a Sphere in a Viscous Fluid. 325 



The small value of the critical radius in all practical cases 

 renders the direct verification of the formula for the terminal 

 velocity difficult. For a particle of sand falling through 

 water at 15° 0. (yu = "0115), taking a=2p, the critical radius 

 is "0085 centim. Unless the radius of the particle is smaller 

 than this, the solution founded upon the type of infinitely 

 slow motion is inapplicable. 



2. Steady Flow in a Uniform Tube. 



In considering the motion of a sphere through a viscous 

 fluid it is instructive to recall the results arrived at in the 

 somewhat analogous case of the flow of a viscous fluid through 

 a uniform pipe or channel. When the motion is steady, the 

 accelerating force must be exactly balanced by the resistance 

 due to fluid viscosity. 



If the flow is rectilinear, the hydraulic gradient — that is the 

 fall in pressure per unit length of the tube — is proportional to 

 the first power of the mean velocity. 



Hagen * and Reynolds f have shown independently that 

 rectilinear flow is only possible in practice when the velocity 

 is less than a certain maximum amount, the " critical velocity," 

 depending on the radius of the tube and the viscosity and 

 density of the fluid. When the velocity exceeds this value 

 the flow becomes turbulent. 



The existence of a definite critical velocity has been 

 questioned. According to Lord Kelvin's view % the stead v 

 motion of a fluid is theoretically stable for infinitely small 

 disturbances for any viscosity however small. This stability 

 may not extend beyond very narrow limits ; so that in practice 

 under finite disturbances the motion would be unstable except 

 for sufficiently viscous fluids. 



This view has been criticized by Lord Rayleigh in a paper 

 " On the Question of the Stability of the Flow of Fluids "§. 



Practically, at least, we seem justified in accepting the idea 

 of a critical velocity at which eddying motion begins. 



The law of resistance in the turbulent regime has been 

 investigated by several observers ||. The results may be 

 expressed by taking the hydraulic gradient proportional to 



* Abhandl. Akad. Wiss. Berlin, 1854, Math. Abt. pp. 17. 



t Phil. Trans, clxxiv. pp. 935-982 (1883). 



\ Phil. Mag. (5) xxiii. pp. 459-464,529-539 (1837) : xxiv. pp. 188-196, 

 272-278, 342-355 (1887). 



§ Phil. Mag. xxxiv. pp. 59-70 (1392). 



|| A. convenient summary and discussion of these results may be found in 

 a paper by G. H. Knibbs, Proceedings Roy. Soc. N. S. W., xxxi. pp. 314- 

 355 (1897). r 



