Viscous Liquid Threads with Inner Fluid Flow Inside Microchannels

Forming droplets are often accompanied by an interconnecting liquid thread. It is postulated that this phenomenon can only exist as long as a pressure gradient exists within the thread, for instance, when a viscous liquid is conveyed via the liquid thread to the forming droplet. We have built a microfluidic setup to form and sustain a liquid thread, which after a length L ends in a droplet. To prevent the droplet from moving up too fast due to buoyancy, we force the droplet to shift along a tilted ceiling, which can be positioned at three different angles. This enables us to keep the gradual lengthening of the liquid thread under control. Based on the Navier-Stokes equation, we are able to predict the axial shape of such a liquid thread as a function of fluid mass density, initial thread radius, initial fluid velocity at the nozzle, fluid viscosity, and surface tension. Although an explicit solution of the governing differential equations is not known, we managed to find an explicit approximating expression for the shape function, which shows excellent agreement with both the measured and the numerically calculated shape functions. An intriguing phenomenon observed in the experiments is the breakup of the thread. This breakup always occurs close to the droplet. Using our approximating solution, we derive a relation that connects, for any time in the development of the thread, its length and the pressure gradient stemming from, among other effects, the shear at the interface of the liquid thread due to motion of the inner liquid. For relatively short thread lengths, this relation is linear on a log-log scale, due to the fact that in this regime, viscosity effects are dominant. However, if the thread length increases, this relation starts to deviate from linear behavior, due to surface tension effects. We show from the experimental results that the thread starts to show unstable behavior as soon as these capillary effects come into play. We show how to predict the thread length at which the capillary instability sets in for any liquid thread system. It is found that the predicted maximum dimensionless thread length is given by Lmax,pred ≈ 12Ca with Ca the capillary number.