1
Homwork, Fluid Dynamics (1.63J/2.21J),
C. C. Mei, 1-cavity.tex, 1999
Cavity collapse
A spherical cavity of initial radius Ro suddenly collapses.Ignore surface tension and water
coressibiiilty. Th ecavity has the pressure po which is smaller than th pressure at infinity p∞ .
Use continuity that
4πr 2 u = constant for r ≥ R(t)
(0.1)
where R(t) is the radius of the cavity, and u(r, t) is the radial velocity. Use also the law of
mometum conservation
!
∂u
∂p
∂u
+u
=−
(0.2)
ρ
∂t
∂r
∂r
Integrate the momentum equation from r = R to r = ∞ to get a differential equation of
the following form
!
p∞
d2 R 3 dR
=−
(0.3)
R 2 +
dt
2 dt
ρ
and show that it can be rewritten as

1 d  3 dR
R
2 dt
dt
!2 

With the initial conditions
=−
∆p d(R3 )
3ρ dt
dR(0)
=0
dt
R(0) = Ro ,
(0.4)
(0.5)
Show that the time to complete collapse is
T =
s
3ρ
2∆p
Z Ro
0
dR
q
Evaluate the integral in terms of Gamma functions.
R2o
R2
−1
(0.6)