Skip to main content Accessibility help

Viscoplastic water entry

  • Maziyar Jalaal (a1), Dave Kemper (a1) and Detlef Lohse (a1) (a2)

The impact of viscoplastic droplets on a free surface of water is studied. The droplet undergoes an elasto-plastic deformation at the early stages of water entry. At large time, the yield stress dominates; therefore, the droplet solidifies and reaches an equilibrium shape. Depending on the impact velocity and the rheology of the droplet, the final morphologies vary from pear-shaped to capsules that contain bubbles. We perform an analysis of the orders of magnitude of the forces and introduce the relevant dimensionless groups. Furthermore, we categorize the final shapes in a phase diagram and analyse their geometrical properties. The process presents a method of making non-spherical beads and capsules with tunable shapes and provides information on the general problem of the impact of highly deformable objects on a liquid surface.

  • View HTML
    • Send article to Kindle

      To send this article to your Kindle, first ensure is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the or variations. ‘’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Viscoplastic water entry
      Available formats
      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Viscoplastic water entry
      Available formats
      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Viscoplastic water entry
      Available formats
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (, which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Corresponding author
Email addresses for correspondence:,
Hide All
Auguste, F., Magnaudet, J. & Fabre, D. 2013 Falling styles of disks. J. Fluid Mech. 719, 388405.
Balmforth, N. & Hewitt, I. 2013 Viscoplastic sheets and threads. J. Non-Newtonian Fluid Mech. 193, 2842.
Balmforth, N. J., Frigaard, I. A. & Ovarlez, G. 2014 Yielding to stress: recent developments in viscoplastic fluid mechanics. Annu. Rev. Fluid Mech. 46, 121146.
Beesabathuni, S. N., Lindberg, S. E., Caggioni, M., Wesner, C. & Shen, A. Q. 2015 Getting in shape: molten wax drop deformation and solidification at an immiscible liquid interface. J. Colloid Interface Sci. 445, 231242.
Berberović, E., van Hinsberg, N. P., Jakirlić, S., Roisman, I. V. & Tropea, C. 2009 Drop impact onto a liquid layer of finite thickness: dynamics of the cavity evolution. Phys. Rev. E 79 (3), 036306.
Bergmann, R., van der Meer, D., Stijnman, M., Sandtke, M., Prosperetti, A. & Lohse, D. 2006 Giant bubble pinch-off. Phys. Rev. Lett. 96 (15), 154505.
Bergmann, R., Van Der Meer, D., Gekle, S., Van Der Bos, A. & Lohse, D. 2009 Controlled impact of a disk on a water surface: cavity dynamics. J. Fluid Mech. 633, 381409.
Bingham, E. C. 1917 An investigation of the laws of plastic flow. Bull. Bur. Stand. 13 (278), 309353.
Bisighini, A., Cossali, G. E., Tropea, C. & Roisman, I. V. 2010 Crater evolution after the impact of a drop onto a semi-infinite liquid target. Phys. Rev. E 82 (3), 036319.
Blackwell, B. C., Deetjen, M. E., Gaudio, J. E. & Ewoldt, R. H. 2015 Sticking and splashing in yield-stress fluid drop impacts on coated surfaces. Phys. Fluids 27 (4), 043101.
Bonn, D., Denn, M. M., Berthier, L., Divoux, T. & Manneville, S. 2017 Yield stress materials in soft condensed matter. Rev. Mod. Phys. 89 (3), 035005.
Cai, Y. 1989 Phenomena of a liquid drop falling to a liquid surface. Exp. Fluids 7 (6), 388394.
Chapman, D. S. & Critchlow, P. 1967 Formation of vortex rings from falling drops. J. Fluid Mech. 29 (1), 177185.
Chen, S. & Bertola, V. 2017 Morphology of viscoplastic drop impact on viscoplastic surfaces. Soft Matt. 13 (4), 711719.
Coussot, P. 2014 Yield stress fluid flows: a review of experimental data. J. Non-Newtonian Fluid Mech. 211, 3149.
Deng, Q., Anilkumar, A. & Wang, T. 2007 The role of viscosity and surface tension in bubble entrapment during drop impact onto a deep liquid pool. J. Fluid Mech. 578, 119138.
Dinkgreve, M., Paredes, J., Denn, M. M. & Bonn, D. 2016 On different ways of measuring the yield stress. J. Non-Newtonian Fluid Mech. 238, 233241.
Dooley, B. S., Warncke, A. E., Gharib, M. & Tryggvason, G. 1997 Vortex ring generation due to the coalescence of a water drop at a free surface. Exp. Fluids 22 (5), 369374.
Elmore, P., Chahine, G. & Oguz, H. 2001 Cavity and flow measurements of reproducible bubble entrainment following drop impacts. Exp. Fluids 31 (6), 664673.
Esmailizadeh, L. & Mesler, R. 1986 Bubble entrainment with drops. J. Colloid Interface Sci. 110 (2), 561574.
Fedorchenko, A. I. & Wang, A.-B. 2004 On some common features of drop impact on liquid surfaces. Phys. Fluids 16 (5), 13491365.
Franz, G. 1959 Splashes as sources of sound in liquids. J. Acoust. Soc. Am. 31 (8), 10801096.
German, G. & Bertola, V. 2009 Impact of shear-thinning and yield-stress drops on solid substrates. J. Phys.: Condens. Matter 21 (37), 375111.
German, G. & Bertola, V. 2010 The free-fall of viscoplastic drops. J. Non-Newtonian Fluid Mech. 165 (13), 825828.
Gielen, M. V., Sleutel, P., Benschop, J., Riepen, M., Voronina, V., Visser, C. W., Lohse, D., Snoeijer, J. H., Versluis, M. & Gelderblom, H. 2017 Oblique drop impact onto a deep liquid pool. Phys. Rev. Fluids 2 (8), 083602.
Hallett, J. & Christensen, L. 1984 Splash and penetration of drops in water. J. Rech. Atmos. 18 (4), 225242.
Herschel, W. & Bulkley, R. 1926 Measurement of consistency as applied to rubber–benzene solutions. Proc. Am. Soc. Test. Mater. 26, 621633.
Hsiao, M., Lichter, S. & Quintero, L. G. 1988 The critical Weber number for vortex and jet formation for drops impinging on a liquid pool. Phys. Fluids 31 (12), 35603562.
Hurd, R. C., Belden, J., Jandron, M. A., Fanning, D. T., Bower, A. F. & Truscott, T. T. 2017 Water entry of deformable spheres. J. Fluid Mech. 824, 912930.
Jalaal, M.2016 Controlled spreading of complex droplets. PhD thesis, University of British Columbia.
Jalaal, M., Balmforth, N. J. & Stoeber, B. 2015 Slip of spreading viscoplastic droplets. Langmuir 31 (44), 1207112075.
Jalaal, M., Seyfert, C., Stoeber, B. & Balmforth, N. 2018 Gel-controlled droplet spreading. J. Fluid Mech. 837, 115128.
Jalaal, M. & Stoeber, B. 2014 Controlled spreading of thermo-responsive droplets. Soft Matt. 10 (6), 808812.
Jørgensen, L., Le Merrer, M., Delanoë-Ayari, H. & Barentin, C. 2015 Yield stress and elasticity influence on surface tension measurements. Soft Matt. 11 (25), 51115121.
Korobkin, A. & Pukhnachov, V. 1988 Initial stage of water impact. Annu. Rev. Fluid Mech. 20 (1), 159185.
Lee, D., Beesabathuni, S. N. & Shen, A. Q. 2015 Shape-tunable wax microparticle synthesis via microfluidics and droplet impact. Biomicrofluidics 9 (6), 064114.
Leng, L. J. 2001 Splash formation by spherical drops. J. Fluid Mech. 427, 73105.
Lhuissier, H., Sun, C., Prosperetti, A. & Lohse, D. 2013 Drop fragmentation at impact onto a bath of an immiscible liquid. Phys. Rev. Lett. 110 (26), 264503.
Liow, J.-L. & Cole, D. 2009 High framing rate PIV studies of an impinging water drop. In 28th International Congress on High-Speed Imaging and Photonics, vol. 7126, p. 71260L. International Society for Optics and Photonics.
Liu, Y. & de Bruyn, J. R. 2018 Start-up flow of a yield-stress fluid in a vertical pipe. J. Non-Newtonian Fluid Mech. 257, 5058.
Lush, P. 1983 Impact of a liquid mass on a perfectly plastic solid. J. Fluid Mech. 135, 373387.
Luu, L.-H. & Forterre, Y. 2009 Drop impact of yield-stress fluids. J. Fluid Mech. 632, 301327.
Macklin, W. & Metaxas, G. 1976 Splashing of drops on liquid layers. J. Appl. Phys. 47 (9), 39633970.
Madlener, K., Frey, B. & Ciezki, H. 2009 Generalized Reynolds number for non-Newtonian fluids. In Progress in Propulsion Physics, vol. 1, pp. 237250. EDP Sciences.
Manglik, R. M., Wasekar, V. M. & Zhang, J. 2001 Dynamic and equilibrium surface tension of aqueous surfactant and polymeric solutions. Exp. Therm. Fluid Sci. 25 (1), 5564.
Miloh, T. 1991 On the initial-stage slamming of a rigid sphere in a vertical water entry. Appl. Ocean Res. 13 (1), 4348.
Morton, D., Rudman, M. & Jong-Leng, L. 2000 An investigation of the flow regimes resulting from splashing drops. Phys. Fluids 12 (4), 747763.
Murphy, D. W., Li, C., d’Albignac, V., Morra, D. & Katz, J. 2015 Splash behaviour and oily marine aerosol production by raindrops impacting oil slicks. J. Fluid Mech. 780, 536577.
Oguz, H. N. & Prosperetti, A. 1990 Bubble entrainment by the impact of drops on liquid surfaces. J. Fluid Mech. 219, 143179.
Peck, B. & Sigurdson, L. 1994 The three-dimensional vortex structure of an impacting water drop. Phys. Fluids 6 (2), 564576.
Peseux, B., Gornet, L. & Donguy, B. 2005 Hydrodynamic impact: numerical and experimental investigations. J. Fluids Struct. 21 (3), 277303.
Peters, I. R., van der Meer, D. & Gordillo, J. M. 2013 Splash wave and crown breakup after disc impact on a liquid surface. J. Fluid Mech. 724, 553580.
Pumphrey, H. C., Crum, L. & Bjørnø, L. 1989 Underwater sound produced by individual drop impacts and rainfall. J. Acoust. Soc. Am. 85 (4), 15181526.
Pumphrey, H. C. & Elmore, P. A. 1990 The entrainment of bubbles by drop impacts. J. Fluid Mech. 220, 539567.
Khaleeq-ur Rahman, M. & Saunders, C. 1988 Corona from splashing water drops. J. Atmos. Terr. Phys. 50 (6), 545555.
Rein, M. 1996 The transitional regime between coalescing and splashing drops. J. Fluid Mech. 306, 145165.
Roberts, G. P. & Barnes, H. A. 2001 New measurements of the flow-curves for carbopol dispersions without slip artefacts. Rheol. Acta 40 (5), 499503.
San Lee, J., Park, S. J., Lee, J. H., Weon, B. M., Fezzaa, K. & Je, J. H. 2015 Origin and dynamics of vortex rings in drop splashing. Nat. Commun. 6, 8187.
Van de Sande, E., Smith, J. M. & Van Oord, J. 1974 Energy transfer and cavity formation in liquid-drop collisions. J. Appl. Phys. 45 (2), 748753.
Saramito, P. 2009 A new elastoviscoplastic model based on the Herschel–Bulkley viscoplastic model. J. Non-Newtonian Fluid Mech. 158 (1), 154161.
Seddon, C. & Moatamedi, M. 2006 Review of water entry with applications to aerospace structures. Intl J. Impact Engng 32 (7), 10451067.
Sigler, J. & Mesler, R. 1990 The behavior of the gas film formed upon drop impact with a liquid surface. J. Colloid Interface Sci. 134 (2), 459474.
Thompson, R. L. & Soares, E. J. 2016 Viscoplastic dimensionless numbers. J. Non-Newtonian Fluid Mech. 238, 5764.
Thomson, J. J. & Newall, H. F. 1886 V. On the formation of vortex rings by drops falling into liquids, and some allied phenomena. Proc. R. Soc. Lond. 39 (239–241), 417436.
Thoraval, M.-J., Li, Y. & Thoroddsen, S. T. 2016 Vortex-ring-induced large bubble entrainment during drop impact. Phys. Rev. E 93 (3), 033128.
Tomita, Y., Saito, T. & Ganbara, S. 2007 Surface breakup and air bubble formation by drop impact in the irregular entrainment region. J. Fluid Mech. 588, 131152.
Truscott, T. T., Epps, B. P. & Belden, J. 2014 Water entry of projectiles. Annu. Rev. Fluid Mech. 46, 355378.
Truscott, T. T., Epps, B. P. & Techet, A. H. 2012 Unsteady forces on spheres during free-surface water entry. J. Fluid Mech. 704, 173210.
Wang, T., Lacík, I., Brissová, M., Anilkumar, A. V., Prokop, A., Hunkeler, D., Green, R., Shahrokhi, K. & Powers, A. C. 1997 An encapsulation system for the immunoisolation of pancreatic islets. Nat. Biotechnol. 15 (4), 358.
Worthington, A. M. 1883 On impact with a liquid surface. Proc. R. Soc. Lond. 34 (220–223), 217230.
Worthington, A. M. 1908 A Study of Splashes. Longmans, Green.
Yarin, A., Rubin, M. & Roisman, I. 1995 Penetration of a rigid projectile into an elastic-plastic target of finite thickness. Intl J. Impact Engng 16 (5–6), 801831.
Zhang, H., Tumarkin, E., Peerani, R., Nie, Z., Sullan, R. M. A., Walker, G. C. & Kumacheva, E. 2006 Microfluidic production of biopolymer microcapsules with controlled morphology. J. Am. Chem. Soc. 128 (37), 1220512210.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *

JFM classification

Type Description Title

Jalaal et al. supplementary movie 1
Panel a in figure 3.

 Video (47 KB)
47 KB

Jalaal et al. supplementary movie 2
Panel b in figure 3.

 Video (43 KB)
43 KB

Jalaal et al. supplementary movie 3
Panel c in figure 3.

 Video (49 KB)
49 KB

Jalaal et al. supplementary movie 4
Panel d in figure 3.

 Video (108 KB)
108 KB

Jalaal et al. supplementary movie 5
Panel e in figure 3.

 Video (190 KB)
190 KB


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed