310-311] TURBULENT FLOW OF A LIQUID. 573 



degrees of mobility. Unless the velocities be very small, the 

 actual motion, in such cases, so far as it admits of being observed, 

 is found to be very different from that represented by our formulae. 

 For example, when a solid of ( easy ' shape moves through a liquid, 

 an irregular, eddying, motion is produced in a layer of the fluid 

 next to the solid, and a widening trail of eddies is left behind, 

 whilst the motion at a distance laterally is comparatively smooth 

 and uniform. 



The mathematical disability above pointed out does not apply 

 to cases of rectilinear flow, such as have been discussed in Arts. 

 288, 289 ; but even here observation shews that the types of 

 motion there investigated, though always theoretically possible, 

 become under certain conditions unstable. The case of flow 

 through a pipe of circular section has been made the subject of 

 a very careful experimental study by Reynolds*, by means of 

 filaments of coloured fluid introduced into the stream. So long 

 as the mean velocity (w ) over the cross-section falls below a 

 certain limit depending on the radius of the pipe and the nature 

 of the fluid, the flow is smooth, and in accordance with Poiseuille's 

 laws ; but when this limit is exceeded the motion becomes wildly 

 irregular, and the tube appears to be filled with interlacing and 

 constantly varying streams, crossing and recrossing the pipe. It 

 was inferred by Reynolds, from considerations of dimensions, that 

 the aforesaid limit must be determined by the ratio of w a to v, 

 where a is the radius, and v the (kinematic) viscosity. This was 

 verified by experiment, the critical ratio being found to be, 

 roughly, 



Thus for a pipe one centimetre in radius the critical velocity for 

 water (z/ = '018) would be 18 cm. per sec. 



Simultaneously with the change in the character of the motion, 

 when the critical ratio is passed, there is a change in the relation 

 between the pressure-gradient (dpjdz) and the mean velocity W Q . 

 So long as w Q a/v falls below the above limit, dp/dz varies as w Q , as 



* "An Experimental Investigation of the Circumstances which determine 

 whether the Motion of Water shall be Direct or Sinuous, and of the Law of Besist- 

 ance in Parallel Channels," Phil. Trans., 1883. 



t The dependence on v was tested by varying the temperature. 



