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 .
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.
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 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 .
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).
(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.
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.
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