^{1}, Marco A. Fontelos

^{2}, Christophe Josserand

^{3,4}and Stéphane Zaleski

^{3,4}

### Abstract

We study the impact of a fluid drop onto a planar solid surface at high speed so that at impact, kinetic energy dominates over surface energy and inertia dominates over viscous effects. As the drop spreads, it deforms into a thin film, whose thickness is limited by the growth of a viscous boundary layer near the solid wall. Owing to surface tension, the edge of the film retracts relative to the flow in the film and fluid collects into a toroidal rim bounding the film. Using mass and momentum conservation, we construct a model for the radius of the deposit as a function of time. At each stage, we perform detailed comparisons between theory and numerical simulations of the Navier–Stokes equation.

It is our pleasure to thank Stéphane Popinet for enlightening discussions on drop impacts. M. A. Fontelos acknowledges the Ministerio de Ciencia for its support through Grant No. MTM2008-03255 and C. Josserand acknowledges the financial support of the Agence Nationale de la Recherche through Grant No. ANR-09-JCJC-0022 (ANR “Deformation”).

I. INTRODUCTION

II. LIQUID SHEET EXPANSION

III. BOUNDARY LAYER

IV. RIM DYNAMICS

V. DISCUSSION

A. Minimal film thickness

B. Maximal spreading radius

C. Retraction dynamics

D. Conclusions

### Key Topics

- Viscosity
- 27.0
- Retraction
- 25.0
- Reynolds stress modeling
- 14.0
- Fluid drops
- 13.0
- Numerical modeling
- 11.0

## Figures

Snapshots of a drop impacting on a solid surface obtained numerically using the markers method for and . Times of the different snapshot correspond to , 0.22, 0.43, 1.3, 2.5, 5.7, 15.8, 21.2, and 60 from left to right and top to bottom, respectively.

Snapshots of a drop impacting on a solid surface obtained numerically using the markers method for and . Times of the different snapshot correspond to , 0.22, 0.43, 1.3, 2.5, 5.7, 15.8, 21.2, and 60 from left to right and top to bottom, respectively.

Time evolution of (a) the radius of the contact area between the drop and the substrate and of (b) the height of the interface on the symmetry axis. The radius and the height are rescaled by the drop radius and time is shown in units of .

Time evolution of (a) the radius of the contact area between the drop and the substrate and of (b) the height of the interface on the symmetry axis. The radius and the height are rescaled by the drop radius and time is shown in units of .

The pressure field corresponding to the same conditions in Fig. 1 for (a) , (b) 0.12, (c) 0.2, and (d) 0.29. The shade scale goes from light to dark for increasing values of the pressure.

The pressure field corresponding to the same conditions in Fig. 1 for (a) , (b) 0.12, (c) 0.2, and (d) 0.29. The shade scale goes from light to dark for increasing values of the pressure.

Pressure at the center of impact as function of for different Reynolds and Weber numbers ranging from 400 to 8000 and from 160 to 80 000, respectively . The behavior predicted by the small time impact theory is the dashed line. For , notice that the pressure drops dramatically.

Pressure at the center of impact as function of for different Reynolds and Weber numbers ranging from 400 to 8000 and from 160 to 80 000, respectively . The behavior predicted by the small time impact theory is the dashed line. For , notice that the pressure drops dramatically.

Top and bottom left: series of drop profiles for different times. Top left for and for varying from 0.8 to 2.7. Bottom left: and for varying from 0.9 to 2.3. Right figures show the same profiles rescaled by the maximum height for and by its square root for , according to Eq. (4). The axis coordinates in the figures correspond to the numerical mesh. The dashed line shows a fit of the converged rescaled profiles , with in dimensionless units.

Top and bottom left: series of drop profiles for different times. Top left for and for varying from 0.8 to 2.7. Bottom left: and for varying from 0.9 to 2.3. Right figures show the same profiles rescaled by the maximum height for and by its square root for , according to Eq. (4). The axis coordinates in the figures correspond to the numerical mesh. The dashed line shows a fit of the converged rescaled profiles , with in dimensionless units.

The similarity profile. The dashed line is .

The similarity profile. The dashed line is .

Numerical profiles of the derivative of vertical velocity on the symmetry axis for , (top figures) and , (bottom figure) at different dimensionless times ranging from 0.2 to 3 in the first case and from 2 to 8 in the second case. The top and bottom left figures show the numerical profiles, while the top and bottom right figures show the same profile rescaled according to formula (14). The thick line in the right figures indicate the approximated theoretical boundary layer .

Numerical profiles of the derivative of vertical velocity on the symmetry axis for , (top figures) and , (bottom figure) at different dimensionless times ranging from 0.2 to 3 in the first case and from 2 to 8 in the second case. The top and bottom left figures show the numerical profiles, while the top and bottom right figures show the same profile rescaled according to formula (14). The thick line in the right figures indicate the approximated theoretical boundary layer .

A sketch of a drop after impact at high speed, so that both Re and We are large. The film thickness is , the radius of the film is , the volume of the rim is , and the total drop radius is .

A sketch of a drop after impact at high speed, so that both Re and We are large. The film thickness is , the radius of the film is , the volume of the rim is , and the total drop radius is .

The total drop radius as function of time of an impacting drop with . The line is the numerical simulation and the dashed line shows the model output. The free parameters [Eq. (26)] have been fitted to obtain the best possible fit of the experiment.

The total drop radius as function of time of an impacting drop with . The line is the numerical simulation and the dashed line shows the model output. The free parameters [Eq. (26)] have been fitted to obtain the best possible fit of the experiment.

(a) The rim radius (dashed line) and the spreading drop radius (solid line) as function of dimensionless time . (b) The mass (normalized by the total mass ) in the rim (solid line) and the one in the film (dotted line) are drawn as functions of time. Both figures show that the rim grows only during the retraction phase of the drop.

(a) The rim radius (dashed line) and the spreading drop radius (solid line) as function of dimensionless time . (b) The mass (normalized by the total mass ) in the rim (solid line) and the one in the film (dotted line) are drawn as functions of time. Both figures show that the rim grows only during the retraction phase of the drop.

The radius as function of time , for a variety of impact parameters (Re,We) (a) (800,400), (b) (800,1600), (c) (800,4000), (d) (800,16000), (e) (400,800), and (f) (8000,800). The continuous line shows the numerical simulation and the dashed lines the results of the simplified model.

The radius as function of time , for a variety of impact parameters (Re,We) (a) (800,400), (b) (800,1600), (c) (800,4000), (d) (800,16000), (e) (400,800), and (f) (8000,800). The continuous line shows the numerical simulation and the dashed lines the results of the simplified model.

The minimum radius of the film as a function of the Reynolds number obtained using the model (circles) for Weber varying between 100 and 4000 and compared to numerical simulations of the full equations (square). The predicted law is the dashed line, while the law is the dotted line for comparison.

The minimum radius of the film as a function of the Reynolds number obtained using the model (circles) for Weber varying between 100 and 4000 and compared to numerical simulations of the full equations (square). The predicted law is the dashed line, while the law is the dotted line for comparison.

A test of the two scaling relations (31) (left) and (32) (right), using as calculated from the model. The rescaled maximum radius ) is plotted (a) as a function of ; the expected behavior for small , , is indicated in dotted line; (b) as function of , again showing the expected behavior for small , (dotted line).

A test of the two scaling relations (31) (left) and (32) (right), using as calculated from the model. The rescaled maximum radius ) is plotted (a) as a function of ; the expected behavior for small , , is indicated in dotted line; (b) as function of , again showing the expected behavior for small , (dotted line).

Retraction velocity measured in the model as the maximum velocity reached during the retraction regime (a) is shown as a function of Weber number for different Reynolds numbers. (b) According to the theory balancing inertia and capillarity [Eq. (33)], all the curves collapse when the retraction rate is considered. The line shows the fit to be compared to the Taylor–Culick retraction rate (33).

Retraction velocity measured in the model as the maximum velocity reached during the retraction regime (a) is shown as a function of Weber number for different Reynolds numbers. (b) According to the theory balancing inertia and capillarity [Eq. (33)], all the curves collapse when the retraction rate is considered. The line shows the fit to be compared to the Taylor–Culick retraction rate (33).

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