The total flux across any section of the pipe is

If now in a length l of the pipe the pressure fall from p, to p2,

we

B=

If B∞, there is no slipping at the boundary, and (22) then reduces to its first term

Poiseuille found in his experiments* on the flow of water through capillary tubes that the time of efflux of a given quantity of water was directly as the length of the tube, inversely as the difference of pressure at the two ends, and inversely as the fourth power of the diameter. These results agree with (22a).

A comparison of the formula (22a) with experiments of this kind would give the means of determining μ.

183. Example 2. To investigate the effect of internal friction on the motion of plane waves of sound. If as in Art. 166 we neglect the squares and products of small quantities, the equation of motion is

To fix the ideas let us suppose that at the plane x = 0 a simple

*Mém. des Sav. Etrangers, t. 9, 1846. Quoted by Wüllner, Experimental physik, t. 1, p. 329.

when x=0. Now (23) is satisfied by the sum of any number of terms of the form

provided

= Cea'+Br..

a2 = c2ß2 + ±μ'aß2.

.(25),

(26).

To obtain a solution consistent with (24), we write a = 2πni, whence

Substituting, and putting (25) in a real form, we find

In (27) we must take the upper or the lower sign according as we consider the waves propagated in the direction of a negative, or x positive. We see that the velocity of propagation is increased by the friction, the increase being greater for higher notes than for lower ones. The amplitude of the waves diminishes as they proceed, the diminution being the more rapid the higher the note. The change in the velocity of propagation depends ultimately on the square of μ, that of the amplitude on the first power of μ, so that the latter effect is much more important than the former. Stokes however has shewn that in all ordinary cases the diminution of amplitude due to friction is insignificant compared with that due to spherical divergence.

The equation (27) does not constitute the complete solution of (23) subject to the condition (24). In fact we may add any number

*Camb. Phil. Trans. Vol. IX. p. 94.

of terms of the form (25), provided the coefficients be so chosen as to make

Thus, writing ẞ= ki in (26), and solving with respect to a, we find

Substituting in (25), and making use of (28), we obtain the solution

.........

§= Σe-3μ'k2 (P cos kc't + Q sin kc't) sin kæ.........................(30), where the summation embraces all values of k. The coefficients P, Q may be determined by Fourier's method so as to make the sum of (27) and (30) satisfy any arbitrary initial conditions. We see that the effect of the initial circumstances gradually disappears, until finally the only sensible part of the disturbance is that due to the forced vibration maintained at x = 0.

It appears from (30) that the effect of friction is to diminish the velocity of propagation of a free wave.

184. Example 3. A sphere moves with uniform velocity in an incompressible viscous fluid; to find the force which must be applied to the sphere in order to maintain this motion*.

Take the centre of the sphere as origin, the direction of motion as axis of x. If we impress on the fluid and the solid a velocity V equal and opposite to that of the sphere, the problem is reduced to one of steady motion. Further, assuming the motion to be symmetrical about the axis of x, we may write, with the same notation as in Art. 103,

whence we have, for the angular velocity w of a fluid element,

If we suppose the motion to be so slow that the squares and pro

* Stokes, Camb. Phil. Trans. Vol. ix. p. 48.

&c. may be neglected, we obtain by elimina

we find on transformation of co-ordinates that the equations (32) reduce to the one equation

whence, substituting the value of a from (31), we obtain

If R,

denote the component velocities at any point of the fluid

along r, and perpendicular to r in the plane of 0, respectively, we have, from the definition of y,

We have still to introduce the conditions to be satisfied at the surface of the sphere. On the hypothesis of no slipping, these are

but we retain for the present the symbols A, B. The resulting value of is simplified if we restore the problem to its original form by removing the impressed velocity - V in the direction of x, i.e. if we add to the term

The resistance experienced by the sphere is most readily calculated by means of the dissipation-function F of Art. 179. Let us take at any point a subsidiary system of rectangular axes, in the directions of R, ✪, and of a normal to the plane of 0, respectively; and let [a], [b], [c], [ƒ], [g], [h] have the same meanings with respect to these axes that a, b, c, f, g, h have with respect to x, y, z. We find by simple calculations