70 HYDKAULICS AND ITS APPLICATIONS 



ART. 26. STEADY FLOW THROUGH A CIRCULAR TUBE, ASSUMING 

 SLIP AT THE BOUNDARIES. 



If we assume slip at the boundaries of a circular tube, some assump- 

 tion must be made as to the magnitude of the slip before any determina- 

 tion of the pressure fall along the tube can be made. Assuming, as 

 seems rational, that the velocity at the boundaries is proportional to the 



tangential stress u ( -= ) we have if u f = velocity at boundaries, then 

 \a rj r ^ a> 



id u\ 7 id u\ Jt a d p , 



u' oc /x U ) or u r = k U ) = ~- . -f- from (4), Art. 25. 



\d r/ r = a \drj r = a 2/x dx 



Using this to determine B in equation (1), Art. 25, and proceeding as 

 before, we finally get 



TT a 4 ( 4 k \ PI PZ , . 

 Q = Q -. r cubic feet per second. 



ART. 27. GENESIS OP EDDY FORMATION. 



The reason lot the sudden change from steady to sinuous motion in a 

 pipe at the critical velocity is not clear. It has been suggested that the 

 change takes place when the shear stress, accompanying the varying rate 

 of flow across the pipe, becomes greater than that which the liquid is 

 capable of withstanding. That this theory is untenable is clear if it be 

 remembered that in viscous flow through a pipe the maximum shear 



occurs at the boundaries, and has the value - ~ (p. 69). For the theory 

 to hold, this shear stress must have the same value in any pipe for the 

 initiation of eddy motion, so that r -~ must be constant. 



Ci X 



But r ^ = 5-^-& = ^-^ (P- 69), and therefore the critical 

 d x TT i* r 



velocity should vary directly as the radius of the pipe. This result is 

 directly opposed to the results of observations, experiments indicating 

 that this velocity is inversely proportional to the radius. 



The law governing the position of the critical point was inferred by 

 Reynolds from a consideration of the general equations of motion for a 

 viscous fluid. 



Here the first equation of (9), p. 61, is 



