Self-similar Worthington jets
Abstract
When a micron-sized bubble bursts, capillary waves deform the cavity into a cone that ejects a Worthington jet. The jet is born by inertial focusing, and the local collapse follows self-similar Euler solutions set by the semiangle β. Writing rj and vj for the dimensionless jet-base radius and velocity, the local Weber number Wej=rj v2j measures inertia relative to capillarity. The theory, supported by accurate numerical simulations gives rj∝τα(β) with α≃0.63 and, hence Wej≫1, with Wej→∞ as rj→0, so inertia increasingly overwhelms capillarity. In simulations, the interface collapses onto a universal shape for more than two decades in dimensionless time when lengths are scaled using our prediction for rj. For water, this gives incipient radii of 𝒪(1) nm, predicting nanometric sea-spray aerosols.
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BibTeX
@misc{gordillo2026selfsimilar,
title = {{Self-similar Worthington jets}},
author = {José M. Gordillo and Javier Rodríguez-Rodríguez and Vatsal Sanjay},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.08972},
eprint = {2607.08972},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.08972},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Self-similar Worthington jets
AU - José M. Gordillo
AU - Javier Rodríguez-Rodríguez
AU - Vatsal Sanjay
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2607.08972
UR - https://arxiv.org/abs/2607.08972
AB - When a micron-sized bubble bursts, capillary waves deform the cavity into a cone that ejects a Worthington jet. The jet is born by inertial focusing, and the local collapse follows self-similar Euler solutions set by the semiangle β. Writing r_j and v_j for the dimensionless jet-base radius and velocity, the local Weber number We_j=r_j v^2_j measures inertia relative to capillarity. The theory, supported by accurate numerical simulations gives r_j∝τ^α(β) with α≃0.63 and, hence We_j≫1, with We_j→∞ as r_j→0, so inertia increasingly overwhelms capillarity. In simulations, the interface collapses onto a universal shape for more than two decades in dimensionless time when lengths are scaled using our prediction for r_j. For water, this gives incipient radii of 𝒪(1) nm, predicting nanometric sea-spray aerosols.
ER -