Escape from end-pinching in Herschel-Bulkley ligaments

TL;DR

This work investigates capillary-driven retraction of Herschel–Bulkley ligaments in air, emphasizing the low-viscosity regime where end-pinching governs detachment. It employs axisymmetric direct numerical simulations with a regularized Herschel–Bulkley constitutive law in Basilisk to map four dynamical outcomes—pinch-off, viscous escape, inertial rebound, and arrest—across the rheological space defined by the flow index $n$ and plastocapillary number $J$, using a time-dependent local Ohnesorge number $Oh_{ ext{loc}}$ as the unifying control parameter. The study identifies four regimes and derives mechanistic criteria (e.g., $J_c\approx 0.04$–$0.07$ for no-neck and $J\gtrsim 0.5$ for motionlessness) tied to neck geometry and vorticity dynamics, including the formation of vortex rings in shear-thinning cases. It generalizes Newtonian end-pinching pathways to viscoplastic fluids, clarifying how yield stress and shear-dependent viscosity reshape capillary-driven retraction and influencing applications in sprays, inkjetting, and emulsification where yield-stress materials are present.

Abstract

Capillary retraction of Herschel-Bulkley ligaments displays a rich set of behaviors that depart significantly from the classical Newtonian picture. We focus here on the low-viscosity regime, where droplet detachment is controlled by the end-pinching mechanism. Using fully resolved axisymmetric simulations, we show that viscoplasticity and shear-dependent rheology reorganize the fundamental routes by which a retracting ligament may pinch off or escape breakup. Four dynamical outcomes are identified: inertial pinch-off, viscous escape driven by an attached vorticity layer, inertial escape caused by vortex-ring detachment, and complete arrest when yielding suppresses motion. These regimes appear in an orderly structure across the rheological parameter space, but their transitions are governed by a single unifying feature of the neck dynamics: whether inertia, viscous diffusion, or yield stress dominates the local collapse. When inertia dominates, the vorticity sheet detaches and rolls into a vortex ring, producing an inertial rebound of the neck. When viscous effects dominate, the vorticity layer remains attached and drives a back-flow that reopens the neck. When yield stress is sufficiently large, both mechanisms are suppressed and the ligament becomes motionless. This framework extends the Newtonian end-pinching and escape mechanisms to viscoplastic fluids and clarifies how non-Newtonian rheology reshapes capillary-driven retraction.

Figures (9)

• Figure 1: (a) Schematic of the initial configuration of the system (top panel) and the computational mesh used in the simulations (bottom panel). (b) Spatio-temporal evolution of a retracting ligament for $L_0=15, Oh_K=0.001,\mathcal{J}=0$ (Newtonian). The red lines correspond to the results of the present study and the solid black lines are collected from notz_dynamics_2004.
• Figure 2: Regime map in terms of the plastocapillary number $\mathcal{J}$ and the flow index parameter $n$ for $Oh_K=0.001$ showing the transitions between the different categories identified in the current study. The insets show a representative case from each of the four regimes. The symbols represent simulations at the different transition lines.
• Figure 3: End-pinching regime. (a) Temporal sequence of the pressure field $p$ (top), azimuthal vorticity $\omega_\theta$ (middle), and effective viscosity $\log_{10}(\mu_{\text{eff}})$ (bottom) during the final stages of breakup, shown at $t = 4.6$, $4.8$, and $4.9$ for $Oh_K=0.001$ and $\mathcal{J}=0$ and $n=1$ (b) Temporal evolution of the local Ohnesorge number $Oh_{\text{loc}}$ at the neck. (c) Normalized minimum radius $r_{\min}$ versus time for several $\mathcal{J}$ at fixed $n = 0.6$
• Figure 4: Shear-thickening reopening regime. (a) Snapshots of pressure $p$, azimuthal vorticity $\omega_\theta$, and $\log_{10}(\mu_{\mathrm{eff}})$ for the case $Oh_K = 0.001$, $\mathcal{J}=0$, and $n=1.5$, with each row showing one of the three fields at times $t = 4.0$, $4.8$, and $5.3$ from left to right. (b) Temporal evolution of the local Ohnesorge number $Oh_{\text{loc}}$ at the neck. (c) Normalized minimum radius $r_{\min}$ versus time for several $\mathcal{J}$ at fixed $n = 1.5$
• Figure 5: Shear-thinning reopening regime. (a) Snapshots of pressure $p$, azimuthal vorticity $\omega_\theta$, and $\log_{10}(\mu_{\mathrm{eff}})$ for the case $Oh_K = 0.001$, $\mathcal{J}=0$, and $n=0.2$, with each row showing one of the three fields at times $t=4.5$, $4.9$, and $5.6$ from left to right. (b) Temporal evolution of the local Ohnesorge number $Oh_{\text{loc}}$. (c) Minimum neck radius $r_{\min}$ over time for several $n$ at fixed $\mathcal{J} = 0$ (d) Temporal evolution of the total kinetic energy $E_k$. (e) Temporal evolution of the surface energy $E_s$