Flow of Viscous Liquids. 365 



i(X 2 +Y 2 + Z 2 ) = (« 2 + ^+^)(J 2 +i ? 2 +n-K-^-^?) 2 - (39) 



If the motion take place in two dimensions, io — O, f = 77 = 0, 

 and 



J (X« + Y*) = (tt« + **){* (40) 



In the case of symmetry round an axis, 



and (39) reduces to 



i (X 2 + Y 2 + Z 2 ) = (u 2 + v 2 + w*) (£ 2 + v* + ? 2 ) • . (41) 



These expressions for the forces necessary to the main- 

 tenance of a motion similar to the infinitely small motion give 

 ns in simple cases an idea of the direction in which the law 

 is first departed from as the motion increases. 



There are very few cases in which the problem of the rapid 

 motion of a viscous fluid has been dealt with. When the 

 motion is in one dimension, the troublesome terms do not 

 present themselves, and the same solution holds good mathe- 

 matically for the steady motion at all velocities. When the 

 motion is so small that the laws appropriate to infinitely 

 small motion hold good as a first approximation, a correction 

 may be calculated. This has been effected by Whitehead *, 

 and in an unpublished paper by Rowland, for the problem, 

 first investigated by Stokes, of a sphere moving with velocity 

 V through viscous liquid. For infinitely small motion the 

 velocity of the fluid in the neighbourhood of the sphere is of 

 order V. It follows from the solution referred to, or may be 

 proved independently by considerations of dimensions, that in 

 the second approximation involving V 2 , the terms are of the 

 order V 2 a/v, a being the radius of the sphere, and v, equal to 

 fi/p, the kinematic coefficient of viscosity. This method of 

 approximation is thus only legitimate when Ya/v is small, a 

 condition of a very restricting character. In the case of 

 water v = *01 C.G.S., and if Ya/v = 'l, it is required that 

 Vo=-001. 



Thus even if a were as small as one millimetre (*1), V 

 should not exceed *01 centimetre per second. With such 

 diameters and velocities as often occur in practice, Ya/v 

 would be a large, instead of a small, quantity ; and a solution 

 founded upon the type of infinitely slow motion is wholly 

 inapplicable. 



We will now recur to the suppositions that the motion is 

 steady, is in two dimensions, and that its square may be 



* Quart. Journ. of Math. vol. xxiii. p. 153 (1889). 



