Bursting bubbles in Herschel-Bulkley fluids: dynamics and jetting transitions

TL;DR

The paper addresses bubble bursting at a gas–liquid interface in Herschel–Bulkley fluids, where yield stress and nonlinear rheology modify the classical inertial–capillary–gravity balance. It combines fully resolved direct numerical simulations with Carbopol experiments to explore how the dimensionless groups $Bo$, $Oh_K$, $J$, and $n$ govern cavity collapse and jet formation. A regime map reveals three outcomes—no jet, jet without droplets, and jet with droplet ejection—showing that shear-thinning promotes jets at low $J$ while yield stress and shear-thickening suppress them, with gravity reducing jet height as $Bo$ increases. These findings provide a framework to predict jetting and droplet outcomes in non-Newtonian gas–liquid interfaces, with wide relevance to natural and industrial processes involving complex fluids.

Abstract

When a bubble rises to a free surface, its bursting dynamics in Newtonian fluids are governed by the interplay between viscous, capillary, and gravitational forces. In this work, we extend this classical problem to Herschel-Bulkley fluids, elucidating the role of viscoplasticity and non-Newtonian rheology in bubble bursting. Using direct numerical simulations validated against experiments, we systematically explore the influence of the key governing dimensionless parameters, such as the Bond number, the Ohnesorge number, the shear-dependent behavior and the plastocapillary number, each varied over several orders of magnitude. Our results reveal that viscoplasticity strongly controls the evolution and interaction of capillary waves within the cavity formed upon bubble rupture. Shear-thinning and shear-thickening effects are significant only for moderate Ohnesorge numbers, while at large Ohnesorge values the free surface dynamics converge to a non-flat equilibrium shape once the internal stresses fall below the yield stress. These findings provide new insights into the coupled effects of viscosity, gravity, yield stress, and shear-dependent rheology in multiphase flows, with broad implications for natural and industrial processes involving gas-liquid interfaces.

Figures (10)

• Figure 1: Rheological characterization of 0.1% Carbopol 981 solution. The blue markers show the experimental steady state curve for the shear stress as a function of shear rate, and the solid curve shows the Herschel-Bulkley fitting.
• Figure 2: Temporal bursting bubble dynamics for a case with $n = 0.4187$, $J=0.1053$, $Bo=1.0990$ and $Oh_K=0.7015$ at (a) 0 ms, (b) 1.6 ms, (c) 2.8 ms, (d) 5.0 ms, (e) 7.5 ms, and (f) 21.6 ms. Numerical prediction of the air-liquid interface is overlapped on yellow in each experimental panel. The scale bar represents 1 mm.
• Figure 3: Temporal evolution of bubble bursting in Herschel--Bulkley fluids for representative combinations of parameters $n=0.8$, $Oh_K=0.01$, and $Bo=0.1$. The three rows correspond to different outcomes identified in the regime map of figure 4: (a–e) no jet with $\mathcal{J}=0.24$, (f–j) jet without droplet formation with $\mathcal{J}=0.22$, and (k–o) jet and droplet ejection with $\mathcal{J}=0.20$. For each case, panels show successive instants during the rupture process, with times nondimensionalised by the capillary timescale $\tau$. Each snapshot displays the velocity field (left) and the deformation–rate tensor modulus $\|\mathbf{\mathcal{D}}\|$ (right), together with the instantaneous interface position.
• Figure 4: Regime map of bubble bursting in Herschel--Bulkley fluids in the $(J,n, Oh_K, Bo)$ space. The horizontal axis represents the plastocapillary number $\mathcal{J}$ and the vertical axis the flow behavior index $n$. Each panel corresponds to a fixed pair of $Oh_K$ and $Bo$, as indicated. The brightest markers indicate cases with jetting and droplet formation, intermediate shading corresponds to jetting without droplet formation, and the darkest markers represent cases where neither a jet nor droplets are observed.
• Figure 5: Effect of $\mathcal{J}$ for $n=(0.4,1.0,1.4)$ with $Bo=0.01$ and $Oh_K=0.1$, corresponding to columns one-to-three, respectively. Panels (a–c) show the jet tip height $\mathcal{H}(t)$. Panels (d–f) and (g–i) display the interfacial shape at $t=0.5$ and $t=2.0$, respectively. The colour indicates the value of $\mathcal{J}$, as specified in the legend in panel (c).