Solid Bodies through Viscous Liquid. 709 



power of the velocity and independence of viscosity are thus 

 inseparably connected " *. 



The general investigation for the sphere moving in any 

 manner (in a straight line) shows that the departure from 

 Stokes's law when the velocity is not very small must be due 

 to the operation of the neglected terms involving the squares 

 of the velocities ; but the manner in which these act has not 

 yet been traced. Observation shows that an essential feature 

 in rapid fluid motion past an obstacle is the formation of a 

 wake in the rear of the obstacle ; but of this the solutions of 

 the approximate equations give no hint. 



Hydrodynamical solutions involving surfaces of discon- 

 tinuity of the kind investigated by Helmholtz and KirchhofE 

 provide indeed for a wake, but here again there are difficulties. 

 Behind a blade immersed transversely in a stream a region of 

 " dead water " is indicated. The conditions of steady motion 

 are thus satisfied ; but, as Helmholtz himself pointed out, the 

 motion thus defined is unstable. Practically the dead and 

 live water are continually mixing; and if there be viscosity, 

 the layer of transition rapidly assumes a finite width inde- 

 pendently of the instability. One important consequence is 

 the development of a suction on the hind surface of the 

 lamina which contributes in no insignificant degree to the 

 total resistance. The amount of the suction does not appear 

 to depend much on the degree of viscosity. When the latter 

 is small, the dragging action of the live upon the dead water 

 extends to a greater distance behind. 



§ 8. If the blade, supposed infinitely thin, be moved edge- 

 ways through the fluid, the case becomes one of " skin- 

 friction."" Towards determining the law of resistance Mr. 

 Lanchester has put forward an argument f which, even if 

 not rigorous, at any rate throws an interesting light upon 

 the question. Applied to the case of two dimensions in order 

 to find the resistance F per unit length of blade, it is some- 

 what as follows. Considering two systems for which the 

 velocity V of the blade is different, let n be the proportional 

 width of corresponding strata of velocity. The momentum 

 communicated to the wake per unit length of travel is as rcV, 

 and therefore on the whole as nY 2 per unit of time. Thus 

 F varies as nV 2 . Again, having regard to the law of viscosity 

 and considering the strata contiguous to the blade, we see 

 that F varies as V/n. Hence, ?iV 2 varies as V/ra, or V varies 

 as n~ 2 , from which it follows that F varies as V 32 . If this 



* Phil. Mag. xxxiv. p. 59 (1892) ; ScientiHc Tapers, iii. p. 576, 

 f Aerodynamics, London, 1907, § 35. 



