# Computational Analysis of Drop Formation and Detachment

This article was contributed by Jelena Dinic and Vivek Sharma, Department of Chemical Engineering, University of Illinois at Chicago, Chicago, IL

## Introduction and Problem Statement

The rapid, repeated, precise creation and deposition of droplets, printing or patterning of small features (say l = 10-3-1 mm), and the formation of thin films with controlled, uniform thickness by spraying, are of great importance to a variety of old and new industrial applications (1-5). The liquid transfer and drop formation/deposition processes involve complex free-surface flows and formation of columnar necks that undergo spontaneous capillary-driven instability, thinning and pinch-off (1-5). Despite the progress made using experimental, theoretical and one-dimensional simulation studies for analyzing drop formation and liquid transfer for simple Newtonian and inelastic fluids, mechanistic understanding of printing and spraying remains a challenge. The primary motivation for the present computation effort is to examine the possibility of using the volume-of-fluid (VOF) approach embedded in the FLOW-3D to obtain mechanistic understanding of pinch-off dynamics of Newtonian fluids. We show that our computational analysis captures the complex interplay of capillary, inertial and viscous stresses that determines the self-similar capillary thinning and pinch-off dynamics. For the drop formation and detachment of Newtonian fluids, we show that the self-similar neck evolution obtained from the computational analysis can be described using the universal scaling laws expected from theory and 1D simulations (1-7) as well as experiments (1, 2, 8-12). Our success in simulating such prototypical flows is a necessary step towards using FLOW-3D for careful computational analysis of the nonlinear dynamics underlying finite-time singularity, satellite drop formation as well as printability in more complex geometries, that are significantly harder to describe or study using 1D models and experiments.

## Modeling Approach and Parameter Space

Simulations of drop formation and detachment from a nozzle were performed using uniform mesh size in FLOW-3D by using the Surface Tension and Gravity models. Drop formation and detachment of a finite volume of fluid involves the interplay of capillarity, inertia, viscosity and gravitational-induced drainage within a neck that connects a pendant growing drop to a nozzle. In the simulation, a finite volume of a Newtonian fluid is issued from a stainless steel nozzle (${{D}_{0}}=2{{R}_{0}}=1.7\,\text{mm}$). Detachment of a newly formed drop occurs once the gravitational force overcomes surface tension force ($mg>2\pi \sigma {{R}_{0}}$). Simulations are divided into two groups to elucidate the dramatic influence of fluid viscosity: low viscosity fluids (e.g., water and glycerol/water mixtures with glycerol content <40% by weight) and high viscosity fluids (e.g. glycerol and glycerol/water mixtures with shear viscosity > 100x Water Viscosity). Properties of the fluids of both groups are listed in Table 1 and 2, respectively.

Simulation of low viscosity fluid (Fluid 2 in Table 1) dripping from a nozzle. Color variable is the velocity magnitude (units: cm/s) and velocity vectors are shown.

Simulation of high viscosity fluid (Fluid 8 in Table 2) dripping from a nozzle. Color variable is the velocity magnitude (units: cm/s) and velocity vectors are shown.

## Discussion of the Simulation Results

Drop formation and detachment was simulated using FLOW-3D for fluids listed in Table 1 and 2, and the neck shape and neck radius evolution over time were analyzed. The shape of the neck and the neck thinning dynamics of low viscosity fluids (see Figure 1) exhibits the characteristic self-similar, inertio-capillary thinning behavior, anticipated by experiments, potential flow theory and 1D simulations (1, 2, 6, 7, 13):

(1) $\displaystyle \frac{{R(t)}}{{{{R}_{0}}}}\approx 0.8{{\left( {\frac{\sigma }{{\rho R_{0}^{3}}}} \right)}^{{\frac{1}{3}}}}{{\left( {{{t}_{c}}-t} \right)}^{{\frac{2}{3}}}}$

Here R(t) is the instantaneous radius of the neck, R0 is outer radius of the nozzle, $\displaystyle \sigma$ is the surface tension, $\displaystyle \rho$ is the density of the fluid and tc is the pinch-off time. Likewise, the radius evolution datasets for these higher viscosity Newtonian fluids appear to show a linear decrease in neck radius with time, and the thinning dynamics follow Papageorgiou’s visco-capillary scaling (8, 9) described by the following expression:

(2) $\displaystyle \frac{R}{{{{R}_{0}}}}=0.0709\frac{\sigma }{{{{\eta }_{s}}{{R}_{0}}}}({{t}_{p}}-t)$

We find that the measured values of capillary velocity (ratio of surface tension and viscosity) are comparable to the values obtained using commercially-available instrument called Capillary Break-up Extensional Rheometer (CaBER) by McKinley and Tripathi (8), and to the capillary velocity computed using their nominal surface tension and viscosity.

FLOW-3D allows for visualization of velocity vectors in the neck during thinning which gives insight into a nature of the flow. In addition, it gives the possibility of determining the moment during thinning after which the flow field within the thinning neck after initially experiencing a combination of shear and extension becomes purely extensional as shown in Figure 1. Additionally, thinning dynamics of low viscosity fluids show a qualitatively different behavior compared to high viscosity Newtonian fluids (see Figure 2). The neck profile for a low viscosity Newtonian fluid, becomes self-similar in agreement with theory (6, 13), in frames leading to the pinch-off.

## Conclusions, Outlook and Ongoing Work

Our preliminary results show that FLOW-3D based computational analysis can be used for simulating prototypical free-surface flows underlying drop formation and detachment. We find that the simulated radius evolution profiles match the scaling laws and pinch-off dynamics that are experimentally-observed and theoretically-predicted for inviscid fluids as well as high viscosity Newtonian fluids.

In contrast with often-used 1D or 2D models, FLOW-3D allows a robust evaluation of the magnitude of the underlying stresses and extensional flow field (both uniformity and magnitude) and the visualization of the flow filed within the thinning liquid filament (see Figure 1 and 2, for example). Stream-wise velocity gradients associated with extensional flow field arise within the columnar necks undergoing capillary-driven thinning. In rheologically-complex fluids, extra elastic stresses as well as non-Newtonian shear and extensional viscosity dramatically alter the nonlinear pinch-off dynamics (2, 10-12). We are currently implementing constitutive models with viscoelasticity and non-Newtonian rheology into FLOW-3D to develop robust computational protocols for assessing processability of complex fluids.

### References

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