Lord Rayleigh on the Resistance of Fluids. 431 



this objection that a mathematically sharp edge is an impossi- 

 bility, inasmuch as the electrical law of flow would require ne- 

 gative pressure in cases where the edge is not perfectly sharp, 

 as may be readily proved from the theory of the simple cir- 

 cular vortex, in which the velocity varies inversely as the dis- 

 tance from the axis. 



The application of these ideas to the problem of the resist- 

 ance of a stream to a plane lamina immersed transversely 

 amounts to a justification of the older theory as at least ap^ 

 proximately correct. Behind the lamina, the fluid is at rest 

 under a pressure equal to that which prevails at a distance, 

 the region of rest being bounded by a surface of separation or 

 discontinuity which joins the lamina tangentially, and is deter- 

 mined mathematically by the condition of constant pressure. 

 On the anterior surface of the lamina there is an augmenta- 

 tion of pressure corresponding to the loss of velocity. 



The relation between the velocity and pressure in a steady 

 stream of incompressible fluid may be obtained immediately 

 by considering the transference of energy along an imaginary 

 tube bounded by stream-lines. In consequence of the steadi- 

 ness of the motion, there must be the same amount of energy 

 transferred in a given time across any one section of the tube 

 as across any other. Now if p and v be the pressure and velo- 

 city respectively at any point, and p be the density of the fluid, 

 the energy corresponding to the passage of the unit of volume 

 is p + ipv 2 , of which the first term represents potential, and 

 the second kinetic energy ; and thus p + \pv 2 must retain the 

 same value at all points of the same stream-line. It is further 

 true, though not required for our present purpose or to be 

 proved so simply, that p + ^pv 2 retains a constant value not 

 merely on the same stream-line, but also when we pass from 

 one stream-line to another, provided that the fluid flows 

 throughout the region considered in accordance with the elec- 

 trical law. 



If u be the velocity of the stream, the increment of pressure 

 due to the loss of velocity is \pu 2 — \pv 2 , and can never exceed 

 ^pu 2 , which value corresponds to a place of rest where the 

 whole of the energy, originally kinetic, has become poten- 

 tial. The old theory of resistances went on the assump- 

 tion that the velocity of the stream was destroyed over the 

 whole of the anterior face of the lamina, and therefore led 

 to the conclusion that the resistance amounted to \pu 2 for 

 each unit of area exposed. It is evident at once that this is 

 an overestimate, since it is only near the middle of the ante- 

 rior face that the fluid is approximately at rest; towards the 

 edge of the lamina the fluid moves outwards with no inconsi- 



