Abstract
At the center of a collapsing hole lies a singularity, a point of infinite curvature where the governing equations break down. It is a topic of fundamental physical interest to clarify the dynamics of fluids approaching such singularities. Here, we use scaling arguments supported by high-fidelity simulations to analyze the dynamics of an axisymmetric hole undergoing capillary collapse in a fluid sheet of small viscosity. We characterize the transitions between the different dynamical regimes —from the initial inviscid dynamics that dominate the collapse at early times to the final Stokes dynamics that dominate near the singularity— and demonstrate that the crossover hole radii for these transitions are related to the fluid viscosity by power-law relationships. The findings have practical implications for the integrity of perforated fluid films, such as bubble films and biological membranes, as well as fundamental implications for the physics of fluids converging to a singularity.
Highlights
At the center of a collapsing hole lies a singularity, a point of infinite curvature where the governing equations break down
We have used force-balance arguments supported by simulations to characterize the crossover between dynamical regimes during the capillary collapse of low-viscosity fluid holes
Our results estimate the transition from the inertial regime that dominates the dynamics at early times to an intermediate regime, and show that the estimated crossover hole size follows a direct power-law relationship with the viscosity
Summary
At the center of a collapsing hole lies a singularity, a point of infinite curvature where the governing equations break down. We use scaling arguments supported by high-fidelity simulations to analyze the dynamics of an axisymmetric hole undergoing capillary collapse in a fluid sheet of small viscosity. If the hole size is small in relation to the sheet thickness, the hole contracts, and the integrity of the sheet is preserved[13] Besides their practical importance for the integrity of liquid films, the dynamics of collapsing holes is of fundamental fluid mechanics interest because it represents an archetypal example of free-surface flow with a finite-time singularity. In this case a singularity of the surface curvature that occurs at the center of the collapsing hole. We have previously benchmarked and successfully applied this numerical scheme to simulate similar free-surface flows, including drop coalescence[32], breakup of Newtonian and non-Newtonian liquid filaments[33], collapse of viscous and inertial pores[34,35], and contraction of surfactant-laden pores and filaments[36,37]
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