Oiiii — stability or Instability of Motions of a Viscous Liquid. 77 



And in Art. 35, pp. 135-138, the more general method is applied to a 

 symmetrical disturbance. The differential equation is of a similar type to 

 that in the preceding case, and is solved in a similar manner; the final 

 result is DUpjfx = 180, D being the diameter of the pipe ; the number 

 obtained by Sharpe is 470. The law of velocity in this instance being 

 U =- C {a? - r^), and that in the last TJ = C{a' - y^), the value I have found 

 for C" is almost double that for 0. 



It is claimed that in each case the numbers I have found are true 

 least values (but vs^ith some reservation as to the effect of end-conditions) ; 

 that below them every disturbance must automatically decrease, and that 

 above them it is possible to prescribe a disturbance which will increase 

 for a time. 



The numbers obtained above give velocities very much below those at 

 which observers have found motions actually to become unstable; this is 

 to be expected. 



Although I cannot profess to have examined the records of the experiments 

 carefully, it seems that the results of Eeynolds^ and of Couette^ are to 

 some extent contradicted by Mallock's.^ The general result of each is that, 

 up to a certain velocity, the motion is certainly stable, and the frictional 

 resistance varies as the velocity : beyond this comes a region in which the 

 motion appears at times to be stable, and at times to be unstable, the average 

 resistance on the whole now increasing more rapidly than the first power 

 of the velocity : if the velocity is still further increased, the motion is 

 permanently eddying and turbulent, and the resistance is, approximately 

 at least, proportional to the square of the velocity. Eeynolds found, from 

 experiments made on pipes of different diameters, and in which the viscosity 

 was varied by varying the temperature, that the motion was certainly stable 

 until I) Ufi //J. = 1900. Couette gives results of experiments* on eight pipes 

 of different diameters, the temperature being approximately constant. The 

 mean value of BU is very nearly 25*4 in C. Gr. S. units, the range being from 

 22 to 28; taking /x/p at 13'''8 C. (the mean temperature) to be '0118, this 

 gives DUplfx = 2150. Moreover, some of Eeynolds' experiments were made 

 with colour-bands — a method which might be expected to reveal eddies which 

 might otherwise escape detection, and thus to give a lower limit for U. 



1 " An experimeiii al investigation of the circumstances which determine whether the motion of 

 water shall be direct or sinuous, and of the law of resistance in parallel channels," Phil. Trans. 1883 ; 

 Scientific Papeis, ii. 



2 " Etudes sur le frottemeiit des liquides," Annales de Chimie et de Physique, 6« Serie xxi., 1890. 



3 " Experimeuto on Fluid Viscosity," Phil. Trans., 187, 1896. 

 *L.c., p. 488. 



K. I. A. PKOC, VOL. XX Vn., SECT. A. [11] 



