Flow and clogging of capillary droplets

The oil–water interfacial surface tension is reported as
𝑔𝑜𝑤 ≈ 10 mJ m−2, “as measured by the ‘‘Dropometer’’ from Droplet Labs,” and is used to describe why droplets tend to remain round and resist deformation in confinement.

The authors define an effective 2D surface tension γ = γ_exp·H from the experimental 3D interfacial tension in the quasi-2D limit, and use it to define the dimensionless group Γ_exp = γ_exp / (g_exp ρ_o σ²), with g_exp = sinθ·g(ρ_w − ρ_o)/ρ_o. Γ_exp is a calculated quantity, not a measured one: because the single measured γ was held fixed, the spread in reported values — Γ_exp ≈ 0.57, 0.88, 3.47 and 5.80 — is produced entirely by changing the microscope tilt angle θ and the droplet diameter σ.

The DP simulations are calibrated against experiments of a single droplet flowing through narrow channels by tuning a dimensionless line tension 𝐺 and a near-wall drag coefficient ratio 𝑏0/𝑏𝑁 to minimize deviations in droplet speed (and comparing shape in parallel). For one calibration case (w = 0.4σ, σ ≈ 3.5 mm, θ ≈ 28°), the paper reports Γ* = 0.16 ± 0.01 and b*₀/b_∞ = 0.064 ± 0.003, giving deviations of Δ_v ≈ 0.09 in speed and Δ_A ≈ 0.01 in shape parameter. Overall the authors state the simulations "recapitulate the experimental results for the droplet shape and speed for flows within narrow channels with an error of less than 10%."

The paper reports a nonmonotonic droplet speed profile as the droplet exits the narrow orifice, including cases where the droplet speed exceeds the terminal speed far from the constriction, with overshoot behavior discussed in terms of the balance between capillarity and driving.

In deformable-particle simulations — not in experiments — the clogging probability becomes nonmonotonic with the dimensionless line tension Γ: at large Γ droplets are nearly rigid and clogging is high; clogging decreases as Γ decreases and droplets become more deformable; and clogging increases again at small Γ, where highly deformable droplets wrap around obstacles. The result is expressed as a clogging decay length λ from a Poisson model P(r) = e^(−r/λ), measured over Γ ≈ 0.04–10 and gap ratios w_ob/σ = 0.2–0.5. The corresponding experiments were qualitative only (see the finding above).

The authors state that varying surface tension in the experiments by more than a factor of 2 is challenging, and they therefore carry out simulations with surface tensions varying by more than a factor of
10^3 to accentuate wrapping behavior.

The paper draws an explicit line between its two experimental programmes: "Note that quantitative experimental studies were performed on flows of capillary droplets through narrow orifices. In contrast, only qualitative experimental studies of droplets flowing through obstacle arrays were performed to illustrate the droplet squeezing and wrapping mechanisms. The experimental studies of droplets in obstacle arrays were only featured in the images in Fig. 5 (a) and (c)." Every quantitative statement about clogging probability, decay length and average droplet speed in obstacle arrays comes from simulation.

The value that best fits the experiments is not the value the Dropometer measured. Across every tilt angle and droplet diameter studied, the authors find "Γ**/Γexp ≈ 0.3" — Γ* ≈ 0.16 against Γ_exp ≈ 0.57, Γ** ≈ 1.7 against 5.80, Γ** ≈ 1.08 against 3.47. Their explanation: the discrepancy "likely stems from the fact that we compare values of the dimensionless line tension in 2D to values of the dimensionless surface tension in 3D." The authors nonetheless describe Γ* as "comparable to Γexp," and name a fully three-dimensional DP model with true surface tension as their first item of future work.

The study uses γ_ow ≈ 10 mJ/m² (measured by the "Dropometer") as the interfacial-tension value that underpins the capillarity arguments and the quasi-2D surface-tension scaling. Note that a single scalar was enough for this purpose: the paper reports it to one significant figure, with no uncertainty, and never varies it.

The paper defines a dimensionless experimental surface-tension quantity Γ_exp from the measured interfacial tension, the tilt angle, the droplet diameter and both fluid densities, and uses it when comparing measured droplet shape and speed to DP simulations.

The reported speed profile through a narrow orifice is nonmonotonic and can exceed the terminal speed after the droplet exits the constriction, with the behaviour discussed in terms of capillary versus driving effects. The overshoot grows as the driving is reduced: the amplitude increases with decreasing tilt angle, because the excess surface energy stored during deformation is set by the orifice width while the kinetic energy at terminal speed falls with the driving force.

Wrapping is shown qualitatively in experiment (Fig. 5a) and quantified in DP simulation: very deformable droplets wrap around obstacles, which decreases average speed in the continuous-flow regime and raises clogging probability in the permanent-clog regime. Both quantitative effects are simulation results. The soft-particle model, which allows droplet–obstacle overlap instead of explicit shape change, cannot reproduce wrapping at all — which is the paper's argument for modelling shape explicitly.

For obstacle arrays, the paper states that permanent clogs can form when w_ob/σ < 1, and that single droplets flow continuously without clogging for w_ob/σ ≫ 1. Continuous-flow results are reported for gap ratios w_ob/σ = 1.0–1.3, and the clogging study for w_ob/σ = 0.2–0.5.