254 EXERCISES. 



11. Prove, and interpret, the following formula for the energy of a 

 liquid moving irrotationally in two dimensions: 2T = p fcf&amp;gt;di}/, the inte 

 gration extending round the boundary. 



12. When a liquid moves irrotationally in two dimensions round a 

 corner where two branches of the boundary meet at an angle a 

 (measured on the side of the liquid) which is &amp;lt;180, shew that the 

 particle at the comer is in a position of permanent or instantaneous rest 

 according as a:f 90 or &amp;gt;90. 



What takes place when a&amp;gt; 180 ? [Stokes.] 



13. A stream of uniform depth and of uniform width 2a flows 

 slowly through a bridge consisting of two equal arches resting on a 

 rectangular pier of width 26, the bridge being so broad that under it the 

 fluid moves uniformly with velocity U. Shew that, after the stream 

 has passed through the bridge, the velocity-potential of the motion is 



a-b rr 2aU 1 mrb mry - 



Ux + 5- 2, -s sin - cos - e a &amp;gt; 

 a TV&quot; L n a a 



the axis of x being in the forward direction of the stream, and the origin 

 at the middle point of the pier. 



Find the equation of the path of any particle of the water. 



[C. Niven, Math. Trip., 1878.] 



14. The transverse section of a uniform prismatic closed vessel is 

 of the form bounded by two intersecting hyperbolas represented by the 

 equations ^2 (x 2 - y 2 ) + x 2 + y 2 = a 2 , v 2 (y 2 - x 2 ) + x 2 + y 2 = b 2 . 



If the vessel be filled with water, and be made to rotate with angular 

 velocity w about its axis, prove that the initial component velocities of 

 any point (x, y) of the water will be 



and - a ~ T2 {2* 3 - 6V + J 2 (a 2 - b 2 ) x}, 



respectively. [Ferrers, Math. Trip., 1872.] 



15. Work out by the method of Art. 95 the solution in the case 

 where a stream of liquid, of given breadth, impinges perpendicularly on 

 an infinite plane lamina. [Kirchhoff.] 



16. An anchor-ring is in motion parallel to its axis in an infinite 

 mass of liquid. Shew by a diagram the arrangement of the stream-lines 

 and equipotential surfaces, firstly when there is, and secondly when there 

 is not, cyclic motion through the ring. 



