1920-21.] Stable Flow of a Fluid in a Uniform Channel. 137 
Pannell,* who have shown that the conclusions arrived at by Osborne 
Reynolds as regards water are independent of the state of the fluid — that, 
in fact, the critical value of \Jl/v is a universal constant for given geometry 
of boundary. From another standpoint, these results have since received 
verification in all hydro- and aero-dynamical experiments, where it is found 
that the resisting force R of a body of given shape in a viscous fluid of 
density p and kinematic viscosity v is always expressible in the form 
where l fixes the scale of the body. In effect these experiments may be 
regarded as justifying the assumption that the properties of real fluids in 
motion should require nothing further for their explanation than the 
assumption that the fluid is viscous and dense. It is important in this 
connection to note that compressibility plays no part in this question, 
and that for the present purpose the air may be justifiably regarded as 
inelastic. Examination of the resistance of projectiles indicates that 
compression effects do not become apparent until the velocity of sound 
is approached, j- 
§ 3. Returning to the flow of a fluid in a channel, let it be supposed 
that a disturbance is communicated to the fluid, say, by dipping a small 
obstacle into the fluid and withdrawing it. For a given shape of obstacle 
inserted in any given manner, for a given speed U of the central stream 
line of the channel, and for given viscosity and density of the fluid, it is 
clear that the vortex distribution immediately resulting from this dis- 
turbance will be physically quite determinate. Let the strength of the 
vorticity at a geometrically given position be k, then k can only be 
dependent on U, l the breadth of the channel, v the kinematic viscosity, 
and p the density, and on nothing else. 
Hence 
K=/(U, Z, V, p). 
Now k being an angular velocity distributed over an area has dimensions 
XL 2 . 
Assuming 
.-. [k] = L 2 T -1 ; [U] = LT -1 ; [v] = LH -1 ; [p] = ML- 3 . 
* Phil. Trans ., A, 214, pp. 199-224, “ Similarity of Motion, in Relation to the Surface 
Friction of Fluids.” 
f Rayleigh’s Scientific Papers , vol. v, p. 534 ; Aeronautical Journal , June 1919, “From 
Model to Full Scale in Aeronautics,” H. Levy. 
