  An analytical method is developed describing the approach to a finite-time
singularity associated with collapse of a narrow fluid layer in an unstable
Hele-Shaw flow. Under the separation of time scales near a bifurcation point, a
long-wavelength mode entrains higher-frequency modes, as described by a version
of Hill's equation. In the slaved dynamics, the initial-value problem is solved
explicitly, yielding the time and analytical structure of a singularity which
is associated with the motion of zeroes in the complex plane. This suggests a
general mechanism of singularity formation in this system.
