FLUID FRICTION 175 



(3) Where a submerged plane moves through still water, the resistance 

 is not proportional to the area of the surface, but, per unit area of the 

 surface, decreases as the length of the latter increases, and approaches a 

 lower limiting value. In the case of flow through a pipe, the length is 

 usually such as to allow of this limiting constant value being attained, so 

 that here the frictional resistance does become practically proportional to 

 the area of the wetted surface. 



(4) Varies with the nature of the wetted surface. 



(5) Varies only slightly with temperature, but is proportional to 

 some power of the density of the fluid, usually slightly less than 

 the first. It is usually assumed to be directly proportional to the 

 density. 



By far the most important series of experiments to determine the 

 resistance to the motion of submerged planes, are those carried out in 1872 

 by Mr. Froude, at Torquay. Here a series of flat boards, having 

 differently prepared surfaces, were held vertically and suspended from a 

 carriage which was driven at an uniform speed, and were thus towed 

 endwise through the still water in a large basin. The carriage was fitted 

 with a dynamometer and automatically recorded the velocity and resist- 

 ance of the board. These boards were T 3 ^ inch wide, 19 inches deep, and 

 varied in length from 1 foot to 50 feet. The top edge was submerged to a 

 depth of 1J inches and the boards were fitted with a cut-water, the 

 resistance to this being determined separately. In these experiments 

 Mr. Froude determined that 



(1) The resistance varies greatly with the condition of the surface, the 

 resistance for boards 50 feet long at a velocity of 10 feet per second being 

 with a coating of 



fVarnish or smooth paint of such composition as] 



is found on the bottom of iron shipB . J ' 



Tinfoil "246 



Fine sand '405 



Calico ... . . '470 



Sand of medium coarseness . . . '488 



(2) The resistance is proportional to v n , where 



(a) depends on the surface, 



(b) decreases, up to a certain limit, with an increase in length, 



(c) is sensibly independent of the velocity. 



(3) The total resistance increases with the length, though the resistance 

 per square foot decreases as the length increases. 



Writing : Resistance = f S v n 



