A hierarchy of models is formulated for the deflection of a thin two-dimensional liquid jet as it passes over a thin air-cushioning layer above a rigid flat impermeable substrate. We perform a systematic derivation of the leading-order equations of motion for the jet in the distinguished limit in which the air pressure jump, surface tension and gravity affect the displacement of the centreline of the jet, but not its thickness or velocity. We identify thereby the axial length scales for centreline deflection in regimes in which the air layer is dominated by viscous or inertial effects. The derived length scales and reduced equations aim to expand the suite of tools available for future analyses of the evolution of lamellae and ejecta in impact problems. Assuming that the jet is sufficiently long that tip and entry effects can be neglected, we demonstrate that the centreline of a constant-thickness jet moving with constant axial speed is destabilised by the air layer for sufficiently small surface tension. Expressions for the fastest-growing modes are obtained in both the viscous-dominated air and inertia-dominated air regimes. For a finite-length jet emanating from a nozzle, we show that, in one particular asymptotic limit, the evolution of the jet centreline is akin to the flapping of an unfurling flag above a thin air layer. We discuss the distinguished limit in which tip retraction can be neglected and perform numerical investigations into the resulting model. We show that the cushioning layer causes the jet centreline to bend, leading to rupture of the air layer. We discuss how our toolbox of models can be adapted and utilised in the context of recent experimental and numerical studies of splash dynamics.
Bibliographical noteKAUST Repository Item: Exported on 2022-06-03
Acknowledged KAUST grant number(s): KUK-C1-013-04
Acknowledgements: The authors would like to thank M.-J. Thoraval, S. T. Thoroddsen, A. A. Korobkin and S. J. Chapman for helpful comments in the course of this work. M.R.M. is grateful to the EPSRC for a studentship that he received while working on part of this study. This publication was based on work supported in part by award no. KUK-C1-013-04, made by King Abdullah University of Science and Technology (KAUST). The authors are grateful to the anonymous referees whose comments improved this paper significantly.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.
ASJC Scopus subject areas
- Mechanics of Materials
- Mechanical Engineering
- Condensed Matter Physics