during the Collapse of a Spherical Cavity, 95 



As formulated by Besant *, the problem is — 

 "An infinite mass of homogeneous incompressible fluid 

 acted upon by no forces is at rest, and a spherical portion 

 of the fluid is suddenly annihilated ; it is required to find 

 the instantaneous alteration of pressure at any point of the 

 mass, and the time in which the cavity will be filled up, 

 the pressure at an infinite distance being supposed to remain 

 constant/'' 



Since the fluid is incompressible, the whole motion is deter- 

 mined by that of the inner boundary. If U be the velocity 

 and R the radius of the boundary at time t, and u the 

 simultaneous velocity at any distance r (greater than R) 

 from the centre, then 



u/u=n 2 y ; (i) 



and if p be the density, the whole kinetic energy of the 

 motion is 



J -8 CO 



.±Trr 2 dr = 27rpWW. ... (2) 



Again, if P be the pressure at infinity and E the initial 

 value of R, the work done is 



47rP 



-PW-R 3 ) (3) 



When we equate (2) and (3) we get 



<"=!(¥-') m 



expressing the velocity of the boundary in terms of the 

 .radius. Also, since J] = dK/dt, 



,_ /(*p\ P° (R S -^R _ B /A>\ f 1 p»d/3 

 '"VlsPJj, W-R»)* - B »V UtJ • }„ (IZtfji 



• • • (5) 

 if /3 = R/R . The time of collapse to a given fraction of 

 the original radius is thus proportional to R />2P~% a result 

 which might have been anticipated by a consideration of 

 4i dimensions." The time r of complete collapse is obtained 

 by making /3 = in (o). An equivalent expression is given 

 by Besant, who refers to Cambridge Senate House Problems 

 of 1847. 



* Besant's ' Hydrostatics and Hydrodynamics,' 1859, § 158. 



