12)

[phi]2 + [psi]2i = 1/8 iRc²sh 2([eta] - [alpha]
+ [xi]i)(ch 2[alpha] - 1)/sh 2([alpha]1 - [alpha]), (13)

satisfying the condition that [psi]1 and [psi]2 are zero over [eta] =
[alpha], and over [eta] = [alpha]1

[psi]1 + ½Rx² = 1/8 Rc²(ch 2[alpha]1 + 1), (14)

[psi]2 + ½Ry² = 1/8 Rc²(ch 2[alpha]1 - 1), (15)

constant values.

In a similar way the more general state of motion may be analysed,
given by

w = m ch 2([zeta] - [gamma]), [gamma] = [alpha] + [beta]i, (16)

as giving a homogeneous strain velocity to the confocal system; to
which may be added a circulation, represented by an additional term
m[zeta] in w.

Similarly, with

x + yi = c[root][sin ([xi] + [eta]i)] (17)

the function

[psi] = Qc sh ½([eta] - [alpha]) sin ½([xi] - [beta]) (18)

will give motion streaming past the fixed cylinder [eta] = [alpha],
and dividing along [xi] = [beta]; and then

x² - y² = c² sin [xi] ch [eta], 2xy = c² cos [xi] sh [eta]. (19)

In particular, with sh [alpha] = 1, the cross-section of [eta] =
[alpha] is

x^4 + 6x²y² + y^4 = 2c^4, or x^4 + y^4 = c^4 (20)

when the axes are turned through 45°.

33. _Example 3._--Analysing in this way the rotation of a rectangle
filled with liquid into the two components of shear, the stream
function [psi]1 is to be made to satisfy the conditions

(i.) [nabla]²[psi]1 = 0,

(ii.) [psi]1 + ½Rx² = ½Ra², or [psi]1 = 0 when x = ±a,

(iii.) [psi]1 + ½Rx² = ½Ra², [psi]1 = ½R(a² - x²), when y = ± b.

Expanded in a Fourier series,

32 __ cos (2n + 1) ½[pi]x/a
a² - x² = ----- a² \ ---------------------, (1)
[pi]³ /__ (2n + 1)³

so that

16 __ cos (2n + 1) ½[pi]x/a · ch(2n + 1) ½[pi]y/a)
[psi]1 = ----- a² \ ---------------------------------------------,
[pi]³ /__ (2n + 1)^3 · ch(2n + 1) ½[pi]b/a

16 __ cos (2n + 1) ½[pi]z/a
w1 = [phi]1 + [psi]1i = iR ----- \ ------------------------------, (2)
[pi]³ /__ (2n + 1)^3 ch(2n + 1) ½[pi]b/a

an elliptic-function Fourier series; with a similar expression for
[psi]2 with x and y, a and b interchanged; and thence [psi] = [psi]1 +
[psi]2.

_Example 4._--Parabolic cylinder, axial advance, and liquid streaming
past.

The polar equation of the cross-section being

r^½ cos ½[theta] = a^½, or r + x = 2a, (3)

the conditions are satisfied by

[psi]´ = Ur sin [theta] - 2Ua^½ r^½ sin ½[theta]
= 2Ur^½ sin ½[theta](r^½ cos ½[theta] - a^½), (4)

[psi] = 2Ua^½ r^½ sin ½[theta] = -U[root][2a(r-x)], (5)

w = -2Ua^½ z^½, (6)

and the resistance of the liquid is 2[pi][rho]aV²/2g.

A relative stream line, along which [psi]´ = Uc, is the quartic curve

(4a²y² - (y - c)^4 4a²y² + (y-c)^4
y - c = [root][2a(r - x)], x = -------------------, r = ---------------, (7)
(4a(y - c)² 4a(y - c)²

and in the absolute space curve given by [psi],

dy (y - c)² 2ac
-- = - --------, x = - ----- 2a log (y - c). (8)
dx 2ay y - c

34. _Motion symmetrical about an Axis._--When the motion of a liquid
is the same for any plane passing through Ox, and lies in the plane, a
function [psi] can be found analogous to that employed in plane
motion, such that the flux across the surface generated by the
revolution of any curve AP from A to P is the same, and represented by
2[pi]([psi] - [psi]0); and, as before, if d[psi] is the increase in
[psi] due to a displacement of P to P´, then k the component of
velocity normal to the surface swept out by PP´ is such that
2[pi]d[psi] = 2[pi]yk.PP´; and taking PP´ parallel to Oy and Ox,

u = -d[psi]/ydy, v = d[psi]/ydx, (1)

and [psi] is called after the inventor, "Stokes's stream or current
function," as it is constant along a stream line (_Trans. Camb. Phil.
Soc._, 1842; "Stokes's Current Function," R. A. Sampson, _Phil.
Trans._, 1892); and d[psi]/yds is the component velocity across ds in
a direction turned through a right angle forward.

In this symmetrical motion

d / 1 d[psi] \ d / 1 d[psi]\
[xi] = 0, [eta] = 0, 2[zeta] = -- ( --- ------ ) + -- ( --- ------ )
dx \ y dx / dy \ y dy /

1 /d²[psi] d²[psi] 1 d[psi]\ 1
= --- ( ------- + ------- - --- ------ ) = - ---[nabla]²[psi], (2)
y \ dx² dy² y dy / y

suppose; and in steady motion,

dH 1 d[psi] dH 1 d[psi]
-- + --- ----- [nabla]²[psi] = 0, -- + --- ------ [nabla]²[psi] = 0, (3)
dx y² dx dy y² dy

so that

2[zeta]/y = -y^(-2)[nabla]²[psi] = dH/d[psi] (4)

is a function of [psi], say [f]´([psi]), and constant along a stream line;

dH/dv = 2q[zeta], H - [f]([psi]) = constant, (5)

throughout the liquid.

When the motion is irrotational,

d[phi] 1 d[psi] d[phi] 1 d[psi]
[zeta] = 0, u = - ------ = - --- ------, v = - ------ = --- ------, (6)
dx y dy dy y dx

d²[psi] d²[psi] 1 d[psi]
[nabla]²[psi] = 0, or ------- + ------- - --- ------ = 0. (7)
dx² dy² y dy

Changing to polar coordinates, x = r cos[theta], y = r sin[theta], the
equation (2) becomes, with cos[theta] = [mu],

d²[psi] d²[psi]
r²------- + (1 - [mu]²) ------- = 2[zeta]r³ sin [theta], (8)
dr² d[mu]²

of which a solution, when [zeta] = 0, is

/ B \ dPn / B \ dPn
[psi] = ( Ar^(n+1) + --- ) (1 - [mu]²) ----- = ( Ar^(n-1) + ------- ) y²-----, (9)
\ r^n / d[mu] \ r^(n+2) / d[mu]

[phi] = {(n + 1)Ar^n - nBr^(-n-1)} Pn, (10)

where Pn denotes the zonal harmonic of the nth order; also, in the
exceptional case of

[psi] = A0 cos[theta], [phi] = A0/r;

[psi] = B0r, [phi] = -B0 log tan ½[theta]
= -½B0 sh(-1) x/y. (11)

Thus cos[theta] is the Stokes' function of a point source at O, and PA
- PB of a line source AB.

The stream function [psi] of the liquid motion set up by the passage
of a solid of revolution, moving with axial velocity U, is such that

1 d[psi] dy
--- ------ = -U --, [psi] + ½Uy² = constant, (12)
y ds ds

over the surface of the solid; and [psi] must be replaced by [psi]´ =
[psi] + ½Uy² in the general equations of steady motion above to obtain
the steady relative motion of the liquid past the solid.

For instance, with n = 1 in equation (9), the relative stream function
is obtained for a sphere of radius a, by making it

[psi]´ = [psi] + ½Uy² = ½U(r² - a³/r) sin² [theta],
[psi] = -½Ua³ sin² [theta]/r; (13)

and then

[phi]´ = Ux(1 + ½a³/r²), [phi] = ½Ua³ cos [theta]/r², (14)

d[phi] a³ d[phi] a³
- ------ = U -- cos [theta], - --------- = ½U -- sin [theta], (15)
dr r³ rd[theta] r³

so that, if the direction of motion makes an angle [psi] with Ox,

tan ([psi] - [theta]) = ½ tan [theta],
tan [psi] = 3 tan [theta]/(2 - tan² [theta]), (16)

Along the path of a liquid particle [psi]´ is constant, and putting it
equal to ½Uc²,

(r² - a³/r) sin² [theta] = c², sin² [theta] = c²r/(r³ - a³), (17)

the polar equation; or

y² = c²r³/(r³ - a³), r³ = a³y²/(y² - c²), (18)

a curve of the 10th degree (C10).

In the absolute path in space

cos [psi] = (2 - 3 sin² [theta])/[root](4 - sin² [theta]),
and sin³ [theta] = (y³ - c²y)/a³, (19)

which leads to no simple relation.

The velocity past the surface of the sphere is

1 d[psi]´ / a³ \ sin² [theta]
------------ ------- = ½U ( 2r + -- ) ------------- = 3/2 U sin [theta], when r = a; (20)
r sin[theta] dr \ r² / r sin [theta]

so that the loss of head is

(9/4 sin² [theta] - 1) U²/2g, having a maximum 5/4 U²/2g, (21)

which must be less than the head at infinite distance to avoid
cavitation at the surface of the sphere.

With n = 2, a state of motion is given by

[psi] = -½Uy²a^4[mu]/r^4, [psi]´ = ½Uy²(1 - a^4[mu]/r^4), (22)

[phi]´ = Ux + [phi], [phi] = -1/3 U(a^4/r³)P2, P2 = 3/2 [mu]² - ½, (23)

representing a stream past the surface r^4 = a^4[mu].

35. A circular vortex, such as a smoke ring, will set up motion symmetrical about an axis, and provide an illustration; a half vortex ring can be generated in water by drawing a semicircular blade a short distance forward, the tip of a spoon for instance. The vortex advances with a certain velocity; and if an equal circular vortex is generated coaxially with the first, the mutual influence can be observed. The first vortex dilates and moves slower, while the second contracts and shoots through the first; after which the motion is reversed periodically, as if in a game of leap-frog. Projected perpendicularly against a plane boundary, the motion is determined by an equal opposite vortex ring, the optical image; the vortex ring spreads out and moves more slowly as it approaches the wall; at the same time the molecular rotation, inversely as the cross-section of the vortex, is seen to increase. The analytical treatment of such vortex rings is the same as for the electro-magnetic effect of a current circulating in each ring.

36. _Irrotational Motion in General._--Liquid originally at rest in a
singly-connected space cannot be set in motion by a field of force due
to a single-valued potential function; any motion set up in the liquid
must be due to a movement of the boundary, and the motion will be
irrotational; for any small spherical element of the liquid may be
considered a smooth solid sphere for a moment, and the normal pressure
of the surrounding liquid cannot impart to it any rotation.