Flow of Viscous Liquids. 371 



importance of the third and second terms in (61), we have to 

 consider the value of 



and the conclusion will depend upon the value of y. If we 

 suppose that ky=l, the ratio is 2L : Sk 2 v ; or, if we denote by 

 Y the undisturbed velocity of the fluid when %==1, Y/Skv, or 

 YX/Gttv, X being the wave-length of the corrugation. With 

 ordinary liquids and moderate values of X, V would have to 

 be very small in order to permit the success of the method of 

 approximation. 



The character of the motion proportional to L 2 is easily seen 

 from the value of v. We have 



v=-^ = ^(f + ikf)e-^co&k^ . . (62) 



d, 



V V 



indicating a motion directed outwards from the wall over the 

 places where the sinuosities encroach upon the fluid, and an 

 inward motion where the sinuosities recede. 



The application of the results towards the explanation of 

 such phenomena as ripple-mark and wave-formation requires 

 a calculation of the forces operative upon the boundary. We 

 will confine ourselves to the first term in /3 and L, as expressed 

 in (55). 



The normal stress, parallel to ?/, is given by 



Q-;+vJ-f-^i • • (63) 



and the tangential stress, parallel to a, is 



M£ + !Mi -&)• • • (64) 



From (34), (55) we find 



p= —4:/jl k/3e~~ hy sin kx, 

 or when y = 0, 



p= — 4/:/3 sin x, simply. 

 Also, when y = 0, 



so that 



Q = (65) 



In like manner, when y = 0, 



U = 2/*{l + 2££cosfo?} (66) 



