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On the deflection of a liquid jet by an air-cushioning layer

Published online by Cambridge University Press:  09 May 2018

Madeleine Rose Moore Open the ORCID record for Madeleine Rose Moore [Opens in a new window] ,
J. P. Whiteley  and
J. M. Oliver
Madeleine Rose Moore*
Affiliation:
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK
J. P. Whiteley
Affiliation:
Department of Computer Science, University of Oxford, Wolfson Building, Parks Road, Oxford OX1 3QD, UK
J. M. Oliver
Affiliation:
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK
*
†Email address for correspondence: [email protected]
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Abstract

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.

JFM classification

Interfacial Flows (free surface): Interfacial Flows (free surface) Wakes/Jets: Jets Interfacial Flows (free surface): Thin films
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 846 , 10 July 2018 , pp. 711 - 751
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Copyright
© 2018 Cambridge University Press 

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Footnotes

Article last updated 07 March 2023

References

Agbaglah, G., Thoraval, M.-J., Thoroddsen, S. T., Zhang, L. V., Fezzaa, K. & Deegan, R. D. 2015 Drop impact into a deep pool: vortex shedding and jet formation. J. Fluid Mech. 764, R1.Google Scholar
Armand, J.-L. & Cointe, R. 1987 Hydrodynamic impact analysis of a cylinder. Trans. ASME J. Offshore Mech. Arctic Engng 111, 109–114.Google Scholar
Connell, B. S. H. & Yue, D. K. P. 2007 Flapping dynamics of a flag in a uniform stream. J. Fluid Mech. 581, 33–67.Google Scholar
Dewynne, J. N., Howell, P. D. & Wilmott, P. 1994 Slender viscous fibres with inertia and gravity. Q. J. Mech. Appl. Maths 47 (4), 541–555.Google Scholar
Dewynne, J. N., Ockendon, J. R. & Wilmott, P. 1992 A systematic derivation of the leading-order equations for extensional flows in slender geometries. J. Fluid Mech. 244, 323–338.Google Scholar
Dombrowski, N. & Hooper, P. C. 1962 The effect of ambient density on drop formation in sprays. Chem. Engng Sci. 17 (4), 291–305.Google Scholar
Driscoll, M. M. & Nagel, S. R. 2011 Ultrafast interference imaging of air in splashing dynamics. Phys. Rev. Lett. 107 (15), 154502.Google Scholar
Driscoll, M. M., Stevens, C. S. & Nagel, S. R. 2010 Thin film formation during splashing of viscous liquids. Phys. Rev. E 82 (3), 036302.Google Scholar
Duchemin, L. & Josserand, C. 2011 Curvature singularity and film-skating during drop impact. Phys. Fluids 23, 091701.Google Scholar
Edwards, C. M., Howison, S. D., Ockendon, H. & Ockendon, J. R. 2008 Non-classical shallow water flows. IMA J. Appl. Maths 73 (1), 137–157.Google Scholar
Eggers, J. & Villermaux, E. 2008 Physics of liquid jets. Rep. Prog. Phys. 71 (3), 036601.Google Scholar
Eriksson, K., Estep, D., Hansbo, P. & Johnson, C. 1996 Computational Differential Equations. Cambridge University Press.Google Scholar
Erneux, T. & Davis, S. H. 1993 Nonlinear rupture of free films. Phys. Fluids 5 (5), 1117–1122.Google Scholar
Filippov, A. & Zheng, Z. 2010 Dynamics and shape instability of thin viscous sheets. Phys. Fluids 22, 023601.Google Scholar
van de Fliert, B. W., Howell, P. D. & Ockendon, J. R. 1995 Pressure-driven flow of a thin viscous sheet. J. Fluid Mech. 292, 359–376.Google Scholar
Gordillo, J. M., Pérez-Saborid, M. & Gañán-Calvo, A. M. 2001 Linear stability of co-flowing liquid–gas jets. J. Fluid Mech. 448, 23–51.Google Scholar
Gordillo, L., Agbaglah, G., Duchemin, L. & Josserand, C. 2011 Asymptotic behaviour of a retracting two-dimensional fluid sheet. Phys. Fluids 23, 122101.Google Scholar
Hagerty, W. W. & Shea, J. F. 1955 A study of the stability of plane fluid sheets. Trans. ASME J. Appl. Mech. 22 (3), 509–514.Google Scholar
Hicks, P. D. & Purvis, R. 2010 Air cushioning and bubble entrapment in three-dimensional droplet impacts. J. Fluid Mech. 649 (1), 135–163.Google Scholar
Howell, P. D.1994 Extensional thin layer flows. DPhil thesis, University of Oxford.Google Scholar
Howell, P. D. 1996 Models for thin viscous sheets. Eur. J. Appl. Maths 7, 321–343.Google Scholar
Howison, S. D., Ockendon, J. R. & Wilson, S. K. 1991 Incompressible water-entry problems at small deadrise angles. J. Fluid Mech. 222, 215–230.Google Scholar
Jian, Z., Josserand, C., Popinet, S., Ray, P. & Zaleski, S. 2018 Two mechanisms of droplet splashing on a solid substrate. J. Fluid Mech. 835, 1065–1086.Google Scholar
Josserand, C., Ray, P. & Zaleski, S. 2016 Droplet impact on a thin liquid film: anatomy of the splash. J. Fluid Mech. 802, 775–805.Google Scholar
Josserand, C. & Thoroddsen, S. T. 2016 Drop impact on a solid surface. Annu. Rev. Fluid Mech. 48, 365–391.Google Scholar
Josserand, C. & Zaleski, S. 2003 Droplet splashing on a thin liquid film. Phys. Fluids 15 (6), 1650–1657.Google Scholar
Kolinski, J. M., Mahadevan, L. & Rubinstein, S. M. 2014 Lift-off instability during the impact of a drop on a solid surface. Phys. Rev. Lett. 112, 134501.Google Scholar
Kolinski, J. M., Rubinstein, S. M., Mandre, S., Brenner, M. P., Weitz, D. A. & Mahadevan, L. 2012 Skating on a film of air: drops impacting on a surface. Phys. Rev. Lett. 108, 074503.Google Scholar
Korobkin, A. A. 1985 Initial asymptotic behavior of a solution to the three-dimensional problem concerning the entry of a blunt body into an ideal fluid. Dokl. Akad. Nauk SSSR 283, 838–842.Google Scholar
Latka, A. 2017 Thin-sheet creation and threshold pressures in drop splashing. Soft Matt. 13 (4), 740–747.Google Scholar
Latka, A., Strandburg-Peshkin, A., Driscoll, M. M., Stevens, C. S. & Nagel, S. R. 2012 Creation of prompt and thin-sheet splashing by varying surface roughness or increasing air pressure. Phys. Rev. Lett. 109 (5), 054501.Google Scholar
Li, X. 1993 Spatial instability of plane liquid sheets. Chem. Engng Sci. 48 (16), 2973–2981.Google Scholar
Li, X. & Tankin, R. S. 1991 On the temporal instability of a two-dimensional viscous liquid sheet. J. Fluid Mech. 226, 425–443.Google Scholar
Liu, T., Tan, P. & Xu, L. 2015 Kelvin–Helmholtz instability in an ultrathin air film causes drop splashing on smooth surfaces. Proc. Natl Acad. Sci. USA 112 (11), 3280–3284.Google Scholar
Mandre, S. & Brenner, M. P. 2012 The mechanism of a splash on a dry solid surface. J. Fluid Mech. 690, 148–172.Google Scholar
Mandre, S., Mani, M. & Brenner, M. P. 2009 Precursors to splashing of liquid droplets on a solid surface. Phys. Rev. Lett. 102, 134502.Google Scholar
Mani, M., Mandre, S. & Brenner, M. P. 2010 Events before droplet splashing on a solid surface. J. Fluid Mech. 647, 163–185.Google Scholar
Marchand, A., Chan, T. S., Snoeijer, J. H. & Andreotti, B. 2012 Air entrainment by contact lines of a solid plate plunged into a viscous fluid. Phys. Rev. Lett. 108 (20), 204501.Google Scholar
Moore, M. R.2014 New mathematical models for splash dynamics. DPhil thesis, University of Oxford.Google Scholar
Moore, M. R., Ockendon, H., Ockendon, J. R. & Oliver, J. M. 2014 Capillary and viscous perturbations to Helmholtz flows. J. Fluid Mech. 742, R1.Google Scholar
Moore, M. R., Ockendon, J. R. & Oliver, J. M. 2013 Air-cushioning in impact problems. IMA J. Appl. Maths 78, 818–838.Google Scholar
Moore, M. R. & Oliver, J. M. 2014 On air cushioning in axisymmetric impacts. IMA J. Appl. Maths 79 (4), 661–680.Google Scholar
Oliver, J. M.2002 Water entry and related problems. DPhil thesis, University of Oxford.Google Scholar
Oliver, J. M. 2007 Second-order Wagner theory for two-dimensional water-entry problems at small deadrise angles. J. Fluid Mech. 572, 59–85.Google Scholar
Palacios, J., Hernández, J., Gómez, P., Zanzi, C. & López, J. 2012 On the impact of viscous drops onto dry smooth surfaces. Exp. Fluids 52 (6), 1449–1463.Google Scholar
Pfingstag, G., Audoly, B. & Boudaoud, A. 2011 Linear and nonlinear stability of floating viscous sheets. J. Fluid Mech. 683, 112–148.Google Scholar
Purvis, R. & Smith, F. T. 2004 Air–water interactions near droplet impact. Eur. J. Appl. Maths 15 (6), 853–871.Google Scholar
Rayleigh, Lord 1878 On the instability of jets. Proc. Lond. Math. Soc. 1 (1), 4–13.Google Scholar
Rein, M. & Delplanque, J.-P. 2008 The role of air entrainment on the outcome of drop impact on a solid surface. Acta Mech. 201, 105–118.Google Scholar
Riboux, G. & Gordillo, J. M. 2014 Experiments of drops impacting a smooth solid surface: a model of the critical impact speed for drop splashing. Phys. Rev. Lett. 113 (2), 024507.Google Scholar
Riboux, G. & Gordillo, J. M. 2015 The diameters and velocities of the droplets ejected after splashing. J. Fluid Mech. 772, 630–648.Google Scholar
Riboux, G. & Gordillo, J. M. 2017 Boundary-layer effects in droplet splashing. Phys. Rev. E 96 (1), 013105.Google Scholar
Roisman, I. V. & Tropea, C. 2002 Impact of a drop onto a wetted wall: description of crown formation and propagation. J. Fluid Mech. 472, 373–397.Google Scholar
de Ruiter, J., Oh, J. M., van den Ende, D. & Mugele, F. 2012 Dynamics of collapse of air films in drop impact. Phys. Rev. Lett. 108 (7), 074505.Google Scholar
Savva, N. & Bush, J. W. M. 2009 Viscous sheet retraction. J. Fluid Mech. 626, 211–240.Google Scholar
Schroll, R. D., Josserand, C., Zaleski, S. & Zhang, W. W. 2010 Impact of a viscous liquid drop. Phys. Rev. Lett. 104 (3), 034504.Google Scholar
Shelley, M. J. & Zhang, J. 2011 Flapping and bending bodies interacting with fluid flows. Annu. Rev. Fluid Mech. 43 (1), 449–465.Google Scholar
Smith, F. T., Li, L. & Wu, G. X. 2003 Air cushioning with a lubrication/inviscid balance. J. Fluid Mech. 482, 291–318.Google Scholar
Söderberg, L. D. & Alfredsson, P. H. 1998 Experimental and theoretical stability investigations of plane liquid jets. Eur. J. Mech. (B/Fluids) 17 (5), 689–737.Google Scholar
Sprittles, J. E. 2017 Kinetic effects in dynamic wetting. Phys. Rev. Lett. 118 (11), 114502.Google Scholar
Squire, H. B. 1953 Investigation of the instability of a moving liquid film. Brit. J. Appl. Phys. 4 (6), 167–169.Google Scholar
Sterling, A. M. & Sleicher, C. A. 1975 The instability of capillary jets. J. Fluid Mech. 68 (3), 477–495.Google Scholar
Süli, E. & Mayers, D. F. 2003 An Introduction to Numerical Analysis. Cambridge University Press.Google Scholar
Tammisola, O., Lundell, F. & Söderberg, L. D. 2012 Surface tension-induced global instability of planar jets and wakes. J. Fluid Mech. 713, 632–658.Google Scholar
Tammisola, O., Sasaki, A., Lundell, F., Matsubara, M. & Söderberg, L. D. 2011 Stabilizing effect of surrounding gas flow on a plane liquid sheet. J. Fluid Mech. 672, 5–32.Google Scholar
Taylor, G. 1959 The dynamics of thin sheets of fluid. Part II. Waves on fluid sheets. Proc. R. Soc. Lond. A 253 (1274), 296–312.Google Scholar
Teng, C. H., Lin, S. P. & Chen, J. N. 1997 Absolute and convective instability of a viscous liquid curtain in a viscous gas. J. Fluid Mech. 332, 105–120.Google Scholar
Thoraval, M.-J., Takehara, K., Etoh, T. G., Popinet, S., Ray, P., Josserand, C., Zaleski, S. & Thoroddsen, S. T. 2012 Von Kármán vortex street within an impacting drop. Phys. Rev. Lett. 108, 264506.Google Scholar
Thoraval, M.-J., Takehara, K., Etoh, T. G. & Thoroddsen, S. T. 2013 Drop impact entrapment of bubble rings. J. Fluid Mech. 724, 234–258.Google Scholar
Thoroddsen, S. T. 2002 The ejecta sheet generated by the impact of a drop. J. Fluid Mech. 451 (1), 373–381.Google Scholar
Thoroddsen, S. T., Takehara, K. & Etoh, T. G. 2010 Bubble entrapment through topological change. Phys. Fluids 22 (5), 051701.Google Scholar
Thoroddsen, S. T., Thoraval, M.-J., Takehara, K. & Etoh, T. G. 2011 Droplet splashing by a slingshot mechanism. Phys. Rev. Lett. 106 (3), 034501.Google Scholar
Trouton, F. T. 1906 On the coefficient of viscous traction and its relation to that of viscosity. Proc. R. Soc. Lond. A 77 (519), 426–440.Google Scholar
Wagner, H. 1932 Über Stoß- und Gleitvorgänge an der Oberfläche von Flüssigkeiten. Z. Angew. Math. Mech. 12, 193–215.Google Scholar
Weber, C. 1931 Zum Zerfall eines Flüssigkeitsstrahles. Z. Angew. Math. Mech. 11 (2), 136–159.Google Scholar
Wilson, S. K. 1991 A mathematical model for the initial stages of fluid impact in the presence of a cushioning fluid layer. J. Engng Maths 25 (3), 265–285.Google Scholar
Xu, L., Zhang, W. W. & Nagel, S. R. 2005 Drop splashing on a dry smooth surface. Phys. Rev. Lett. 94 (18), 184505.Google Scholar
Yakimov, Y. L. 1973 Influence of atmosphere at falling of bodies on water. Izv. Akad. Nauk SSSR Mekh. Zhidk. Gaza 5 (3), 3–7.Google Scholar
Yarin, A. L. & Weiss, D. A. 1995 Impact of drops on solid surfaces: self-similar capillary waves, and splashing as a new type of kinematic discontinuity. J. Fluid Mech. 283 (1), 141–173.Google Scholar
Zhang, L. V., Toole, J., Fezzaa, K. & Deegan, R. D. 2012a Evolution of the ejecta sheet from the impact of a drop with a deep pool. J. Fluid Mech. 690 (5), 5–15.Google Scholar
Zhang, L. V., Toole, J., Fezzaa, K. & Deegan, R. D. 2012b Splashing from drop impact into a deep pool: multiplicity of jets and the failure of conventional scaling. J. Fluid Mech. 703, 402–413.Google Scholar