SCIENCE. 



[N. S. Vol. VIII. No. 202. 



the velocity exceeds a limit which is very 

 easily reached, eddies are formed and the 

 motion entirely changes its character. Kor- 

 teweg has made some suggestive remarks on 

 this point in a paper on the stability of the 

 motion of viscous fluids. He remarks that 

 the existence and formation of eddies was 

 generally supposed to be due to unstable solu- 

 tions of the equations of motion, von Helm- 

 holtz, however, had found that when a solu- 

 tion with given boundary conditions in a 

 simply-connected region is obtained, that 

 solution is unique. Hence, we cannot at- 

 tribute the formation of eddies to other 

 solutions which in fact do not exist. Korte- 

 weg further proves that this unique solution 

 is stable. Hence, neglecting squares and 

 products of the velocities, it is evident that 

 having got a unique stable solution, eddies 

 cannot be formed. Concerning these eddies, ' 

 he says : " When, on the contrary, squares 

 and higher powers of the velocities are 

 taken into account I have my reasons for 

 supposing that even in the case of a sphere 

 moving with uniform velocity — if such a 

 state of steady motion can be reached — the 

 motion must finally become unstable. " Lord 

 Eayleigh has also attempted to examine 

 how viscosity affects the motion, and how 

 eddies, when formed, are maintained. 



" Lord Kelvin concludes that the linear 

 flow of a fluid through a pipe or of a stream 

 over a plane bed is stable for very small 

 disturbances, but that for disturbances of 

 more than a certain amplitude the motion 

 becomes unstable, the limits of stabil- 

 ity being smaller the smaller the viscos- 

 ity." ( Lamb's Hydrodynamics. ) It is 

 possible that a remark made by Klein in 

 his lectures on the top may have some 

 bearing in this case. He points out that 

 near the limit of stability, obtained as usual 

 by neglecting the second and higher powers 

 of small quantities, instability really takes 

 place when we include them. It is, indeed, 

 possible that all viscous fluid motions as at 



present investigated are really unstable 

 and that eddies are always formed. It is 

 instructive to read Reynolds' experiments 

 of 1883 as to the point at which, with in- 

 creasing velocity, stream lines appear to 

 break up into eddies. 



A warning must be given against laying 

 too great a stress on the equations of mo- 

 tion. They are formed under certain sup- 

 positions as to the character of the inter- 

 nal friction of the fluid, but we have no se- 

 curity that these suppositions represent the 

 facts. At the same time most of the differ- 

 ent assumptions made lead to the same 

 equations, so that only a very fundamental 

 alteration would affect the equations of 

 motion. 



Another difficulty arises with the skin 

 friction at the surface of a body moving in 

 the fluid. The difficulty arises from the 

 fact that the action along the wall in con- 

 tact with the fluid is treated quite dififer- 

 ently in the cases of no friction and very 

 small friction. The mathematical char- 

 acter of the motion may be completely al- 

 tered. The difference is this. With non- 

 viscous fluids we assume a finite slip of the 

 fluid along the wall. If even the smallest 

 coefficient of friction be introduced, a finite 

 slip cannot be consistently allowed. The 

 whole subject is very obscure. It is now 

 believed by some prominent physicists that 

 no such finite slip actually takes place at 

 all. 



A somewhat easier problem is the decay 

 of waves proceeding along water, owing to 

 the influence of viscosity. In several short 

 papers during the last three years in the 

 Comptes Eendus, Boussinesq has treated the 

 various kinds of waves, in many cases re- 

 ducing his results to numbers. A practical 

 application is a determination of the time 

 necessary for the sea raised by a storm to 

 subside. Hough has treated a similar 

 problem, namely, the effects of viscosity on 

 ocean currents and on tides of long period. 



