^{1}and John W. M. Bush

^{1,a)}

### Abstract

We present the results of an experimental investigation of fluid drops impacting an inclined rigid surface covered with a thin layer of high viscosity fluid. We deduce the conditions under which droplet bouncing, splitting, and merger arise. Particular attention is given to rationalizing the observed contact time and coefficients of restitution, the latter of which require a detailed consideration of the drop energetics.

We thank the National Science Foundation (NSF) for financial support through Grant No. CBET-0966452. We gratefully acknowledge Jacy Bird, Lydia Bourouiba, and Ruben Rosales for fruitful discussions.

I. INTRODUCTION

II. EXPERIMENTAL SETUP

III. EXPERIMENTAL RESULTS

A. Phenomenology

B. Bouncing as a black box

C. Energy transfers and the normal coefficient of restitution

IV. DISCUSSION

V. CONCLUSION

### Key Topics

- Fluid drops
- 91.0
- Viscosity
- 21.0
- Energy transfer
- 10.0
- Capillary waves
- 9.0
- Drop coalescence
- 7.0

## Figures

A drop impacts an inclined coated surface and splits into four droplets. (a) Superposition of successive frames, taken at 300 fps, reveals the trajectory of the droplets. (b) The initial drop, before impact (time *t* = −7 ms). (c) The satellite droplets formed after impact (time *t* = 16 ms). Dimensionless parameters are , , and α = 14°.

A drop impacts an inclined coated surface and splits into four droplets. (a) Superposition of successive frames, taken at 300 fps, reveals the trajectory of the droplets. (b) The initial drop, before impact (time *t* = −7 ms). (c) The satellite droplets formed after impact (time *t* = 16 ms). Dimensionless parameters are , , and α = 14°.

Impact scenarios at different Weber number , as revealed through a superposition of successive frames separated by δ*t* milliseconds. (a) , δ*t* = 10 ms. (b) , δ*t* = 6 ms. (c) , δ*t* = 4 ms. (d) , δ*t* = 4 ms. Other dimensionless parameters are , and α = 14°. A detailed analysis of these impacts is found in Fig. 4.

Impact scenarios at different Weber number , as revealed through a superposition of successive frames separated by δ*t* milliseconds. (a) , δ*t* = 10 ms. (b) , δ*t* = 6 ms. (c) , δ*t* = 4 ms. (d) , δ*t* = 4 ms. Other dimensionless parameters are , and α = 14°. A detailed analysis of these impacts is found in Fig. 4.

Shade code used in our presentation of experimental results. Different symbols correspond to different Ohnesorge numbers (between 0.007 and 0.35), while the shade indicates the value of the inclination angle α. Experiments with slightly different values of are grouped together with the same symbol; the dashed rectangle indicates the region described by the symbol. In subsequent figures, is denoted by bold symbols. The satellite droplets have a smaller , and so are represented by plain symbols.

Shade code used in our presentation of experimental results. Different symbols correspond to different Ohnesorge numbers (between 0.007 and 0.35), while the shade indicates the value of the inclination angle α. Experiments with slightly different values of are grouped together with the same symbol; the dashed rectangle indicates the region described by the symbol. In subsequent figures, is denoted by bold symbols. The satellite droplets have a smaller , and so are represented by plain symbols.

Bouncing, splitting, and merging: (a) , (b) , (c) , (d) and (e) , (f) . The other parameters are fixed for each droplet impact (α = 14°, and ). The frames are taken at identical times after impact, normalized by the capillary time τ_{σ} = 12.4 ms and indicated in the left column.

Bouncing, splitting, and merging: (a) , (b) , (c) , (d) and (e) , (f) . The other parameters are fixed for each droplet impact (α = 14°, and ). The frames are taken at identical times after impact, normalized by the capillary time τ_{σ} = 12.4 ms and indicated in the left column.

Volume of the satellite droplets Ω_{ s } normalized by the volume of the impacting drop Ω, as a function of the incident normal Weber number . Other parameters are , and α = 14°. (Blue circle) Main droplet. (Red circle) Worthington satellite droplets, ejected above the main drop at *t* ≃ 0.7τ_{σ}. (Green circle) Satellite droplets from the pinch off, ejected below the main drop at *t* ∈ [1.16, 1.48]τ_{σ}. (Black square) Total volume ejected after impact.

Volume of the satellite droplets Ω_{ s } normalized by the volume of the impacting drop Ω, as a function of the incident normal Weber number . Other parameters are , and α = 14°. (Blue circle) Main droplet. (Red circle) Worthington satellite droplets, ejected above the main drop at *t* ≃ 0.7τ_{σ}. (Green circle) Satellite droplets from the pinch off, ejected below the main drop at *t* ∈ [1.16, 1.48]τ_{σ}. (Black square) Total volume ejected after impact.

Normalized contact time *t* _{ c }/τ_{σ} as a function of the incident normal Weber number in the case of complete bouncing. Symbols are defined in Figure 3.

Normalized contact time *t* _{ c }/τ_{σ} as a function of the incident normal Weber number in the case of complete bouncing. Symbols are defined in Figure 3.

Bouncing is seen as a black box that modifies the trajectory of the center-of-mass of the droplet, changing the velocity from to .

Bouncing is seen as a black box that modifies the trajectory of the center-of-mass of the droplet, changing the velocity from to .

Normalized time delay Δ*t*/*t* _{ c } as a function of the incident normal Weber number . Symbols are defined in Figure 3.

Normalized time delay Δ*t*/*t* _{ c } as a function of the incident normal Weber number . Symbols are defined in Figure 3.

Slip length Δ*L*, normalized by the prediction Δ*L* _{0} [Eq. (2)], as a function of the incident normal Weber number *We* _{1n }. Symbols are defined in Figure 3.

Slip length Δ*L*, normalized by the prediction Δ*L* _{0} [Eq. (2)], as a function of the incident normal Weber number *We* _{1n }. Symbols are defined in Figure 3.

Normal coefficient of restitution: Ratio of the normal Weber number after and before impact, as a function of . The dashed curve corresponds to Eq. (3) with . Symbols are defined in Figure 3.

Normal coefficient of restitution: Ratio of the normal Weber number after and before impact, as a function of . The dashed curve corresponds to Eq. (3) with . Symbols are defined in Figure 3.

Prefactor as defined in the scaling law (3). Symbols are defined in Figure 3.

Prefactor as defined in the scaling law (3). Symbols are defined in Figure 3.

Tangential coefficient of restitution: ratio of the tangential Weber number after and before impact, as a function of . Symbols are defined in Figure 3.

Tangential coefficient of restitution: ratio of the tangential Weber number after and before impact, as a function of . Symbols are defined in Figure 3.

Corrected tangential coefficient of restitution [defined in Eq. (4)], as a function of . The dashed line corresponds to Eq. (5). Symbols are defined in Figure 3.

Corrected tangential coefficient of restitution [defined in Eq. (4)], as a function of . The dashed line corresponds to Eq. (5). Symbols are defined in Figure 3.

Time evolution of the energy, for : (a) and (b) and (c) and (d) . In (a) and (c), the snapshots are taken every 3 ms, the third frame corresponding to the impact time (*t* = 0). In (b) and (d), the dashed and solid lines represent the surface energy of the droplet and the mechanical energy (kinetic + gravity) of its center-of-mass.

Time evolution of the energy, for : (a) and (b) and (c) and (d) . In (a) and (c), the snapshots are taken every 3 ms, the third frame corresponding to the impact time (*t* = 0). In (b) and (d), the dashed and solid lines represent the surface energy of the droplet and the mechanical energy (kinetic + gravity) of its center-of-mass.

Stored energy *E* _{ S }, normalized by 4π*R* ^{2}σ, as a function of the input Weber number , for . The solid line corresponds to the scaling law . Error bars are smaller than the symbol size.

Stored energy *E* _{ S }, normalized by 4π*R* ^{2}σ, as a function of the input Weber number , for . The solid line corresponds to the scaling law . Error bars are smaller than the symbol size.

Output Weber number as a function of the stored energy *E* _{ S } normalized by 4π*R* ^{2}σ. The solid line corresponds to the scaling law . Error bars are smaller than the symbol size.

Output Weber number as a function of the stored energy *E* _{ S } normalized by 4π*R* ^{2}σ. The solid line corresponds to the scaling law . Error bars are smaller than the symbol size.

Snapshots of the impacts corresponding to the experiments depicted in Fig. 14 [ , (a) and (b) ]. Frames are separated by 1.33 ms. Time is increasing from left to right on the first row, then from right to left on the second row, in order to reveal the strong time asymmetry at .

Snapshots of the impacts corresponding to the experiments depicted in Fig. 14 [ , (a) and (b) ]. Frames are separated by 1.33 ms. Time is increasing from left to right on the first row, then from right to left on the second row, in order to reveal the strong time asymmetry at .

Detail of the compression phase for the experiment depicted in Figs. 14(c) and 14(d) and Fig. 17(b) [ and ]. Frames are separated by 0.33 ms.

Detail of the compression phase for the experiment depicted in Figs. 14(c) and 14(d) and Fig. 17(b) [ and ]. Frames are separated by 0.33 ms.

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