Turbulent Flow in Pipes and Channels. #25 



and discussed in Part III.) indicate the somewhat surprising- 

 result that the parabolic distribution calculated from the 

 usual viscous fluid theory is only a rough approximation to 

 a much more complicated state of affairs. 



Section 3. Stead?./ Motion in a Pipe of Circular 

 Cross- Section. 



Taking the axis of z to coincide with that of the circular 

 cylinder, the pressure-gradient in this direction is constant 

 and is given by 



*pftz=( Pi - Pl )/l, (9) 



P2—P1 being the pressure-difference measured at points a 

 distance I apart. The equation for steady motion is 9 



^r^)^-r( P2 - Pl )/l, . . . (10) 



of which the general solution is 



W=-^(p 2 - j p 1 )/(4^)+Alogr+B . . (11) 



Under conditions of finite velocity along the axis r = 

 and no slipping at the boundary r = a, we have 



W=(a 2 -»- 2 )( ft -)b 1 )/(V)- • • • (12) 



The maximum velocity at r=0 is given by 



W =a\p,-p 1 )/(4pd), .... (13) 



and the total flow is 



C a 

 3>=l W .27rrdr=7Ta*{p 2 - 1 J l )/($fjLl) = i7ra 2 \\\ ) , (14) 



Jo 

 and the mean velocity over the cross-section is 



W = iW (15) 



This solution forms the basis of most of the practical 

 methods of determining the coefficients of viscosity of a 

 liquid or of a gas ; it is also the starting point of some of 

 the theoretical treatments on the stability of stream-line flow, 

 and, until the writer's own experiments on the subject, all 

 observations on the laws of flow were confined to the case 

 of pipes of circular cross-section. It is generally assumed 

 that the theory of viscosity obtains its application to reality 

 in that equation (14) contains all the laws found experi- 

 mentally by Poiseuille 10 ; although these laws have been 



9 Lamb. < Hydrodynamics,' 1906, r. 543. 

 10 Poiseuille,*' Compfes Rendits, vol. xv. p. 1167 (1842). 



