394 HYDRAULICS AND ITS APPLICATIONS 



involved if the frictional resistance be calculated on the assumption that 

 the wetted surface is equivalent to that of a plane of equal area and length 

 in the direction of motion, and moving at the same speed. The frictional 

 resistance of the ship may thus be calculated from Froude's experimental 

 results on the resistance of plane surfaces (Art. 61). 



With an inviscid fluid, the particles displaced laterally by the prow 

 would move over surrounding particles without any frictional losses and 

 without any tendency to eddy formation, and would immediately return 

 to exert a pressure on the stern equivalent to that on the bows. Thus 

 the only resistance to uniform motion through such a fluid would be that 

 due to wave formation, and, with a deeply immersed body, would be 

 zero. A geometrical construction for the stream lines in a perfect fluid 

 has been deduced by Professor Rankine, while Professor Hele-Shaw has 

 verified the accuracy of this construction by experiments on a viscous 

 fluid flowing past an obstacle, the motion taking place between parallel 

 glass plates at a very small distance apart. 1 In this case the motion is 

 governed almost entirely by viscosity, all eddy motion is prevented, and, 

 as proved by Professor Sir Gr. G. Stokes, 1 the effect as regards stream-line 

 formation is the same as in the case of a perfect fluid. 



This may be shown as follows : 2 



At all points in the same horizontal plane in a perfect fluid we have, if u 



v w 2 

 be the velocity of flow, ^ -j- x = const. 



If the stream lines are curved, and if & p be the change in pressure, due to 

 centrifugal action, across a stream tube* of radial width 5 r, and of radius 

 of curvature r, we have (p. 95) 



d^_ Wjtf 

 d r g r 

 and on substituting this value in (1), 



+ ^ = 0. (2) 



r T d r 



In the case of a viscous fluid flowing with stream line motion between 

 two parallel plates, if s be the direction of flow at any instant, we have 



(p. 67) ^=0; '|^ = 0; d -f- = M r^; while if u be the mean velocity 

 d y cl z d s d y 



1 British Association Report, 1898 ; also " Transactions Inst. Naval Architects," 18!>8. 

 vol. 86. 

 a Dunkerlcy, "Transactions Inst. Naval Architects," 1900, vol. 42, p. 227. 



