a = b = c, A0 = B0 = C0 = 1/3, [alpha] = [beta] = [gamma] = ½, (29)
so that the effective inertia of a sphere is increased by half the
weight of liquid displaced; and in frictionless air or liquid the
sphere, of weight W, will describe a parabola with vertical
acceleration
W - W´
------- g. (30)
W + ½W´
Thus a spherical air bubble, in which W/W´ is insensible, will begin
to rise in water with acceleration 2g.
45. When the liquid is bounded externally by the fixed ellipsoid
[lambda] = [lambda]1, a slight extension will give the velocity
function [phi] of the liquid in the interspace as the ellipsoid
[lambda] = 0 is passing with velocity U through the confocal position;
[phi] must now take the form x([psi] + N), and will satisfy the
conditions in the shape
_
abc /[lambda]1 abcd[lambda]
------ + | ----------------
A + B1 + C1 a1b1c1 _/ [lambda] (a² + [lambda])P
[phi] = Ux ----------------- = Ux ------------------------------------------, (1)
B0 + C0 - B1 - C1 abc /[lambda]1 abcd[lambda]
1 - ------ - | ----------------
a1b1c1 _/0 (a² + [lambda])P
and any confocal ellipsoid defined by [lambda], internal or external
to [lambda] = [lambda]1, may be supposed to swim with the liquid for
an instant, without distortion or rotation, with velocity along Ox
B_[lambda] + C_[lambda] - B1 - C1
U ----------------------------------
B0 + C0 - B1 - C1
Since - Ux is the velocity function for the liquid W´ filling the
ellipsoid [lambda] = 0, and moving bodily with it, the effective
inertia of the liquid in the interspace is
A0 + B1 + C1
----------------- W´. (2)
B0 + C0 - B1 - C1
If the ellipsoid is of revolution, with b = c,
A + 2B1
[phi] = ½Ux -------, (3)
B0 - B1
and the Stokes' current function [psi] can be written down
B - B1
[psi] = - ½Uy² -------; (4)
B0 - B1
reducing, when the liquid extends to infinity and B1 = 0, to
A B
[phi] = ½Ux --, [psi] = - ½Uy² --; (5)
B0 B0
so that in the relative motion past the body, as when fixed in the
current U parallel to xO,
/ A \ / B \
[phi]´ = ½Ux ( 1 + -- ), [psi]´ = ½Uy² ( 1 - -- ). (6)
\ B0 / \ B0 /
Changing the origin from the centre to the focus of a prolate
spheroid, then putting b² = pa, [lambda] = [lambda]´a, and proceeding
to the limit where a = [oo], we find for a paraboloid of revolution
p B p
B = ½ -------------, -- = -------------, (7)
p + [lambda]´ B0 p + [lambda]´
y²
------------- = p + [lambda]´ - 2x, (8)
p + [lambda]´
with [lambda]´ = 0 over the surface of the paraboloid; and then
[psi]´ = ½U [y² - p[root](x² + y²) + px]; (9)
[psi] = -½Up [[root](x² + y²) - x]; (10)
[phi] = -½Up log [ [root](x² + y²) + x]. (11)