Dynamics of a viscoelastic liquid filament connected to two mobile droplets

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 a 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 a system. We derive time evolution equations for the radius and the length of the filament taking into account the fluid flux from the filament to the 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 (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 [“Coalescence of two viscoelastic droplets connected by a string,” Phys. Fluids 20, 043101 (2008)] in a four-roller mill device.