246 UNIVERSITY OF ^^RGINIA PUBLICATIONS 



the main flow rises and falls with expanding and contracting cross sections 

 alternately. It is readily understood that there is no such thing here 

 as a rigid definite or fixed boundary in the liquid itself but quite a definite 

 zone as it were between the main body of the flow in the direction of the 

 axis and the permanent eddies in the neighborhood of the rough boundary. 



Such a typical case is the condition of flow artificially produced, in 

 some of Darcy and Bazin's experiments in which a wooden flume was 

 constructed with laths 1 X | inches nailed crosswise on the bottom and 

 sides 2 inches apart. Thus the artificial barriers act as a series of sub- 

 merged dams over which the water flows alternately rising and falling 

 in sinuous curves of much regularity near the boundary while between 

 these dams the water is whirled in a series of vortex rings. In a natural 

 boundary such as that of the surface of a pipe, brick or cement conduit 

 or smooth earthen canal the same thing takes place with a difference not 

 so much of kind as of degree. In every regular artificial chamiel, fairly 

 smooth, water flows through a series of more or less uniformly distributed 

 cross sections which act as alternate contractions and expansions of the 

 channel. 



The form of this liquid boundary of the main chamiel flow depends 

 on three things; first the geometrical topography of the rigid material 

 surface boundary which for a given channel is constant, the mean slope 

 of the hachure lines of this surface fixes to some definite extent the mean 

 undulations of the liquid boundary; second the mean velocity with which 

 the water flows in the main channel, the swifter be the main current the 

 closer will the hquid boundary be to the rigid boundary, the more flat 

 will be the lines of flow of the liquid boundary and the less will be the 

 slope of these lines to the axis of flow; third the relative quantities of water 

 flowing in the main channel and outside its boundary. The further the 

 particles of water are from the disturbing influences of the rigid bound- 

 ary the less their energies are disturbed bj' these influences and this is 

 measured by some increasing function of the mean hydraulic radius and 

 consequently the mean slope of the liquid boundary is relatively dimin- 

 ished by an increase of the mean hydraulic radius. 



3. In view of the above observations we now postulate the hypotheti- 

 cal case of a straight horizontal pipe whose inner boundary is a surface of 

 revolution whose meridian section is a pair of symmetrical sinuous curves, 

 as shown in the figure, such that there are 2n cross sections alternately 

 contractions and expansions uniformly distributed throughout the pipe 

 length L at a distance apart AL. Let fi be the area of the expanded sec- 

 tion at B of radius R, and w that of a contracted section at A or C of radius 



