  A filament of liquid is usually unstable and breaks up into small droplets,
while a filament of polymer solution is known to be quite stable against such
instability, and they form a stable configuration of filament connecting two
spherical droplets. If the droplets are fixed in space, the liquid flows from
the filament region to the droplet region to reduce the surface energy and the
filament gets thinner. If the whole liquid is placed in another viscous fluid,
the droplets approach each other, and the filament can get thicker. Here we
study the dynamics of such system. We derive time evolution equations for the
radius and the length of the filament taking into account the fluid flux from
filament to droplets and the motion of the droplets. We will show that (a) if
the centers of the droplets are fixed, the filament thins following the
classical prediction of Entov and Hinch, and that (b) if the droplets are
mobile (subject to the Stokes drag in the viscous medium), the thinning of the
filament is suppressed, and under certain conditions, the filament thickens.
This theory explains the phenomena observed by Yang and Xu [Phys. Fluids 20,
043101 (2008)] in four-roller mill device.
