Skip to main content Accessibility help

A theory on the spreading of impacting droplets

  • José Manuel Gordillo (a1), Guillaume Riboux (a1) and Enrique S. Quintero (a1)


Here we provide a self-consistent analytical solution describing the unsteady flow in the slender thin film which is expelled radially outwards when a drop hits a dry solid wall. Thanks to the fact that the fluxes of mass and momentum entering into the toroidal rim bordering the expanding liquid sheet are calculated analytically, we show here that our theoretical results closely follow the measured time-varying position of the rim with independence of the wetting properties of the substrate. The particularization of the equations describing the rim dynamics at the instant the drop reaches its maximal extension which, in analogy with the case of Savart sheets, is characterized by a value of the local Weber number equal to one, provides an algebraic equation for the maximum spreading radius also in excellent agreement with experiments. The self-consistent theory presented here, which does not make use of energetic arguments to predict the maximum spreading diameter of impacting drops, provides us with the time evolution of the thickness and of the velocity of the rim bordering the expanding sheet. This information is crucial in the calculation of the diameters and of the velocities of the droplets ejected radially outwards for drop impact velocities above the splashing threshold.


Corresponding author

Email address for correspondence:


Hide All
Antonini, C., Amirfazli, A. & Marengo, M. 2012 Drop impact and wettability: from hydrophilic to superhydrophobic surfaces. Phys. Fluids 24 (10), 102104.
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Clanet, C., Béguin, C., Richard, D. & Quéré, D. 2004 Maximal deformation of an impacting drop. J. Fluid Mech. 517, 199208.
Culick, F. E. C. 1960 Comments on a ruptured soap film. J. Appl. Phys. 31, 11281129.
Eggers, J., Fontelos, M. A., Josserand, C. & Zaleski, S. 2010 Drop dynamics after impact on a solid wall: theory and simulations. Phys. Fluids 22, 062101.
Gordillo, J. M., Lhuissier, H. & Villermaux, E. 2014 On the cusps bordering liquid sheets. J. Fluid Mech. 754, R1.
Josserand, C. & Thoroddsen, S. T. 2016 Drop impact on a solid surface. Annu. Rev. Fluid Mech. 48 (1), 365391.
Laan, N., de Bruin, K. G., Bartolo, D., Josserand, C. & Bonn, D. 2014 Maximum diameter of impacting liquid droplets. Phys. Rev. Appl. 2, 044018.
Laan, N., de Bruin, K. G., Slenter, D., Wilhelm, J., Jermy, M. & Bonn, D. 2015 Bloodstain pattern analysis: implementation of a fluid dynamic model for position determination of victims. Sci. Rep. 5, 11461.
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.
Lejeune, S., Gilet, T. & Bourouiba, L. 2018 Edge effect: liquid sheet and droplets formed by drop impact close to an edge. Phys. Rev. Fluids 3, 083601.
Lv, C., Hao, P., Zhang, X. & He, F. 2016 Drop impact upon superhydrophobic surfaces with regular and hierarchical roughness. Appl. Phys. Lett. 108, 141602.
Quintero, E. S., Riboux, G. & Gordillo, J. M. 2019 Splashing of droplets impacting superhydrophobic substrates. J. Fluid Mech. (submitted).
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, 024507.
Riboux, G. & Gordillo, J. M. 2015 The diameters and velocities of the droplets ejected after splashing. J. Fluid Mech. 772, 630648.
Riboux, G. & Gordillo, J. M. 2016 Maximum drop radius and critical Weber number for splashing in the dynamical Leidenfrost regime. J. Fluid Mech. 803, 516527.
Riboux, G. & Gordillo, J. M. 2017 Boundary–layer effects in droplet splashing. Phys. Rev. E 96, 013105.
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, 052104.
Roisman, I. V., Rioboo, R. & Tropea, C. 2002 Normal impact of a liquid drop on a dry surface: model for spreading and receding. Proc. R. Soc. Lond. A 458 (2022), 14111430.
Stow, C. D., Hadfield, M. G. & Ziman, J. M. 1981 An experimental investigation of fluid flow resulting from the impact of a water drop with an unyielding dry surface. Proc. R. Soc. Lond. A 373 (1755), 419441.
Taylor, G. I. 1959 The dynamics of thin sheets of fluid. III. Desintegration of fluid sheets. Proc. R. Soc. Lond. A 253, 1274.
Tran, T., Staat, H. J. J., Prosperetti, A., Sun, C. & Lohse, D. 2012 Drop impact on superheated surfaces. Phys. Rev. Lett. 108, 036101.
Tsai, P., Hendrix, M. H. W., Dijkstra, R. R. M., Shui, L. & Lohse, D. 2011 Microscopic structure influencing macroscopic splash at high Weber number. Soft Matt. 7, 1132511333.
Villermaux, E. & Bossa, B. 2011 Drop fragmentation on impact. J. Fluid Mech. 668, 412435.
Visser, C. W., Frommhold, P. E., Wildeman, S., Mettin, R., Lohse, D. & Sun, C. 2015 Dynamics of high-speed micro-drop impact: numerical simulations and experiments at frame-to-frame times below 100 ns. Soft Matt. 11, 17081722.
Wang, Y. & Bourouiba, L. 2017 Drop impact on small surfaces: thickness and velocity profiles of the expanding sheet in the air. J. Fluid Mech. 814, 510534.
Wang, Y., Dandekar, R., Bustos, N., Poulain, S. & Bourouiba, L. 2018 Universal rim thickness in unsteady sheet fragmentation. Phys. Rev. Lett. 120, 204503.
Wildeman, S., Visser, C. W., Sun, C. & Lohse, D. 2016 On the spreading of impacting drops. J. Fluid Mech. 805, 636655.
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification


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