MOTION IN RELATION TO THE SURFACE FRICTION OF FLUIDS. 
201 
Comparing this equation with the theoretical value of dp/dx for stream-line motion, i.e., 
dp _ 32 /ulv 
dx d? 
it will be seen that under conditions of stream-line motion 
r fvd\ vd 
f \~) ” 7 ' 
For velocities above the critical value, and as high as could be carried in his apparatus, 
Reynolds considered that the function on the right-hand side of the equation could be 
expressed by the relation 
f/vd\ fvd\ n 
where n has a value which is constant for any given pipe, but may vary from 1 '75 
to 2 according to the roughness of the surface. As will be seen later, this conclusion 
has not been verified in the present experiments, which show definitely that when the 
range in speed is considerable the index law fails to represent the results. 
It has been shown by Lord Rayleigh that by the Principle of Dynamical Similarity 
the relation expressed by equation (l) is only a particular case of a general law of 
resistance of bodies immersed in fluids moving relatively to them, under the assumption 
that this resistance depends only on the linear dimensions of the body and on the 
velocity, density, and kinematical viscosity of the fluid. This relation may be 
expressed as 
E = ^ ! f(^).(2) 
where R is the resistance per unit area and F is a function of the one variable vd/v. 
From the foregoing it appears that similarity of motion in fluids at constant values 
of the variable vd/v will exist, provided the surfaces relative to which the fluids move 
are geometrically similar, which similarity; as Lord Rayleigh has pointed out, must 
extend to those irregularities in the surfaces which constitute roughness. In view of 
the practical value of the ability to apply this principle to the prediction of the 
resistance of aircraft from experiments on models, experimental investigation of the 
conditions under which similar motions can be produced under practical conditions 
becomes of considerable importance, and during the last three years the accuracy of 
the assumptions made in the derivation of equation (2) has been tested at the National 
Physical Laboratory under varying conditions. By the use of colouring matter to 
reveal the eddy systems at the back of similar inclined plates in streams of air and 
water, photographs of the systems existing in the two fluids when the value of vd/v 
was the same for each, have been obtained, and their comparison has revealed a 
remarkable similarity in the motions.* For the case of surface friction, experiments 
Report of Advisory Committee for Aeronautics,’ 1911-12, p. 97. 
2 D 
YOL. CCXIY.—A. 
