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.