RECENT PKOGRESS IN HTDEODYNAMICS. 65 



The results are used to determine the effect of the wire by which it is 

 suspended on the time of oscillation of a ball-pendulum. If it be pro- 

 posed to determine the state of motion of the fluid due to a uniform 

 translation of the cylinder, it will be found that a steady motion of the 

 fluid will be impossible, but that as the time increases the quantity of 

 fluid carried forward with the cylinder continually increases. In fact, if 

 the differential equation for the stream function be integrated on the 

 supposition of steady motion, it will be found that, though the integral 

 takes a simple finite form, there are not arbitrary constants sufficient to 

 satisfy the conditions. 



Plane and Disc. — The determination of the motion of a disc in a viscous 

 fluid is important, as it forms a useful method to determine experimen- 

 tally the value of the coefficient of viscosity for different fluids. The 

 method was first employed by Coulomb, but first received mathematical 

 treatment by Stokes (1850), and afterwards (1861) by Meyer. Stokes * 

 begins by investigating the motion when an infinite plane oscillates in its 

 own plane, so that its displacement at any time is given by c sin nt. The 

 displacement at any point in the fluid at a distance x is then given by 

 c sin {nt — .r\/(?i/2^4)} x exp — x v/(ii./2^). A given phase is therefore 

 propagated with a velocity ^/2im. This for air, treated as incompres- 

 sible, andfor a time of vibration of one second, is about "2908 inch per 

 second. The solution for a disc oscillating in its own plane can then be 

 obtained by treating each element of it as a portion of the above plane, a 

 solution which is exact if the squares of the velocities are neglected, except 

 in so far as the action of the rim is concerned. In this way he finds the 

 change in the'time of vibration due to viscosity, and also the logarithmic 

 decrement of the arc of oscillation, with a correction to be applied, 

 because the observations are made soon after the disc is set in motion, 

 and before the motions, due to the starting in a fluid at rest, have dis- 

 appeared. The results obtained enable him to discuss the observations of 

 Coulomb. 



Without a knowledge of Stokes' work, O. E. Meyer ^ attacked the 

 same problem about ten years later. He supposes the angular velocity 

 of the fluid to depend only on the distance from the disc, and not on the 

 distance from the axis of rotation. This is equivalent to Stokes' appli- 

 cation of the motion for a plane to that for a disc. He also determines 

 the logarithmic decrement. 



The Sphere. — The analytical difficulties connected with this surface 

 may be considered to have been surmounted, chiefly through the work of 

 Stokes, Meyer, and Oberbeck. In his second paper on viscosity of 

 fluids Stokes ' attacks the problem of the motion by expressing the velo- 

 cities in terms of the stream function — first introduced by him — ana 

 making the determination of this stream function the basis of the investi- 

 gation. The motion is supposed so small that squares and products of 

 the velocities may be neglected. In this case the stream function -^ must 

 satisfy the differential equation ^ V^(^^ X ofi~^dlJt)\p = 0, the solution 

 of which can be represented in the form ;// = i//, + \p.2 where v'^i/'i = 0, 

 (v^ + pfi~^d/dt)\l^.2 ^ O- The complete solution of this is not entered 



' ' On the effect of the internal friction of fluids on tlie motion of pendulums,' 

 Carnb Phil. Trans, ix. part 2, p. 8 (1850). 



^ ' Ueber die Reibung der Fliissigkeiten,' Borcli. lix. p. 229. 



^ I use — V" to denote the operation d^jda.'- + d-jdi/- + d-jdz", or, in this case, 

 d-jdx- + d-jdw- — m- ^djdw, as v denotes the vector operator idjdx + jdjdy + Jidjdz. 

 1882. F 



