Canon Moseley on the steady Flow of a Liquid, 31 



these papers) that state of steady motion to which it would 

 attain in a channel of uniform section whose internal surface 

 was everywhere of the same degree of roughness, and whose di- 

 rection was straight and its shape constant. The liquid would 

 under these circumstances flow in filaments parallel to one an- 

 other and to the sides of the channel, having different velocities. 

 By equation (2) *, 



where 

 U = work done per unit of time on the liquid which enters 



the pipe by the pressure of that in the reservoir. 

 U, = work carried away per unit of time by the liquid which 



flows from the extremity of the pipe. 

 U2= work expended on the various resistances which are op- 

 posed to the descent of the liquid in the reservoir and 

 to its passage from the reservoir through its aperture into 

 the pipe. 

 113= work expended on the resistance of the internal surface 



of the pipe to the flow of the liquid along it. 

 114= internal work of the resistance of the films to the flowing 



of each film over the surface of the next in succession. 

 Let a plane be imagined to intersect the liquid in a direction 

 at right angles to that in which it flows, and let the point in the 

 plane where the filament of maximum velocity intersects it be 

 taken as the origin of the coordinates ; also let x and y be the 

 rectangular coordinates of the point in this plane where any other 

 filament intersects it, x being horizontal. Let vhe, the velocity 

 of this filament. Its displacement in the unit of time over the 

 filament immediately above it will then be represented by 



— (-7- Wy, the negative sign being taken because v diminishes as 



y increases. 



But the resistance opposed to the shearing by the horizontal 

 face of the filament is /t/Aa? ; //. representing the unit of resistance, 

 and / the length of the filament. Therefore the work done in 

 the unit of time by the resistance of the lower of these two fila- 

 ments to the motion of the upper is represented by 



In the same manner the work done by the resistance of the 

 side surface of the filament to the motion of the next contiguous 

 filament sidewise is represented by 



* Phil. Mag. September 1871. 



