Hostname: page-component-cb9f654ff-w5vf4 Total loading time: 0 Render date: 2025-08-28T18:49:22.541Z Has data issue: false hasContentIssue false
  • English
  • Français

Scaling maximum spreading of droplet impacting on flexible substrates

Published online by Cambridge University Press:  07 March 2023

Yufei Ma Open the ORCID record for Yufei Ma [Opens in a new window]  and
Haibo Huang Open the ORCID record for Haibo Huang [Opens in a new window]
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
*
†Email address for correspondence: huanghb@ustc.edu.cn
Get access Rights & Permissions [Opens in a new window]

Abstract

We numerically study the impact of a droplet on superhydrophobic flexible plates, aiming to understand how the flexible substrate influences the maximum spreading of the droplet. Compared with the rigid case, the vertical movement of the flexible substrate due to droplet impact reduces the maximum spreading. Besides, the average acceleration $a$ during droplet spreading changes significantly. Arising from energy conservation, we rescale the acceleration $a$ for cases with different bending stiffness $K_B$ and mass ratio $M_r$. Moreover, through theoretical analysis, we propose a scaling for the droplet's maximum spreading diameter ratio $\beta _{max}$. In the scaling, based on the derived $a$, an effective Weber number $We_m$ is well defined, which accounts for the substrate properties without any adjustable parameters. In the ($\beta _{max}, We_m$) plane, the two-dimensional numerical results of different $K_B$, $M_r$ and rigid cases all collapse into a single curve, as do the experimental and three-dimensional (3-D) results. In particular, the collapsed 3-D data can be well represented by the universal rescaling of $\beta _{max}$ proposed by Lee et al. (J. Fluid Mech., vol. 786, 2016, R4). Furthermore, an a posteriori energy analysis confirms the validation of our a priori scaling law.

JFM classification

Drops and Bubbles: Drops Interfacial Flows (free surface): Contact lines

Information

Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 958 , 10 March 2023 , A35
Check for updates
Copyright
© The Author(s), 2023. 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

REFERENCES

Bergeron, V., Bonn, D., Martin, J.Y. & Vovelle, L. 2000 Controlling droplet deposition with polymer additives. Nature 405 (6788), 772–775.CrossRefGoogle ScholarPubMed
Clanet, C., Béguin, C., Richard, D. & Quéré, D. 2004 Maximal deformation of an impacting drop. J. Fluid Mech. 517, 199–208.CrossRefGoogle Scholar
Derby, B. 2010 Inkjet printing of functional and structural materials: fluid property requirements, feature stability, and resolution. Annu. Rev. Mater. Res. 40, 395–414.10.1146/annurev-matsci-070909-104502CrossRefGoogle Scholar
Dorschner, B., Chikatamarla, S.S. & Karlin, I.V. 2018 Fluid-structure interaction with the entropic lattice Boltzmann method. Phys. Rev. E 97 (2), 023305.CrossRefGoogle ScholarPubMed
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
Eggers, J., Fontelos, M.A., Josserand, C. & Zaleski, S. 2010 Drop dynamics after impact on a solid wall: theory and simulations. Phys. Fluids 22 (6), 062101.10.1063/1.3432498CrossRefGoogle Scholar
Fakhari, A. & Bolster, D. 2017 Diffuse interface modeling of three-phase contact line dynamics on curved boundaries: a lattice Boltzmann model for large density and viscosity ratios. J. Comput. Phys. 334, 620–638.CrossRefGoogle 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.CrossRefGoogle Scholar
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 (18), 184502.CrossRefGoogle ScholarPubMed
Hua, R.N., Zhu, L.D. & Lu, X.Y. 2014 Dynamics of fluid flow over a circular flexible plate. J. Fluid Mech. 759, 56–72.CrossRefGoogle Scholar
Josserand, C. & Thoroddsen, S.T. 2016 Drop impact on a solid surface. Annu. Rev. Fluid Mech. 48, 365–391.10.1146/annurev-fluid-122414-034401CrossRefGoogle Scholar
Joung, Y.S. & Buie, C.R. 2015 Aerosol generation by raindrop impact on soil. Nat. Commun. 6 (1), 1–9.CrossRefGoogle ScholarPubMed
Kim, S., Wu, Z., Esmaili, E., Dombroskie, J.J. & Jung, S. 2020 How a raindrop gets shattered on biological surfaces. Proc. Natl Acad. Sci. USA 117 (25), 13901–13907.CrossRefGoogle ScholarPubMed
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
Lee, J.B., Laan, N., de Bruin, K.G., Skantzaris, G., Shahidzadeh, N., Derome, D., Carmeliet, J. & Bonn, D. 2016 Universal rescaling of drop impact on smooth and rough surfaces. J. Fluid Mech. 786, R4.10.1017/jfm.2015.620CrossRefGoogle 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.CrossRefGoogle ScholarPubMed
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, 425–445.CrossRefGoogle Scholar
Pasandideh-Fard, M., Qiao, Y., Chandra, S. & Mostaghimi, J. 1996 Capillary effects during droplet impact on a solid surface. Phys. Fluids 8 (3), 650–659.CrossRefGoogle Scholar
Pegg, M., Purvis, R. & Korobkin, A. 2018 Droplet impact onto an elastic plate: a new mechanism for splashing. J. Fluid Mech. 839, 561–593.CrossRefGoogle Scholar
Roisman, I.V. 2009 Inertia dominated drop collisions. II. An analytical solution of the Navier–Stokes equations for a spreading viscous film. Phys. Fluids 21 (5), 052104.CrossRefGoogle 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, 8–32.CrossRefGoogle Scholar
Soto, D., De Lariviere, A.B., Boutillon, X., Clanet, C. & Quéré, D. 2014 The force of impacting rain. Soft Matt. 10 (27), 4929–4934.CrossRefGoogle ScholarPubMed
Vasileiou, T., Gerber, J., Prautzsch, J., Schutzius, T.M. & Poulikakos, D. 2016 Superhydrophobicity enhancement through substrate flexibility. Proc. Natl Acad. Sci. USA 113 (47), 13307–13312.10.1073/pnas.1611631113CrossRefGoogle ScholarPubMed
Wang, Y.F., Wang, Y.B., He, X., Zhang, B.X., Yang, Y.R., Wang, X.D. & Lee, D.J. 2022 Scaling laws of the maximum spreading factor for impact of nanodroplets on solid surfaces. J. Fluid Mech. 937, A12.CrossRefGoogle Scholar
Weisensee, P.B., Tian, J., Miljkovic, N. & King, W.P. 2016 Water droplet impact on elastic superhydrophobic surfaces. Sci. Rep. 6 (1), 1–9.CrossRefGoogle ScholarPubMed
Wildeman, S., Visser, C.W., Sun, C. & Lohse, D. 2016 On the spreading of impacting drops. J. Fluid Mech. 805, 636–655.CrossRefGoogle Scholar
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.CrossRefGoogle ScholarPubMed
Ye, X. & Van Der Meer, D. 2021 Hydrogel sphere impact cratering, spreading and bouncing on granular media. J. Fluid Mech. 929, A24.CrossRefGoogle Scholar