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Droplet impact onto an elastic plate: a new mechanism for splashing

  • Michael Pegg (a1), Richard Purvis (a1) and Alexander Korobkin (a1)

During a droplet impact onto a substrate, splashing is known to be caused by the presence of surrounding gas or by surface roughness. Impact occurring in a vacuum onto a smooth rigid wall results in droplet spreading, rather than development of a corona or prompt splash. Here we present an analytical and numerical study of a third potential splashing mechanism, namely elastic deformation of the substrate. An axisymmetric Wagner-style model of droplet impact is formulated and solved using the method of normal modes, together with asymptotic analysis and numerical methods. We highlight the effect that a flexible substrate brings to the contact line velocity and jet behaviour, demonstrating that oscillation of the substrate can cause blow-up of the splash jet which is absent for a rigid substrate and indicate the onset of splashing.

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Alizadeh, A., Bahadur, V., Shang, W., Zhu, Y., Buckley, D., Dhinojwala, A. & Sohal, M. 2013 Influence of substrate elasticity on droplet impact dynamics. Langmuir 29 (14), 45204524.
Blocken, B. & Carmeliet, J. 2004 A reivew of wind-driven rain research in building science. J. Wind Engng Ind. Aerodyn. 92 (13), 10791130.
Ellis, A. S., Smith, F. T. & White, A. H. 2011 Droplet impact on to a rough surface. Q. J. Mech. Appl. Maths 64 (2), 107139.
Gradshteyn, I. S. & Ryzhik, I. M. 2007 Table of Integrals, Series and Products, 7th edn. Academic Press.
Howison, S. D., Ockendon, J. R., Oliver, J. M., Purvis, R. & Smith, F. T. 2005 Droplet impact on a thin fluid layer. J. Fluid Mech. 542, 123.
Howison, S. D., Ockendon, J. R. & Wilson, S. K. 1991 Incompressible water-entry problems at small deadrise angles. J. Fluid Mech. 222, 215230.
Howland, C. J., Antkowiak, A., Castrejón-Pita, J. R., Howison, S. D., Oliver, J. M., Style, R. W. & Castrejón-Pita, A. A. 2016 It’s harder to splash on soft solids. Phys. Rev. Lett. 117, 184502.
von Karman, Th. 1929 The impact on seaplane floats during landing. Nat. Advisory Comittee Aeronaut. 321, 309313.
Khabakhpasheva, T. I. & Korobkin, A. A. 2013 Elastic wedge impact onto a liquid surface: Wagner’s solution and approximate models. J. Fluids Struct. 36, 3249.
Khabakhpasheva, T. I. & Korobkin, A. A. 2016 Liquid drop impact on a vibrating substrate. In Proceedings of 3rd International Conference on Violent Flows, Osaka, Japan.
Korobkin, A. A. 1982 Formulation of penetration problem as a variational inequality. Dinamika Sploshnoi Sredy 58, 7379.
Korobkin, A. A. 1996 Water Impact Problems in Ship Hydrodynamics, chap. 7, pp. 323371. Computational Mechanics Publications.
Korobkin, A. A. 1997 Liquid-Solid Impact. Siberian Branch of the Russian Academy.
Korobkin, A. A. 1998 Wave impact on the centre of an Euler beam. J. Appl. Mech. Tech. Phys. 39 (5), 770781.
Korobkin, A. A. & Scolan, Y. M. 2003 Energy distribution from vertical impact of a three-dimensional solid body onto the flat free surface of an ideal fluid. J. Fluids Struct. 17 (2), 275286.
Korobkin, A. A. & Khabakhpasheva, T. I. 2006 Regular wave impact onto an elastic plate. J. Engng Maths 55, 127159.
Korobkin, A. A. & Scolan, Y. M. 2006 Three-dimensional theory of water impact. Part 2. Linearized Wagner problem. J. Fluid Mech. 549, 343373.
Leissa, A. W.1969 Vibration of plates. Tech. Rep., Scientific and Technical Information Division, National Aeronautics and Space Administration.
Maitra, T., Antonini, C., Tiwari, M. K., Mularczyk, A., Imeri, Z., Schoch, P. & Poulikakos, D. 2014a Supercooled water drops impacting superhydrophobic textures. Langmuir 30 (36), 1085510861.
Maitra, T., Tiwari, M. K., Antonini, C., Schoch, P., Jung, S., Eberle, P. & Poulikakos, D. 2014b On the nanoengineering of superhydrophobic and impalement resistant surface textures below the freezing temperature. Nano Lett. 14 (1), 172182.
Mangili, S., Antonini, C., Marengo, M. & Amirfazli, A. 2012 Understanding the drop impact phenomenon on soft PDMS substrates. Soft Matt. 8 (39), 1004510054.
Martin, G. D., Hoath, S. D. & Hutchings, I. M. 2008 Inkjet printing: the physics of manipulating liquid jets and drops. J. Phys.: Conf. Ser. 105, 012001.
Moore, M. R., Howison, S. D., Ockendon, J. R. & Oliver, J. M. 2012 Three-dimensional oblique water-entry problems at small deadrise angles. J. Fluid Mech. 711, 259280.
Moore, M. R., Ockendon, J. R. & Oliver, J. M. 2013 Air-cushioning in impact problems. IMA J. Appl. Maths 78 (4), 818838.
Moore, M. R. & Oliver, J. M. 2014 On air cushioning in axisymmetric impacts. IMA J. Appl. Maths 79 (4), 661.
Nguyen, T. V., Takahashi, H. & Shimoyama, I. 2017 MEMS-based pressure sensor with a superoleophobic membrane for measuring droplet vibration. In 2017 19th International Conference on Solid-State Sensors, Actuators and Microsystems (TRANSDUCERS), pp. 11521155. IEEE.
Oliver, J. M.2002 Water entry and related problems. PhD thesis, St Anne’s College, University of Oxford.
Pepper, R. E., Courbin, L. & Stone, H. A. 2008 Splashing on elastic membranes: the importance of early-time dynamics. Phys. Fluids 20 (8), 082103.
Philippi, J., Lagree, P. Y. & Antkowiak, A. 2016 Drop impact on a solid surface: short time self-similarity. J. Fluid Mech. 795, 96138.
Reinhard, M., Korobkin, A. A. & Cooker, M. J. 2013 Water entry of a flat elastic plate at high horizontal speed. J. Fluid Mech. 724, 123153.
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.
Scolan, Y. M. 2004 Hydroelastic behaviour of a conical shell impacting on a quiescent-free surface of an incompressible liquid. J. Sound Vib. 227 (1–2), 163203.
Scolan, Y. M. & Korobkin, A. A. 2001 Three-dimensional theory of water impact. Part 1. Inverse Wagner problem. J. Fluid Mech. 440, 293326.
Semenov, Y. A., Wu, G. X. & Korobkin, A. A. 2015 Impact of liquids with different densities. J. Fluid Mech. 766, 527.
Sneddon, I. N. 1966 Mixed Boundary Value Problems in Potential Theory. North-Holland.
Tkacheva, L. A. 2008 Impact of a box with an elastic bottom on a thin liquid layer. J. Appl. Math. Mech. 72 (4), 427436.
Vasileiou, T., Schutzius, T. M. & Poulikakos, D. 2017 Imparting icephobicity with substrate flexibility. Langmuir 33 (27), 67086718.
Wagner, H. 1932 Über Stoss- und Gleitvergänge an der Oberfläche von Flüssigkeiten. Z. Angew. Math. Mech. 12 (4), 193215.
Weisensee, P. B., Tian, J., Miljkovik, N. & King, W. P. 2016 Water droplet impact on elastic superhydrophobic surfaces. Sci. Rep. 6, 30328.
Wu, G. X. 2007 Two-dimensional liquid column and liquid droplet impact on a solid wedge. Q. J. Mech. Appl. Maths 60, 496511.
Xu, G. D., Wu, G. X. & Duan, W. Y. 2011 Axisymmetric liquid block impact on a solid surface. Appl. Ocean Res. 33 (4), 366374.
Xu, L. 2007 Liquid drop splashing on smooth, rough, and textured surfaces. Phys. Rev. E 75 (5).
Xu, L., Zhang, W. W. & Nagel, S. R. 2005 Drop splashing on a dry smooth surface. Phys. Rev. Lett. 94 (18), 184505.
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Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
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