Shape and stability in liquid threads and jets : a link to droplet formation

This thesis explores relevant fluid dynamic processes for the formation of uniformly sized droplets in microfluidic systems. Growing droplets made from a bulk source have often liquid threads or jets in between to supply liquid to the droplet. Liquid threads and jets are however known to be instable and finding parameters determining their instability/stability will possibly promote a more controlled formation of uniformly sized droplets. Different droplet formation processes in microfluidic devices are explained, such as cross-flow, co-flow and flow focussing. Dimensionless numbers (introduced in chapter 1) represent the ratio of relevant forces or pressures acting on the fluids and/or their interfaces. These forces and pressures originate from their related fluid dynamic parameters, such as viscosity, interfacial tension, mass density and velocity of the fluid within a specific fluidic confinement with a certain length scale. We show that the dimensionless Reynolds, Weber and Capillary numbers can be associated with the stability of liquid threads and/or jets and provide insight in droplet formation processes. The phenomenon of spontaneous droplet formation at low flow rates of an inner fluid confined in a microfluidic channel is studied in chapter 2. A short overview of known processes of spontaneous droplet formation with micro-engineered microfluidic devices is presented. We have studied the process of auto breakup with rectangular and round glass capillaries, the latter provided with micro-corrugations and uniform sized droplets were obtained, but only if the outer fluid is able to enter the capillary during droplet formation. The process of auto breakup is described by a new analytical model described in chapter 3. The model states that the instability of a liquid thread is induced by the decrease of a local liquid thread pressure inside the capillary near the growing droplet. Predicted droplet sizes have been experimentally verified accurately, and also the predicted breakup length inside a micro-corrugated capillary has been verified. The model states that viscous flow stabilises the liquid thread and that auto breakup happens as long as the capillary number is below a critical capillary number of 0.0625. Above 0.0625 droplets grow infinitely large. Auto breakup is however already hampered at Capillary numbers above 0.03, because between 0.03 and 0.0625 no well controlled droplet sizes could be obtained by auto breakup. This is explained by the observed formation of a partially collapsed inner liquid thread that remains open and supplies the growing droplet with inner fluid. In chapter 4 the formation and stability of a liquid thread in free surface flow feeding a large growing droplet is demonstrated and discussed. The shape of the liquid thread is positively tapering (towards the droplet) and can be described accurately by a Navier-Stokes based ordinary differential equation (ODE) assuming steady state, axisymmetry and an averaged fluid velocity over the cross section of the liquid thread. The axial shape of a viscous liquid thread is concave and its radial dimension has initially a cubic dependence with respect to the axial dimension. A driving force to stabilise the liquid thread was identified, which is a pressure gradient Q = Q0/L – Q1. Q0 is the pressure drop over thread length L, and Q1 is interfacial based dissipation of energy of the outer fluid. The maximum length of the liquid thread is predicted to be reached when Q goes towards 0 as the ratio Q0/Q1. Shape and stability of emanating liquid jets, which appear after impact of falling droplets from a deep liquid, is presented in chapter 5. During rise and fall of the jet due to gravity, the jet is additionally decelerated towards the liquid surface by a tensile retraction force from the surface tension force exerted on the jet surface by the liquid bath. The retracting force generates an inertial deceleration pressure inside the jet that is balanced by the local Laplace pressure, herewith defining its local curvature and therefore also the shape of the complete jet. A deceleration based Young-Laplace equation is introduced and the predicted shape is experimentally verified for different fluids. Furthermore, the size of droplets forming on the tip of the jet can also be explained by the found pressure balance between the local Laplace pressure and the inertial deceleration of the jet (including the forming droplet). In general we found that the stability of a liquid thread or jet seems correlated with an applied pressure difference that is distributed between the begin and end of the thread or jet. Studying auto breakup (chapters 2 and 3) of a confined liquid thread it was found that only when the applied pressure is high enough the liquid thread is stable and infinitely large droplets are formed. For the free surface flow liquid thread (chapter 4) it was found that breakup happens when the applied pressure gradient over the length of the thread goes to zero. For the emanating jet (chapter 5) an inertial pressure difference between the base and tip of the jet comes into existence that opposes the squeezing Laplace pressure that wants to break up the liquid jet. Furthermore we found that the last stages of droplet breakup from a liquid thread or jet appeared to follow universal pinch-off, and also that micro-thread formation is observed between droplet and liquid thread or jet.