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XXXI. A proposed Hydraulic Experiment, 

 By Lord Rayleigh, OJL, F.R.S* 



IN an early paperf Stokes showed " that in the case 

 o£ a homogeneous incompressible fluid, whenever 

 udcc 4- vdy + wdz is an exact differential, not only are 

 the ordinary equations of fluid motion satisfied, but the 

 equations obtained when friction is taken into account are 

 satisfied likewise. It is only the equations of condition 

 which belong to the boundaries of the fluid that are violated." 

 In order to satisfy these also, it is only necessary to suppose 

 that every part of the solid boundaries is made to move with 

 the velocity which the fluid in irrotational motion would 

 there assume. There is no difficulty in the supposition itself; 

 but the only case in which it could readily be carried into 

 effect with tolerable completeness is for the two-dimensional 

 motion of fluid between coaxial cylinders, themselves made 

 to rotate in the same direction with circumferential velocities 

 which are inversely as the radii. Experiments upon these 

 lines, but not I think quite satisfying the above conditions, 

 have been made by Conette and Mallock. It would appear 

 that, except at low velocities, the simple steady motion 

 becomes unstable. 



But the point of greatest interest is not touched in the 

 above example. It arises when fluid passing along a uniform 

 or contracting pipe, or channel, arrives at a place where the 

 pipe expands. It is known that if the expansion be suffi- 

 ciently gradual, the fluid generally speaking follows the 

 walls, or, as it is often expressed, the pipe flows full ; and 

 the loss of velocity accompanying the increased section is 

 represented by an augmentation of pressure, approximately 

 according to Bernoulli's law. On the other hand, if in 

 order to effect the conversion of velodhVv into pressure more 

 rapidly, the expansion be made too violently, the fluid refuses 

 to follow the walls, eddies result, and mechanical energy is 

 lost by fluid friction. According to W. Froude's generally 

 accepted view, the explanation is to be sought in the loss of 

 velocity near the walls in consequence of fluid friction, which 

 is such that the fluid in question is unable to penetrate into 

 what should be the region of higher pressure beyond. 



It would be a difficult matter to satisfy the necessary 



* Communicated by the Author. 



t Camb. Trans, vol. ix. p. [S], 1850 ; Math, and Phys. Papers, vol. iii. 

 p. 73. 



Y2 



