^{1}and Paul D. Metcalfe

^{2}

### Abstract

We study the impact of a line mass onto a liquid-gas interface. At early times we find a similarity solution for the interfacial deformation and show how the resulting surface tension force slows the fall of the mass. We compute the motion beyond early times using a boundary integral method, and find conditions on the weight and impact speed of the mass that determine whether it sinks or is trapped by the interface. We find that for given impact speed there is a critical weight above which the mass sinks, and we investigate the asymptotic behavior of this critical weight in the limits of small and large impact speeds. Below this critical weight, the mass is trapped by the interface and subsequently floats. We also compare our theoretical results with some simple tabletop experiments. Finally, we discuss the implications of our work for the vertical jumps of water-walking arthropods.

D.V. is supported by the EPSRC and Trinity College, Cambridge. We are grateful to Robert Whittaker and Stephen Morris for discussions and to Herbert Huppert for comments on earlier drafts of this paper.

I. INTRODUCTION

II. THEORETICAL FORMULATION

III. EARLY TIME SIMILARITY SOLUTION

A. Far-field behavior

B. Modification of ballistic motion by surface tension

IV. LATE TIMES: BOUNDARY INTEGRAL SIMULATIONS

A. The numerical method

B. Numerical results

V. IMPACT INDUCES SINKING

A. The limit

B. The limit

C. A composite expansion

VI. EXPERIMENTAL RESULTS

VII. BIOLOGICAL DISCUSSION

VIII. CONCLUSIONS

### Key Topics

- Surface tension
- 25.0
- Interface dynamics
- 12.0
- Boundary value problems
- 9.0
- Kinematics
- 8.0
- Boundary integral methods
- 7.0

## Figures

Setup for the impact of a line of weight per unit length onto a liquid surface.

Setup for the impact of a line of weight per unit length onto a liquid surface.

The short-time similarity solution for the interfacial profile . Here, so that the resulting profile can be rescaled to give that for any .

The short-time similarity solution for the interfacial profile . Here, so that the resulting profile can be rescaled to give that for any .

Main figure: The algebraic decay of observed in numerical solutions of Eqs. (15) and (16) is the same as that expected from Eq. (22). Here (and so ). Inset: The numerically computed wavelength of capillary waves (×) decreases with in accordance with Eq. (23) (solid curve).

Main figure: The algebraic decay of observed in numerical solutions of Eqs. (15) and (16) is the same as that expected from Eq. (22). Here (and so ). Inset: The numerically computed wavelength of capillary waves (×) decreases with in accordance with Eq. (23) (solid curve).

Schematic illustration of the contour used in our boundary integral simulations.

Schematic illustration of the contour used in our boundary integral simulations.

Comparison of the interface shape obtained from boundary integral simulations (points) with that predicted by the short-time similarity solution discussed in Sec. III (curve). The interface is pictured in similarity coordinates , at time . Here and .

Comparison of the interface shape obtained from boundary integral simulations (points) with that predicted by the short-time similarity solution discussed in Sec. III (curve). The interface is pictured in similarity coordinates , at time . Here and .

The correction to the ballistic motion caused by surface tension for a mass with and . The results of the boundary integral method (solid curve) agree with the leading-order asymptotic prediction Eq. (26) for (dashed line).

The correction to the ballistic motion caused by surface tension for a mass with and . The results of the boundary integral method (solid curve) agree with the leading-order asymptotic prediction Eq. (26) for (dashed line).

The two distinct sinking mechanisms for a cylinder with *finite* radius: (a) for a hydrophilic cylinder surface , sinking occurs when the two contact lines meet at the top of the cylinder. (b) For a hydrophobic cylinder surface , sinking occurs when the menisci merge above the cylinder.

The two distinct sinking mechanisms for a cylinder with *finite* radius: (a) for a hydrophilic cylinder surface , sinking occurs when the two contact lines meet at the top of the cylinder. (b) For a hydrophobic cylinder surface , sinking occurs when the menisci merge above the cylinder.

Regime diagram showing the regions of parameter space for which a line mass is observed to float or sink. The dashed line shows the composite expansion Eq. (41), which gives to within 15% for intermediate values of .

Regime diagram showing the regions of parameter space for which a line mass is observed to float or sink. The dashed line shows the composite expansion Eq. (41), which gives to within 15% for intermediate values of .

Replotting of the boundary between floating and sinking, , for . The numerically determined values of (points) agree well with the general form suggested in Eq. (36), which arises from symmetry considerations. The solid line, , is plotted as a guide for the eye.

Replotting of the boundary between floating and sinking, , for . The numerically determined values of (points) agree well with the general form suggested in Eq. (36), which arises from symmetry considerations. The solid line, , is plotted as a guide for the eye.

Replotting of the boundary between floating and sinking, , for . The numerically determined values of (points) show the scaling predicted in Eq. (40).

Replotting of the boundary between floating and sinking, , for . The numerically determined values of (points) show the scaling predicted in Eq. (40).

(Color online) The experimentally determined regime diagram showing values of and for which impacting objects were observed to float (blue 엯) or to sink (red ×) upon impact. Here and . The solid line shows the theoretically computed curve , which separates floating from sinking for a line mass impacting an ideal liquid. A typical error bar is included for illustration.

(Color online) The experimentally determined regime diagram showing values of and for which impacting objects were observed to float (blue 엯) or to sink (red ×) upon impact. Here and . The solid line shows the theoretically computed curve , which separates floating from sinking for a line mass impacting an ideal liquid. A typical error bar is included for illustration.

## Tables

Parameter values investigated in the eight sets of experiments presented here. The nondimensional weight per unit length, , and the dimensionless cylinder radius, , are dependent on the value of the interfacial tension . The dependence of on isopropanol concentration is taken from the literature.^{23}

Parameter values investigated in the eight sets of experiments presented here. The nondimensional weight per unit length, , and the dimensionless cylinder radius, , are dependent on the value of the interfacial tension . The dependence of on isopropanol concentration is taken from the literature.^{23}

Typical values from the literature for the jumping of two species of water-walking arthropod.

Typical values from the literature for the jumping of two species of water-walking arthropod.

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