Drop formation culminates when a thinning fluid filament breaks, yet the continuum theory of this event is incomplete. Córdoba, Córdoba, Fefferman and Fontelos (PNAS 2002; companion paper in Adv. Math. 2004) proved that a viscous jet surrounded by another fluid, governed by the two-phase Navier–Stokes equations with surface tension, cannot break up through the uniform collapse of a regular filament: the enclosed volume stays above an explicit positive function of time. What remains open is whether the continuum equations predict genuine finite-time break-up at all, i.e. pinch-off of the interface to zero thickness at isolated points, possibly through violent non-uniform bending and twisting not covered by the uniform-collapse theorem. Later literature has settled adjacent questions — pinch-off is impossible for axisymmetric, freely suspended drops in the inertialess Stokes regime; engineered splash (interface self-intersection) singularities exist for one-phase free boundaries even with surface tension but are impossible for two-fluid Euler — while the singularity structure is confirmed only by asymptotics, numerics and experiment. The rigorous finite-time pinch-off question for the capillary jet remains open.

Background

Drop formation from liquid jets has been studied since the nineteenth century: a long column of fluid is unstable to axisymmetric, surface-tension-driven disturbances (the Rayleigh–Plateau instability), because the interfacial area, and hence the surface energy, decreases when a cylinder develops bulges and necks. The final stage, in which a neck thins to zero thickness and the fluid column separates, is the pinch-off or break-up singularity. In the classical continuum description one considers two incompressible Newtonian fluids — a viscous jet of fluid 1 surrounded by an ambient fluid 2, with viscosities μi\mu_i and densities ρi\rho_i — separated by a moving interface ∂Ω(t)\partial\Omega(t), governed in each phase by the Navier–Stokes equations, with continuity of velocity and the stress-balance condition [Tij(1)−Tij(2)]nj=σHni[T^{(1)}_{ij}-T^{(2)}_{ij}]n_j=\sigma H n_i at the interface, where Tij(k)=−pδij+μk(∂ivj+∂jvi)T^{(k)}_{ij}=-p\delta_{ij}+\mu_k(\partial_i v_j+\partial_j v_i) is the Newtonian stress tensor, σ>0\sigma>0 the surface-tension coefficient, HH the mean curvature of the interface and nn its normal.

Whether this system actually predicts break-up is a long-standing open question of mathematical fluid dynamics. Physically the answer is believed to be yes, and the local structure of the event is believed to be universal: Eggers (1993) constructed, asymptotically and numerically, a self-similar mechanism for break-up at isolated points in which inertia, viscosity and surface tension balance, giving universal scaling laws for the minimum neck radius as the pinch time is approached; the one-dimensional slender-jet approximation of the Navier–Stokes equations (Eggers and Dupont, 1994) reproduces it, and experiments and high-resolution simulations support it. Rigorous results, however, cover only the neighboring regimes. Córdoba and Fefferman (2001) proved that the volume of a 'regular tube' transported by a 3D incompressible flow cannot vanish in finite time under an integrability condition on the velocity. Córdoba, Córdoba, Fefferman and Fontelos (2002, 2004) extended this to the two-phase problem with surface tension: the uniform collapse of a fluid tube (a filament whose thickness h(t)h(t), with h(t)→0h(t)\to 0, bounds above and below the distance from every interface point to a fixed smooth curve) is impossible, and the enclosed volume satisfies an explicit positive lower bound for all time. Consequently the neck of a jet cannot reach zero thickness in finite time unless the filament bends and twists so violently that no part of it forms a regular tube; a possible break-up must therefore occur at isolated pinch points or through such violent non-uniform collapse. In the inertialess limit the situation is rigid: Eggers and Fontelos (2005) proved that an axisymmetric, freely suspended drop in the Stokes regime (viscosity so large that inertia and ambient forcing are negligible) can never pinch off, whatever the initial condition — inertia is essential. For the related phenomenon of interface self-intersection ('splash' singularities), one-phase free-boundary problems are now known to develop finite-time splash even with surface tension, while two-fluid Euler interfaces provably cannot splash. Experiments show filaments thinning almost uniformly down to micron diameters, a scale at which molecular (non-continuum) effects may enter, which makes it important to know what the continuum equations alone predict.

Problem Statement

Determine, for the continuum two-phase Navier–Stokes equations with surface tension (two incompressible Newtonian fluids, with velocity-continuity and mean-curvature stress-balance conditions at the moving interface), whether the break-up of a viscous capillary jet occurs in finite time.

Starting from smooth initial data representing a capillary jet or filament of fluid 1 surrounded by fluid 2 (for example, a perturbation of a fluid cylinder subject to the surface-tension instability), prove either that the interface pinches to zero thickness at isolated points in finite time — so that the connected fluid domain changes topology — or that no finite-time break-up can occur, including through the non-uniform, highly bent or convoluted collapse that the uniform-collapse theorem of Córdoba–Córdoba–Fefferman–Fontelos does not cover.

The self-similar break-up mechanism proposed by Eggers, described in Background, is the accepted asymptotic picture of what such a pinch-off would look like; this mechanism is context for the question, not a required deliverable. The engineered one-phase self-intersection configurations, the strictly one-dimensional slender-jet models, and the inertialess Stokes regime address neighboring questions rather than this one.

References

  1. Antonio Córdoba, Diego Córdoba, Charles Fefferman, Marco A. Fontelos, 'Drops: The collapse of capillary jets', Proceedings of the National Academy of Sciences 99(17), 11006–11007 (2002). DOI: 10.1073/pnas.152318399, https://doi.org/10.1073/pnas.152318399
  2. A. Córdoba, D. Córdoba, C. L. Fefferman, M. A. Fontelos, 'A geometrical constraint for capillary jet breakup', Advances in Mathematics 187, 228–239 (2004). DOI: 10.1016/j.aim.2003.08.009, https://doi.org/10.1016/j.aim.2003.08.009
  3. Diego Córdoba, Charles Fefferman, 'On the Collapse of Tubes Carried by 3D Incompressible Flows', Communications in Mathematical Physics 222, 293–298 (2001). DOI: 10.1007/s002200100502, https://doi.org/10.1007/s002200100502
  4. Jens Eggers, 'Universal pinching of 3D axisymmetric free-surface flow', Physical Review Letters 71, 3458–3460 (1993). DOI: 10.1103/PhysRevLett.71.3458, https://doi.org/10.1103/PhysRevLett.71.3458
  5. Jens Eggers, 'Nonlinear dynamics and breakup of free-surface flows', Reviews of Modern Physics 69, 865–930 (1997). DOI: 10.1103/RevModPhys.69.865, https://doi.org/10.1103/RevModPhys.69.865
  6. Jens Eggers, Todd F. Dupont, 'Drop formation in a one-dimensional approximation of the Navier–Stokes equation', Journal of Fluid Mechanics 262, 205–221 (1994). DOI: 10.1017/S0022112094000480, https://doi.org/10.1017/S0022112094000480 (preprint arXiv:physics/0110081, https://arxiv.org/abs/physics/0110081)
  7. Marco A. Fontelos, 'On the evolution of thin viscous jets: filament formation', Mathematical Methods in the Applied Sciences 27, 1197–1220 (2004). DOI: 10.1002/mma.489, https://doi.org/10.1002/mma.489
  8. Jens Eggers, Marco A. Fontelos, 'Isolated inertialess drops cannot break up', Journal of Fluid Mechanics 530, 177–180 (2005). DOI: 10.1017/S0022112005003678, https://doi.org/10.1017/S0022112005003678 (preprint arXiv:physics/0408099, posted under the title 'Very viscous drops cannot break up', https://arxiv.org/abs/physics/0408099)
  9. Angel Castro, Diego Córdoba, Charles L. Fefferman, Francisco Gancedo, Javier Gómez-Serrano, 'Splash singularity for water waves', Proceedings of the National Academy of Sciences 109, 733–738 (2012). DOI: 10.1073/pnas.1115948108, https://doi.org/10.1073/pnas.1115948108
  10. Angel Castro, Diego Córdoba, Charles Fefferman, Francisco Gancedo, Javier Gómez-Serrano, 'Finite time singularities for water waves with surface tension', Journal of Mathematical Physics 53, 115622 (2012). DOI: 10.1063/1.4765339, https://doi.org/10.1063/1.4765339
  11. Angel Castro, Diego Córdoba, Charles Fefferman, Francisco Gancedo, Javier Gómez-Serrano, 'Splash singularities for the free boundary Navier–Stokes equations', Annals of PDE 5(1), article 12 (2019). DOI: 10.1007/s40818-019-0068-1, https://doi.org/10.1007/s40818-019-0068-1 (preprint arXiv:1504.02775, https://arxiv.org/abs/1504.02775)
  12. Daniel Coutand, Steve Shkoller, 'On the finite-time splash and splat singularities for the 3-D free-surface Euler equations', Communications in Mathematical Physics 325, 143–183 (2014). DOI: 10.1007/s00220-013-1855-2, https://doi.org/10.1007/s00220-013-1855-2
  13. Charles Fefferman, Alexandru D. Ionescu, Victor Lie, 'On the absence of "splash" singularities in the case of two-fluid interfaces', Duke Mathematical Journal 165(3), 417–462 (2016). DOI: 10.1215/00127094-3166629, https://doi.org/10.1215/00127094-3166629 (preprint arXiv:1312.2917, https://arxiv.org/abs/1312.2917)
  14. Yoshikazu Giga, Zhongyang Gu, 'On existence of a collapsed bubble with surface tension in viscous incompressible fluid', arXiv:2606.31266 [math.AP] (2026). DOI: 10.48550/arXiv.2606.31266, https://arxiv.org/abs/2606.31266
  15. Jens Eggers, Marco A. Fontelos, 'Singularities: Formation, Structure, and Propagation', Cambridge University Press (2015). ISBN 9781107098411. DOI: 10.1017/CBO9781316161692, https://doi.org/10.1017/CBO9781316161692
  16. Michael C. Dallaston, Chengxi Zhao, James E. Sprittles, Jens Eggers, 'Stability of similarity solutions of viscous thread pinch-off', Physical Review Fluids 6, 104004 (2021). DOI: 10.1103/PhysRevFluids.6.104004, https://doi.org/10.1103/PhysRevFluids.6.104004 (preprint arXiv:2106.06642, https://arxiv.org/abs/2106.06642)
  17. Marie Corpart, Miguel Ángel Herrada, Antoine Deblais, Daniel Bonn, 'Approach to similarity in the pinch-off of a viscous liquid thread', Journal of Fluid Mechanics 1019, A15 (2025). DOI: 10.1017/jfm.2025.10582, https://doi.org/10.1017/jfm.2025.10582
  18. Michael Moseler, Uzi Landman, 'Formation, Stability, and Breakup of Nanojets', Science 289, 1165–1169 (2000). DOI: 10.1126/science.289.5482.1165, https://doi.org/10.1126/science.289.5482.1165