The corresponding formulae for the external space will be as in (4) and (6), above.

Again, on comparison of (6) and (23) it appears that the continuity of stress requires

We have thus five equations to determine A, B, A', B', U when K is given. Solving, we find

The total force which must act on the sphere in the direction of x-negative, in order to maintain it at rest in the stream, is

If we put μ' = ∞ we reproduce the results relating to a solid sphere.

To adapt the results to the case of motion under gravity (supposed to act in the direction of x-negative), we must put

K = g(p' - p),

..(31) where p' is the internal density. The terminal velocity is then given by (28). If p' < p, U is negative, indicating that the globule ascends relatively to the surrounding fluid. In the case of a bubble of gas ascending through water we may put, with sufficient accuracy, p' = 0, μ' = 0, whence

gpa2
3
μ

(32)

3o. A variation of the problem of the solid sphere is afforded if we allow for the possibility of slipping of the fluid over the surface, assuming the empirical law referred to in Art. 327.

The formulae (6) give, for the normal stress on a sphere of radius r, the expression

the three components of which may be written, in virtue of Art. 336 (14),

..(33)

.(34)

Subtracting these from the expressions in (6) we find, for the components of tangential stress,

At the surface r = a, the radial velocity must vanish, and the expressions in (4) will become components of tangential velocity.

We must have, therefore,

and it appears on reference to (5) that these satisfy the condition of zero radial velocity.

For B

=∞, this agrees with (15). If ẞ were = 0, the resultant would be 4μUа.

4o. The problem of a rotating sphere in an infinite mass of liquid is solved by assuming

the axis of z being that of rotation. At the surface r = a we must have

=

wa3; cf. Art. 334.

if o be the angular velocity of the sphere. This gives A

338. Problems relating to flow about a sphere, in planes through an axis. of symmetry, have been usually treated, as by Stokes originally, by means of the current-function . It may be useful, therefore, to give a few indications of the method.

Putting y = cos 9, z= sin 9, and accordingly

In the case of steady motion, we have, from Art. 336 (1), v2ŋ = 0, v2 = 0, and therefore

In the case of uniform flow at infinity we must have whence

.(8)

.(9) .

.(10)

.(11)

The component velocities along and at right angles to the radius vector are

The rates of elongation in the directions of r and 6, and at right angles to these two, are found by simple calculations to be

The force on the sphere may be calculated directly from the stress-formulae, or may be inferred from the rate of dissipation of energy. It follows from (13) and (14) that the function of Art. 329 (8) takes the form

To find the total rate of dissipation of energy in the fluid we must multiply this by 2πr sin er808r, and integrate from

and from r = a tor. The result is

=

0 to 8 =

On the hypothesis of no slipping at the surface r = a we find from (12)

The force (-P, say) which must be applied to the sphere to maintain the motion is found by equating the rate of dissipation of energy to PU. Substituting in (17) from (18) we find

as before.

P = 6πμαU,

.(21)

If there is slipping, with a coefficient ẞ of sliding friction, the conditions to be satisfied for ra are

in the original form of the problem where the sphere is regarded as at rest. Substituting from (12) and (14) we find

There is in this case an additional dissipation of energy by sliding friction at the surface of the sphere, of amount 302 per unit area. If we integrate this over the surface, the result is, by (22) and (14),

If we add this to (17), and insert the values of A and B from (23), we find, on equating the total dissipation to PU,

P = 6πμαU.

βα + 2μ
βα + 3μ'

.(25)

in agreement with Basset's result (Art. 337 (38)).

339. The problem of the steady translation of an ellipsoid in a viscous liquid can be solved in terms of the gravitation-potential of the solid, regarded as homogeneous and of unit density.

the gravitation-potential is given, at external points, by Dirichlet's formula*

.(1)

* Crelle, t. xxxii. p. 80 (1846) [Werke, t. ii. p. 11]; see also Kirchhoff, Mechanik, c. xviii., and Thomson and Tait (2nd ed.), Art. 494m.

L. H.

3888

it has been shewn in Art. 113 that this satisfies v2x = 0.

If the fluid be streaming past the ellipsoid, regarded as fixed, with the general velocity U in the direction of x, we assume*

These satisfy the equation of continuity, in virtue of the relations

It remains to shew that by a proper choice of A, B we can make u, v, w = 0 at the surface (1). The conditions v = 0, w = O require

With the help of this relation, the condition u = O reduces to

(11)

...(12)

where the suffix denotes that the lower limit in the integrals (6) and (7) is to be replaced by zero.

Hence

whence it appears, on comparison with the equations (4) of the preceding Art., that the disturbance is the same as would be produced by a sphere of radius R, determined by