100 MOTION OF A LIQUID IN TWO DIMENSIONS. [CHAP. IV 



Discontinuous Motions. 



73. We have, in the preceding pages, had several instances of 

 the flow of a liquid round a sharp projecting edge, and it appeared 

 in each case that the velocity there was infinite. This is indeed a 

 necessary consequence of the assumed irrotational character of the 

 motion, whether the fluid be incompressible or not, as may be 

 seen by considering the configuration of the equipotential surfaces 

 (which meet the boundary at right angles) in the immediate 

 neighbourhood. 



The occurrence of infinite values of the velocity may be 

 avoided by supposing the edge to be slightly rounded, but even 

 then the velocity near the edge will much exceed that which 

 obtains at a distance great in comparison with the radius of 

 curvature. 



In order that the motion of a fluid may conform to such 

 conditions, it is necessary that the pressure at a distance should 

 greatly exceed that at the edge. This excess of pressure is 

 demanded by the inertia of the fluid, which cannot be guided 

 round a sharp curve, in opposition to centrifugal force, except by 

 a distribution of pressure increasing with a very rapid gradient 

 outwards. 



Hence unless the pressure at a distance be very great, the 

 maintenance of the motion in question would require a negative 

 pressure at the corner, such as fluids under ordinary conditions 

 are unable to sustain. 



To put the matter in as definite a form as possible, let us 

 imagine the following case. Let us suppose that a straight tube, 

 whose length is large compared with the diameter, is fixed in the 

 middle of a large closed vessel filled with frictionless liquid, and 

 that this tube contains, at a distance from the ends, a sliding 

 plug, or piston, P, which can be moved in any required manner by 

 extraneous forces applied to it. The thickness of the walls of 

 the tube is supposed to be small in comparison with the diameter; 

 and the edges, at the two ends, to be rounded off, so that there are 

 no sharp angles. Let us further suppose that at some point of the 

 walls of the vessel there is a lateral tube, with a piston Q, by 

 means of which the pressure in the interior can be adjusted at will. 



