326 Prof. L.Vessot King on 



amply veiified in the case of tubes of capillary diameters, 

 a marked deviation is easily noticeable as soon as the 

 diameters exceed a few millimetres, even at velocities 

 considerably below what is taken to represent the " critical 

 velocity." 



Section 4. Shearing Motion between Parallel Planes. 



Measuring y perpendicular to the moving planes at a 

 distance d apart, the equation for steady motion is d 2 TJIdij 2 = 0, 

 of which the solution appropriate to U = at y=0, and 

 U= U at y = d is 



U = Uo3//d, (16) 



which, it will be noticed, does not involve the viscosity. 

 The shearing stress between the planes per unit area is 



F= /t (<ro/^) = /i L 1 o/rf (17) 



In this case experiment measures the shearing stress F ; 

 it may be remarked here that in these circumstances the 

 fundamental law of viscosity relates to the traction between 

 the fluid and a solid boundary and not to the traction 

 between layers of fluid, as would be revealed by an experi- 

 mental analysis of the velocity-gradients themselves. 



The simple case just considered is not realizable experi- 

 mentally, although it is made the starting point of one of 

 the important cases of laminar flow examined theoretically 

 from the standpoint of stability. The appropriate experi- 

 mental arrangements involve the measurement of the tractions 

 between circular disks, as in Maxwell's classical experiments, 

 or between concentric spheres or cylinders ; observations 

 are generally carried out by oscillation methods, although 

 an apparatus has recently been constructed by Gilchrist 11 

 which enables the torque clue to viscous shear between 

 relatively rotating coaxial cylinders to be measured. Taking 

 the case of a cylinder of radius a, rotating with angular 

 velocity co inside a coaxial cylinder of radius b, the angular 

 velocity in the fluid at radius r is given by 12 



cor=(a/r) . (b 2 - r 2 ) I (b 2 - a 2 ) . » a, . . (18) 



while the couple exerted between the cylinders is, per unit 

 length, 



L=-4t7Tfia 2 a) .b 2 /(b 2 r-a 2 ). . . . (19) 



Gilchrist points out, in a review of the best experimental 

 data available at the date of his paper, that small discrepancies 



11 Gilchrist, Phvs. Rev. vol. viii. p. 124 (1913). 



12 Lamb, < Hydrodynamics,' 1906, p. 546. 



