Hostname: page-component-7f64f4797f-l842n Total loading time: 0 Render date: 2025-11-05T04:04:13.466Z Has data issue: false hasContentIssue false

Early bounce of droplets on elastic plates: the role of momentum morphology and predictive modelling

Published online by Cambridge University Press:  03 November 2025

Yufei Ma
Affiliation:
Department of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui 230026, PR China
Haibo Huang*
Affiliation:
Department of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui 230026, PR China
*
Corresponding author: Haibo Huang, huanghb@ustc.edu.cn

Abstract

This study investigates droplet impact on elastic plates using a two-phase lattice Boltzmann method in both two-dimensional (2-D) and three-dimensional (3-D) configurations, with a focus on rebound dynamics and contact time. The 2-D simulations reveal three distinct rebound modes – conventional bounce, early bounce and rim rising – driven by fluid–structure interaction. Among them, the early bounce mode uniquely achieves a significant reduction in contact time, occurring only at moderate plate oscillation frequency. Momentum analysis shows a non-monotonic relationship between vertical momentum transfer and rebound efficiency: increased momentum does not necessarily promote rebound if it concentrates in a central jet, which contributes minimally to lift-off. This introduces a novel rebound mechanism governed by momentum distribution morphology rather than total magnitude. A theoretical model treating the droplet–plate system as coupled oscillators is developed to predict contact time in the early bounce regime, showing good agreement with numerical results. The mechanism and model are further validated through fully 3-D simulations, confirming the robustness of the findings.

Information

Type
JFM Papers
Copyright
© The Author(s), 2025. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Abolghasemibizaki, M., Dilmaghani, N., Mohammadi, R. & Castano, C.E. 2019 Viscous droplet impact on nonwettable textured surfaces. Langmuir 35 (33), 1075210761.10.1021/acs.langmuir.9b01109CrossRefGoogle ScholarPubMed
Antonini, C., Bernagozzi, I., Jung, S., Poulikakos, D. & Marengo, M. 2013 Water drops dancing on ice: how sublimation leads to drop rebound. Phys. Rev. Lett. 111, 014501.10.1103/PhysRevLett.111.014501CrossRefGoogle ScholarPubMed
Bhushan, B., Jung, Y.C. 2011 Natural and biomimetic artificial surfaces for superhydrophobicity, self-cleaning, low adhesion, and drag reduction. Prog. Mater. Sci. 56 (1), 1108.10.1016/j.pmatsci.2010.04.003CrossRefGoogle Scholar
Biance, A.-L., Chevy, F., Clanet, C., Lagubeau, G. & Quéré, D. 2006 On the elasticity of an inertial liquid shock. J. Fluid Mech. 554, 4766.10.1017/S0022112006009189CrossRefGoogle Scholar
Bird, J.C., Dhiman, R., Kwon, H.-M. & Varanasi, K.K. 2013 Reducing the contact time of a bouncing drop. Nature 503 (7476), 385388.10.1038/nature12740CrossRefGoogle ScholarPubMed
Blossey, R. 2003 Self-cleaning surfacesvirtual realities. Nat. Mater. 2 (5), 301306.10.1038/nmat856CrossRefGoogle ScholarPubMed
Chantelot, P., Coux, M., Clanet, C. & Quéré, D. 2018 Drop trampoline. Europhys. Lett. 124 (2), 24003.10.1209/0295-5075/124/24003CrossRefGoogle Scholar
Cheng, X., Sun, T.-P. & Gordillo, L. 2022 Drop impact dynamics: impact force and stress distributions. Annu. Rev. Fluid Mech. 54 (1), 5781.10.1146/annurev-fluid-030321-103941CrossRefGoogle Scholar
Clanet, C., Béguin, C., Richard, D. & Quéré, D. 2004 Maximal deformation of an impacting drop. J. Fluid Mech. 517, 199208.10.1017/S0022112004000904CrossRefGoogle Scholar
Connell, B.S.H. & Yue, D.K.P. 2007 Flapping dynamics of a flag in a uniform stream. J. Fluid Mech. 581, 3367.10.1017/S0022112007005307CrossRefGoogle Scholar
Derby, B. 2010 Inkjet printing of functional and structural materials: fluid property requirements, feature stability, and resolution. Annu. Rev. Mater. Res. 40, 395414.10.1146/annurev-matsci-070909-104502CrossRefGoogle Scholar
Dhiman, R., McDonald, Aé G. & Chandra, S. 2007 Predicting splat morphology in a thermal spray process. Surf. Coatings Technol. 201 (18), 77897801.10.1016/j.surfcoat.2007.03.010CrossRefGoogle Scholar
Doyle, J.F. 2001 Nonlinear Analysis of Thin-Walled Structures: Statics, Dynamics, and Stability. Springer Science & Business Media.10.1007/978-1-4757-3546-8CrossRefGoogle Scholar
Dressaire, E., Sauret, A., Boulogne, F. & Stone, H.A. 2016 Drop impact on a flexible fiber. Soft Matter 12, 200208.10.1039/C5SM02246KCrossRefGoogle ScholarPubMed
Fan, Y., Tan, Y., Dou, Y., Huang, S. & Tian, X. 2023 Reducing the contact time of bouncing droplets on superhydrophobic surfaces: foundations, strategies and applications. Chem. Engng J. 476, 146485.10.1016/j.cej.2023.146485CrossRefGoogle Scholar
Gabbard, C.T., Agüero, E.A., Cimpeanu, R., Kuehr, K., Silver, E., Barotta, J.-W., Galeano-Rios, C.A. & Harris, D.M. 2025 Drop rebound at low Weber number. J. Fluid Mech. 1019, A25.10.1017/jfm.2025.10589CrossRefGoogle Scholar
Gart, S., Mates, J.E., Megaridis, C.M. & Jung, S. 2015 Droplet impacting a cantilever: a leaf-raindrop system. Phys. Rev. Appl. 3 (4), 044019.10.1103/PhysRevApplied.3.044019CrossRefGoogle Scholar
Gauthier, A., Symon, S., Clanet, C. & Quéré, D. 2015 Water impacting on superhydrophobic macrotextures. Nat. Commun. 6 (1), 8001.10.1038/ncomms9001CrossRefGoogle ScholarPubMed
de Goede, T.C., de Bruin, K.G., Shahidzadeh, N. & Bonn, D. 2019 Predicting the maximum spreading of a liquid drop impacting on a solid surface: effect of surface tension and entrapped air layer. Phys. Rev. Fluids 4 (5), 053602.10.1103/PhysRevFluids.4.053602CrossRefGoogle Scholar
Hua, R.-N., Zhu, L. & Lu, X.-Y. 2014 Dynamics of fluid flow over a circular flexible plate. J. Fluid Mech. 759, 5672.10.1017/jfm.2014.571CrossRefGoogle Scholar
Huang, L., Song, J., Wang, X., Zhao, C., Liu, Z. & Liu, J. 2018 Soft elastic superhydrophobic cotton: a new material for contact time reduction in droplet bouncing. Surf. Coatings Technol. 347, 420426.10.1016/j.surfcoat.2018.05.019CrossRefGoogle Scholar
Josserand, C. & Thoroddsen, S.T. 2016 Drop impact on a solid surface. Annu. Rev. Fluid Mech. 48, 365391.10.1146/annurev-fluid-122414-034401CrossRefGoogle Scholar
Joung, Y.S. & Buie, C.R. 2015 Aerosol generation by raindrop impact on soil. Nat. Commun. 6 (1), 19.10.1038/ncomms7083CrossRefGoogle ScholarPubMed
Kim, J.-H., Rothstein, J.P. & Shang, J.K. 2018 Dynamics of a flexible superhydrophobic surface during a drop impact. Phys. Fluids 30 (7), 072102.10.1063/1.5028127CrossRefGoogle Scholar
Kraj, A.G. & Bibeau, E.L. 2010 Phases of icing on wind turbine blades characterized by ice accumulation. Renew. Energy 35 (5), 966972.10.1016/j.renene.2009.09.013CrossRefGoogle Scholar
Kreder, M.J., Alvarenga, J., Kim, P. & Aizenberg, J. 2016 Design of anti-icing surfaces: smooth, textured or slippery? Nat. Rev. Materials 1 (1), 115.Google Scholar
Laan, N., de Bruin, K.G., Bartolo, D., Josserand, C. & Bonn, D. 2014 Maximum diameter of impacting liquid droplets. Phys. Rev. Appl. 2 (4), 044018.10.1103/PhysRevApplied.2.044018CrossRefGoogle Scholar
Li, J., Hou, Y., Liu, Y., Hao, C., Li, M., Chaudhury, M.K., Yao, S. & Wang, Z. 2016 Directional transport of high-temperature Janus droplets mediated by structural topography. Nat. Phys. 12 (6), 606612.10.1038/nphys3643CrossRefGoogle Scholar
Li, X., Ma, X. & Lan, Z. 2010 Dynamic behavior of the water droplet impact on a textured hydrophobic/superhydrophobic surface: the effect of the remaining liquid film arising on the pillars’ tops on the contact time. Langmuir 26 (7), 48314838.10.1021/la903603zCrossRefGoogle ScholarPubMed
Liang, G. & Mudawar, I. 2017 Review of drop impact on heated walls. Intl J. Heat Mass Transfer 106, 103126.10.1016/j.ijheatmasstransfer.2016.10.031CrossRefGoogle Scholar
Liang, H., Xu, J.R., Chen, J.X., Wang, H.L., Chai, Z.H. & Shi, B.C. 2018 Phase-field-based lattice Boltzmann modeling of large-density-ratio two-phase flows. Phys. Rev. E 97 (3), 033309.10.1103/PhysRevE.97.033309CrossRefGoogle ScholarPubMed
Lin, S., Shi, D., Zhou, L., Chen, L. & Li, Z. 2024 Probing the contact time of droplet impacts: From the Hertz collision to oscillation regimes. Phys. Rev. E 110, L053101.10.1103/PhysRevE.110.L053101CrossRefGoogle ScholarPubMed
Liu, Y., Andrew, M., Li, J., Yeomans, J.M. & Wang, Z. 2015 Symmetry breaking in drop bouncing on curved surfaces. Nat. Commun. 6 (1), 10034.10.1038/ncomms10034CrossRefGoogle ScholarPubMed
Liu, Y., Moevius, L., Xu, X., Qian, T., Yeomans, J.M. & Wang, Z. 2014 Pancake bouncing on superhydrophobic surfaces. Nat. Phys. 10 (7), 515519.10.1038/nphys2980CrossRefGoogle ScholarPubMed
Lohse, D. 2022 Fundamental fluid dynamics challenges in inkjet printing. Annu. Rev. Fluid Mech. 54 (1), 349382.10.1146/annurev-fluid-022321-114001CrossRefGoogle Scholar
Lv, C., Hao, P., Zhang, X. & He, F. 2016 Drop impact upon superhydrophobic surfaces with regular and hierarchical roughness. Appl. Phys. Lett. 108 (14), 141602.10.1063/1.4945662CrossRefGoogle Scholar
Ma, Y. & Huang, H. 2023 Scaling maximum spreading of droplet impacting on flexible substrates. J. Fluid Mech. 958, A35.10.1017/jfm.2023.124CrossRefGoogle Scholar
Ma, Y. & Huang, H. 2024 Clinching 1/2 scaling: deciphering spreading data of droplet impact. Phys. Rev. Fluids 9, 113601.10.1103/PhysRevFluids.9.113601CrossRefGoogle Scholar
Ma, Y. & Huang, H. 2025 Spreading and retraction dynamics of drop impact onto elastic surfaces. Phys. Rev. Fluids 10, 053607.10.1103/PhysRevFluids.10.053607CrossRefGoogle Scholar
Maitra, T., Antonini, C., Tiwari, M.K., Mularczyk, A., Imeri, Z., Schoch, P. & Poulikakos, D. 2014 a Supercooled water drops impacting superhydrophobic textures. Langmuir 30 (36), 1085510861.10.1021/la502675aCrossRefGoogle ScholarPubMed
Maitra, T., Antonini, C., Tiwari, M.K., Mularczyk, A., Imeri, Z., Schoch, P. & Poulikakos, D. 2014 b Supercooled water drops impacting superhydrophobic textures. Langmuir 30 (36), 1085510861.10.1021/la502675aCrossRefGoogle ScholarPubMed
Massinon, M. & Lebeau, F. 2012 Experimental method for the assessment of agricultural spray retention based on high-speed imaging of drop impact on a synthetic superhydrophobic surface. Biosyst. Engng 112 (1), 5664.10.1016/j.biosystemseng.2012.02.005CrossRefGoogle Scholar
Mishchenko, L., Hatton, B., Bahadur, V., Taylor, J.A., Krupenkin, T. & Aizenberg, J. 2010 Design of ice-free nanostructured surfaces based on repulsion of impacting water droplets. ACS Nano 4 (12), 76997707.10.1021/nn102557pCrossRefGoogle ScholarPubMed
Mohseni, M. & Amirfazli, A. 2013 A novel electro-thermal anti-icing system for fiber-reinforced polymer composite airfoils. Cold Reg. Sci. Technol. 87, 4758.10.1016/j.coldregions.2012.12.003CrossRefGoogle Scholar
Molacek, J. & Bush, J.W.M. 2012 A quasi-static model of drop impact. Phys. Fluids 24 (12), 127103.10.1063/1.4771607CrossRefGoogle Scholar
Moqaddam, A.M., Chikatamarla, S.S. & Karlin, I.V. 2017 Drops bouncing off macro-textured superhydrophobic surfaces. J. Fluid Mech. 824, 866885.10.1017/jfm.2017.306CrossRefGoogle Scholar
Nuyttens, D., Baetens, K., De Schampheleire, M. & Sonck, B. 2007 Effect of nozzle type, size and pressure on spray droplet characteristics. Biosyst. Engng 97 (3), 333345.10.1016/j.biosystemseng.2007.03.001CrossRefGoogle Scholar
Pan, J.-H., Ni, M.-J. & Zhang, N.-M. 2018 A consistent and conservative immersed boundary method for MHD flows and moving boundary problems. J. Comput. Phys. 373, 425445.10.1016/j.jcp.2017.12.034CrossRefGoogle Scholar
Quéré, D. 2008 Wetting and roughness. Annu. Rev. Mater. Res. 38 (1), 7199.10.1146/annurev.matsci.38.060407.132434CrossRefGoogle Scholar
Regulagadda, K., Bakshi, S. & Das, S.K. 2018 Droplet ski-jumping on an inclined macro-textured superhydrophobic surface. Appl. Phys. Lett. 113 (10), 103702.10.1063/1.5048301CrossRefGoogle Scholar
Richard, D., Clanet, C. & Quéré, D. 2002 Contact time of a bouncing drop. Nature 417 (6891), 811811.10.1038/417811aCrossRefGoogle ScholarPubMed
Richard, D. & Quéré, D. 2000 Bouncing water drops. Europhys. Lett. 50 (6), 769.10.1209/epl/i2000-00547-6CrossRefGoogle Scholar
Sanjay, V., Chantelot, P. & Lohse, D. 2023 When does an impacting drop stop bouncing? J. Fluid Mech. 958, A26.10.1017/jfm.2023.55CrossRefGoogle Scholar
Shao, J.Y., Shu, C. & Chew, Y.T. 2013 Development of an immersed boundary-phase field-lattice Boltzmann method for Neumann boundary condition to study contact line dynamics. J. Comput. Phys. 234, 832.10.1016/j.jcp.2012.08.040CrossRefGoogle Scholar
Shiri, S. & Bird, J.C. 2017 Heat exchange between a bouncing drop and a superhydrophobic substrate. Proc. Natl Acad. Sci. 114 (27), 69306935.10.1073/pnas.1700197114CrossRefGoogle Scholar
Soto, D., De Larivière, Aélie B., Boutillon, X., Clanet, C. & Quéré, D. 2014 The force of impacting rain. Soft Matter 10 (27), 49294934.10.1039/C4SM00513ACrossRefGoogle ScholarPubMed
Sousa, S.C.L., Mendes, A.d.O., Fiadeiro, P.T. & Ramos, A.M.M. 2014 Dynamic interactions of pigment-based inks on chemically modified papers and their influence on inkjet print quality. Indust. Engng Chem. Res. 53 (12), 46604668.10.1021/ie403595fCrossRefGoogle Scholar
Sun, Q., et al. 2019 Surface charge printing for programmed droplet transport. Nat. Mater. 18 (9), 936941.10.1038/s41563-019-0440-2CrossRefGoogle ScholarPubMed
Takaki, R. & Adachi, K. 1985 Vibration of a flattened drop. ii. normal mode analysis. J. Phys. Soc. Japan 54 (7), 24622469.10.1143/JPSJ.54.2462CrossRefGoogle Scholar
Thenarianto, C., Koh, X.Q., Lin, M., Jokinen, V. & Daniel, D. 2023 Energy loss for droplets bouncing off superhydrophobic surfaces. Langmuir 39 (8), 31623167.10.1021/acs.langmuir.2c03449CrossRefGoogle ScholarPubMed
Upadhyay, G., Kumar, V. & Bhardwaj, R. 2021 Bouncing droplets on an elastic, superhydrophobic cantilever beam. Phys. Fluids 33 (4), 042104.10.1063/5.0047868CrossRefGoogle Scholar
Vasileiou, T., Gerber, J., Prautzsch, J., Schutzius, T.M. & Poulikakos, D. 2016 Superhydrophobicity enhancement through substrate flexibility. Proc. Natl Acad. Sci. USA 113 (47), 1330713312.10.1073/pnas.1611631113CrossRefGoogle ScholarPubMed
Villermaux, E. & Bossa, B. 2011 Drop fragmentation on impact. J. Fluid Mech. 668, 412435.10.1017/S002211201000474XCrossRefGoogle Scholar
Voigt, C.C., Schneeberger, K., Voigt-Heucke, S.L. & Lewanzik, D. 2011 Rain increases the energy cost of bat flight. Biol. Lett.-UK 7 (5), 793795.10.1098/rsbl.2011.0313CrossRefGoogle ScholarPubMed
Wang, L., Wang, R., Wang, J. & Wong, T.-S. 2020 Compact nanoscale textures reduce contact time of bouncing droplets. Sci. Adv. 6 (29), eabb2307.10.1126/sciadv.abb2307CrossRefGoogle ScholarPubMed
Wei, X., Jia, Z., Sun, Z., Farzaneh, M. & Guan, Z. 2016 Effect of the parameters of the semiconductive coating on the anti-icing performance of the insulators. IEEE Trans. Power Delivery 31 (4), 14131421.10.1109/TPWRD.2014.2337012CrossRefGoogle Scholar
Weisensee, P.B., Tian, J., Miljkovic, N. & King, W.P. 2016 Water droplet impact on elastic superhydrophobic surfaces. Sci. Rep. 6 (1), 19.10.1038/srep30328CrossRefGoogle ScholarPubMed
Wisdom, K.M., Watson, J.A., Qu, X., Liu, F., Watson, G.S. & Chen, C.-H. 2013 Self-cleaning of superhydrophobic surfaces by self-propelled jumping condensate. Proc. Natl Acad. Sci. 110 (20), 79927997.10.1073/pnas.1210770110CrossRefGoogle ScholarPubMed
Xiong, Y.F., Huang, H.B. & Lu, X.Y. 2020 Numerical study of droplet impact on a flexible substrate. Phys. Rev. E 101 (5), 053107.10.1103/PhysRevE.101.053107CrossRefGoogle ScholarPubMed
Yarin, A.L. 2006 Drop impact dynamics: splashing, spreading, receding, bouncing. Annu. Rev. Fluid Mech. 38 (1), 159192.10.1146/annurev.fluid.38.050304.092144CrossRefGoogle Scholar
Ye, Q. & Domnick, J. 2017 Analysis of droplet impingement of different atomizers used in spray coating processes. J. Coat. Technol. Res. 14 (2), 467476.10.1007/s11998-016-9867-4CrossRefGoogle Scholar
Zhan, H., Lu, C., Liu, C., Wang, Z., Lv, C. & Liu, Y. 2021 Horizontal motion of a superhydrophobic substrate affects the drop bouncing dynamics. Phys. Rev. Lett. 126 (23), 234503.10.1103/PhysRevLett.126.234503CrossRefGoogle ScholarPubMed
Zhang, Z., Chi, Y., Shang, L., Zhang, P. & Zhao, Z. 2016 On the role of droplet bouncing in modeling impinging sprays under elevated pressures. Intl J. Heat Mass Transfer 102, 657668.10.1016/j.ijheatmasstransfer.2016.06.052CrossRefGoogle Scholar