Skip to main content Accessibility help

The transient force profile of low-speed droplet impact: measurements and model

  • Benjamin R. Mitchell (a1), Joseph C. Klewicki (a2), Yannis P. Korkolis (a3) and Brad L. Kinsey (a1)


The transient force exerted by a low-speed liquid droplet impinging onto a flat rigid surface is investigated experimentally. The measurements employ a high-sensitivity piezo-electric sensor, along with a high-speed camera, and cover four decades in droplet Reynolds number and greater than two decades in Weber number. Across these ranges, the peak of individual force profiles span from 3 mN to over 1300 mN. Once normalised, the force–time profiles support the existence of an inertially dominated self-similar regime. Within this regime, previous numerical and theoretical studies predict a $\sqrt{t}$ dependence of impact normal force during the initial pre-peak rise. While our measurements confirm this finding, they also indicate that, after the peak force the profiles exhibit an exponential decay. This long-time decay law suggests treatment of the momentum transport from the droplet using a lumped model. An observed linear dependence between the force and force decay rate supports this approach. The reason for the efficacy of treating this system via a lumped model apparently connects to the physics right at the surface that limit the rate of momentum transport from the droplet to the surface. This is explored by estimating the momentum transfer by solely using the deforming droplet shape, but under the condition of negligible momentum gradients within the droplet. The short- and long-time solutions are combined and the resulting model equation is shown to accurately cover the entire force–time profile.


Corresponding author

Email address for correspondence:


Hide All
Ahmad, M., Schatz, M. & Casey, M. V. 2013 Experimental investigation of droplet size influence on low pressure steam turbine blade erosion. Wear 303 (1), 8386.
Amirzadeh, B., Louhghalam, A., Raessi, M. & Tootkaboni, M. 2017 A computational framework for the analysis of rain-induced erosion in wind turbine blades. J. Wind Engng Ind. Aerodyn. 163, 3343.
Association, Glycerine Producers’ 1963 Physical Properties of Glycerine and its Solutions. Glycerine Producers’ Association.
Baker, E. A. & Hunt, G. M. 1986 Erosion of waxes from leaf surfaces by simulated rain. New Phytol. 102 (1), 161173.
Beard, K. V. 1976 Terminal velocity and shape of cloud and precipitation drops aloft. J. Atmos. Sci. 33 (5), 851864.
Clift, R., Grace, J. R. & Weber, M. E. 1978 Bubbles, Drops and Particles. Academic Press.
Coleman, H. W. & Steele, W. G. 2009 Experimentation, Validation, and Uncertainty Analysis for Engineers. Wiley.
Dean, R., Nelson, D. F., Brown, M. L., Couch, R. W. & Blanchard, M. W.2008 Method and apparatus for forming high-speed liquid. US Patent 7,380,918.
Dickerson, A. K., Shankles, P. G., Madhavan, N. M. & Hu, D. L. 2012 Mosquitoes survive raindrop collisions by virtue of their low mass. Proc. Natl Acad. Sci. USA 109 (25), 98229827.
Dorf, R. C. 2004 The Engineering Handbook. CRC press.
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.
Fyall, A. A. 1966 Practical aspects of rain erosion of aircraft and missiles. Phil. Trans. R. Soc. Lond. A 260 (1110), 161167.
Gordillo, L., Sun, T.-P. & Cheng, X. 2018 Dynamics of drop impact on solid surfaces: evolution of impact force and self-similar spreading. J. Fluid Mech. 840, 190214.
Grinspan, A. S. & Gnanamoorthy, R. 2010 Impact force of low velocity liquid droplets measured using piezoelectric pvdf film. Colloids Surf. A 356 (1), 162168.
Haferl, S. & Poulikakos, D. 2003 Experimental investigation of the transient impact fluid dynamics and solidification of a molten microdroplet pile-up. Intl J. Heat Mass Transfer 46 (3), 535550.
Haley, P. J. & Miksis, M. J. 1991 The effect of the contact line on droplet spreading. J. Fluid Mech. 223, 5781.
Haller, K. K., Poulikakos, D., Ventikos, Y. & Monkewitz, P. 2003 Shock wave formation in droplet impact on a rigid surface: lateral liquid motion and multiple wave structure in the contact line region. J. Fluid Mech. 490, 114.
Haller, K. K., Ventikos, Y., Poulikakos, D. & Monkewitz, P. 2002 Computational study of high-speed liquid droplet impact. J. Appl. Phys. 92 (5), 28212828.
Huang, L., Folkes, J., Kinnell, P. & Shipway, P. H. 2012 Mechanisms of damage initiation in a titanium alloy subjected to water droplet impact during ultra-high pressure plain waterjet erosion. J. Mater. Process. Technol. 212 (9), 19061915.
Josserand, C. & Thoroddsen, S. T. 2016 Drop impact on a solid surface. Annu. Rev. Fluid Mech. 48, 365391.
Josserand, C. & Zaleski, S. 2003 Droplet splashing on a thin liquid film. Phys. Fluids 15 (6), 16501657.
Lagubeau, G., Fontelos, M. A., Josserand, C., Maurel, A., Pagneux, V. & Petitjeans, P. 2012 Spreading dynamics of drop impacts. J. Fluid Mech. 713, 5060.
Li, J., Zhang, B., Guo, P. & Lv, Q. 2014 Impact force of a low speed water droplet colliding on a solid surface. J. Appl. Phys. 116 (21), 214903.
Marshall, H. P., Conway, H. & Rasmussen, L. A. 1999 Snow densification during rain. Cold Reg. Sci. Technol. 30 (1), 3541.
Mitchell, B. R., Bate, T. E., Klewicki, J. C., Korkolis, Y. P. & Kinsey, B. L. 2017 Experimental investigation of droplet impact on metal surfaces in reduced ambient pressure. Procedia Manufacturing 10, 730736.
Mitchell, B. R., Nassiri, A., Locke, M. R., Klewicki, J. C., Korkolis, Y. P. & Kinsey, B. L. 2016 Experimental and numerical framework for study of low velocity water droplet impact dynamics. In ASME 2016 11th International Manufacturing Science and Engineering Conference, American Society of Mechanical Engineers.
Mongruel, A., Daru, V., Feuillebois, F. & Tabakova, S. 2009 Early post-impact time dynamics of viscous drops onto a solid dry surface. Phys. Fluids 21 (3), 032101.
Nearing, M. A., Bradford, J. M. & Holtz, R. D. 1986 Measurement of force versus time relations for waterdrop impact. Soil Sci. Soc. Am. 50 (6), 15321536.
Ortega-Jimenez, V. M., Badger, M., Wang, H. & Dudley, R. 2016 Into rude air: hummingbird flight performance in variable aerial environments. Phil. Trans. R. Soc. Lond. B 371, 20150387.
Philippi, J., Lagrée, P. & Antkowiak, A. 2016 Drop impact on a solid surface: short-time self-similarity. J. Fluid Mech. 795, 96135.
Pizzola, P. A., Roth, S. & De Forest, P. R. 1986 Blood droplet dynamics i. J. Forensic Sci. 31 (1), 3649.
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.
Rioboo, R., Marengo, M. & Tropea, C. 2002 Time evolution of liquid drop impact onto solid, dry surfaces. Exp. Fluids 33 (1), 112124.
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.
Roisman, I. V., Berberović, E. & Tropea, C. 2009 Inertia dominated drop collisions. i. on the universal flow in the lamella. Phys. Fluids 21 (5), 052103.
Römkens, M. J. M., Helming, K. & Prasad, S. N. 2002 Soil erosion under different rainfall intensities, surface roughness, and soil water regimes. Catena 46 (2), 103123.
Saylor, J. R. & Jones, B. K. 2005 The existence of vortices in the wakes of simulated raindrops. Phys. Fluids 17 (3), 031706.
Šikalo, Š., Marengo, M., Tropea, C. & Ganić, E. N. 2002 Analysis of impact of droplets on horizontal surfaces. Expl Therm. Fluid Sci. 25 (7), 503510.
Šikalo, Š., Tropea, C. & Ganić, E. N. 2005 Dynamic wetting angle of a spreading droplet. Expl. Therm. Fluid Sci. 29 (7), 795802.
Szakáll, M., Mitra, S. K., Diehl, K. & Borrmann, S. 2010 Shapes and oscillations of falling raindrops. Atmospheric Res. 97 (4), 416425.
Taylor, T. D. & Acrivos, A. 1964 On the deformation and drag of a falling viscous drop at low reynolds number. J. Fluid Mech. 18 (3), 466476.
Thoraval, M.-J., Takehara, K., Etoh, T. G., Popinet, S., Ray, P., Josserand, C., Zaleski, S. & Thoroddsen, S. T. 2012 von Kármán vortex street within an impacting drop. Phys. Rev. Lett. 108 (26), 264506.
Thoroddsen, S. T. & Sakakibara, J. 1998 Evolution of the fingering pattern of an impacting drop. Phys. Fluids 10 (6), 13591374.
Trefethen, L. 1969 Surface tension in fluid mechanics. Lubricating Oil 25, 3538.
Tsai, P., Pacheco, S., Pirat, C., Lefferts, L. & Lohse, D. 2009 Drop impact upon micro- and nanostructured superhydrophobic surfaces. Langmuir 25 (20), 1229312298.
Weiss, D. A. & Yarin, A. L. 1999 Single drop impact onto liquid films: neck distortion, jetting, tiny bubble entrainment, and crown formation. J. Fluid Mech. 385, 229254.
Wildeman, S., Visser, C. W., Sun, C. & Lohse, D. 2016 On the spreading of impacting drops. J. Fluid Mech. 805, 636655.
Xu, L., Zhang, W. W. & Nagel, S. R. 2005 Drop splashing on a dry smooth surface. Phys. Rev. Lett. 94 (18), 184505.
Yarin, A. L. & Weiss, D. A. 1995 Impact of drops on solid surfaces: self-similar capillary waves, and splashing as a new type of kinematic discontinuity. J. Fluid Mech. 283, 141173.
Yu, Y. & Hopkins, C. 2018 Experimental determination of forces applied by liquid water drops at high drop velocities impacting a glass plate with and without a shallow water layer using wavelet deconvolution. Exp. Fluids 59, 123.
Zhang, B., Li, J., Guo, P. & Lv, Q. 2017 Experimental studies on the effect of Reynolds and weber numbers on the impact forces of low-speed droplets colliding with a solid surface. Exp. Fluids 58 (9), 125.
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification

Related content

Powered by UNSILO
Type Description Title
Supplementary materials

Mitchell et al. supplementary material
Mitchell et al. supplementary material 1

 Unknown (3.6 MB)
3.6 MB

The transient force profile of low-speed droplet impact: measurements and model

  • Benjamin R. Mitchell (a1), Joseph C. Klewicki (a2), Yannis P. Korkolis (a3) and Brad L. Kinsey (a1)


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.