A poorly elastic shock of a Leidenfrost drop has been modeled by an imaginary spring [4, 6], which is a linear spring model with two mass points that represent the mass of the drop at both ends of the spring. Here, we extend the imaginary spring model by adding a damping term:

$$\begin{array}{}{\displaystyle \left\{\begin{array}{}\frac{1}{2}m\frac{{\mathrm{d}}^{2}{x}_{1}}{\mathrm{d}{t}^{2}}=-\frac{1}{2}mg-k\u03f5-c\frac{\mathrm{d}\u03f5}{\mathrm{d}t}\\ \frac{1}{2}m\frac{{\mathrm{d}}^{2}{x}_{2}}{\mathrm{d}{t}^{2}}=-\frac{1}{2}mg+k\u03f5+c\frac{\mathrm{d}\u03f5}{\mathrm{d}t}+F,\end{array}\right.}\end{array}$$(14a,14b)

where *x*_{1} and *x*_{2} are the heights of the bottom and top of the spring above the plate respectively, and

$$\begin{array}{}{\displaystyle \u03f5={x}_{1}-{x}_{2}-{D}_{0}}\end{array}$$(15)

is the strain of the spring, *m* is the mass of the drop, *D*_{0} is the initial vertical length of the drop, *k* is the stiffness of the spring, *c* is the damping coefficient of the spring (*c* ≥ 0), and *F* is the external force loaded at the bottom of the spring. Figure 2 shows a schematic diagram of the imaginary damped spring model. Note that by combining Eqs. (14a) and (14b), the momentum equation for the centroid of the spring,
$\begin{array}{}{\displaystyle {x}_{c}=\frac{1}{2}({x}_{1}+{x}_{2}),}\end{array}$ can be represented as

$$\begin{array}{}{\displaystyle m\frac{{\mathrm{d}}^{2}{x}_{c}}{\mathrm{d}{t}^{2}}=-mg+F,}\end{array}$$(16)

which plots the free-fall and bounce-back of the spring.

Let us define the regime in which the drop is in contact with the vapor film over the plate as regime I. In regime I, the height of the bottom of the spring is considered to be fixed (*x*_{2} = 0); therefore,

$$\begin{array}{}{\displaystyle \frac{{\mathrm{d}}^{2}{x}_{1}}{\mathrm{d}{t}^{2}}=\frac{{\mathrm{d}}^{2}{\u03f5}_{\mathrm{I}}}{\mathrm{d}{t}^{2}},}\end{array}$$(17)

and

$$\begin{array}{}{\displaystyle \frac{{\mathrm{d}}^{2}{x}_{2}}{\mathrm{d}{t}^{2}}=0,}\end{array}$$(18)

where *ϵ*_{I} is the strain in regime I. Equations (14a) and (14b) then become:

$$\begin{array}{}{\displaystyle \left\{\begin{array}{}\frac{{\mathrm{d}}^{2}{\u03f5}_{\mathrm{I}}}{\mathrm{d}{t}^{2}}=-g-\frac{{k}_{\mathrm{I}}}{m}{\u03f5}_{\mathrm{I}}-\frac{{c}_{\mathrm{I}}}{m}\frac{\mathrm{d}{\u03f5}_{\mathrm{I}}}{\mathrm{d}t}\\ F=\frac{1}{2}\left(mg-{k}_{\mathrm{I}}{\u03f5}_{\mathrm{I}}-{c}_{\mathrm{I}}\frac{\mathrm{d}{\u03f5}_{\mathrm{I}}}{\mathrm{d}t}\right),\end{array}\right.}\end{array}$$(19a,19b)

where

$$\begin{array}{}{\displaystyle {k}_{\mathrm{I}}=2k,\phantom{\rule{thickmathspace}{0ex}}{c}_{\mathrm{I}}=2c.}\end{array}$$(20)

By solving Eq. (19a), we obtain

$$\begin{array}{}{\displaystyle {\u03f5}_{\mathrm{I}}=-{A}_{\mathrm{I}}{\mathrm{e}}^{-{\zeta}_{\mathrm{I}}{\omega}_{\mathrm{I}}{t}_{\mathrm{I}}}\mathrm{s}\mathrm{i}\mathrm{n}\left({\omega}_{\mathrm{d},\mathrm{I}}{t}_{\mathrm{I}}+{\psi}_{\mathrm{I}}\right)-\frac{mg}{{k}_{\mathrm{I}}},}\end{array}$$(21)

where *t*_{I} is the time after the impact, *A*_{I} is the initial amplitude of the oscillation,
$\begin{array}{}{\displaystyle {\zeta}_{\mathrm{I}}=\frac{{c}_{\mathrm{I}}}{2\sqrt{m{k}_{\mathrm{I}}}}}\end{array}$ is the damping ratio,
$\begin{array}{}{\displaystyle {\omega}_{\mathrm{I}}=\sqrt{\frac{{k}_{\mathrm{I}}}{m}}}\end{array}$ is the undamped angular frequency of the spring,
$\begin{array}{}{\displaystyle {\omega}_{\mathrm{d},\mathrm{I}}=\sqrt{1-{\zeta}_{\mathrm{I}}^{2}}{\omega}_{\mathrm{I}}}\end{array}$ is the under-damped harmonic oscillator, and *ψ*_{I} is the phase at the impact.

The time span from the lift-up to the next impact of the drop is defined as regime II. In regime II, the bottom height of the spring is no longer fixed (*x*_{2} ≥ 0) and there is no external force loaded on the bottom mass point (*F* = 0). Differentiation of Eq. (15) gives

$$\begin{array}{}{\displaystyle \frac{{\mathrm{d}}^{2}{\u03f5}_{\mathrm{I}\mathrm{I}}}{\mathrm{d}{t}^{2}}=\frac{{\mathrm{d}}^{2}{x}_{1}}{\mathrm{d}{t}^{2}}-\frac{{\mathrm{d}}^{2}{x}_{2}}{\mathrm{d}{t}^{2}},}\end{array}$$(22)

where *ϵ*_{II} is the strain in regime II. The combination of Eqs. (14a), (14b), and (22) gives

$$\begin{array}{}{\displaystyle \frac{{\mathrm{d}}^{2}{\u03f5}_{\mathrm{I}\mathrm{I}}}{\mathrm{d}{t}^{2}}=-\frac{{k}_{\mathrm{I}\mathrm{I}}}{m}{\u03f5}_{\mathrm{I}\mathrm{I}}-\frac{{c}_{\mathrm{I}\mathrm{I}}}{m}\frac{\mathrm{d}{\u03f5}_{\mathrm{I}\mathrm{I}}}{\mathrm{d}t},}\end{array}$$(23)

where

$$\begin{array}{}{\displaystyle {k}_{\mathrm{I}\mathrm{I}}=4k,\phantom{\rule{thickmathspace}{0ex}}{c}_{\mathrm{I}\mathrm{I}}=4c.}\end{array}$$(24)

By solving Eq. (23), we obtain

$$\begin{array}{}{\displaystyle {\u03f5}_{\mathrm{I}\mathrm{I}}={A}_{\mathrm{I}\mathrm{I}}{\mathrm{e}}^{-{\zeta}_{\mathrm{I}\mathrm{I}}{\omega}_{\mathrm{I}\mathrm{I}}{t}_{\mathrm{I}\mathrm{I}}}\mathrm{c}\mathrm{o}\mathrm{s}\left({\omega}_{\mathrm{I}\mathrm{d},\mathrm{I}}{t}_{\mathrm{I}\mathrm{I}}+{\psi}_{\mathrm{I}\mathrm{I}}\right),}\end{array}$$(25)

where *t*_{II} is the time after the lift-up, *A*_{II} is the amplitude of the oscillation,
$\begin{array}{}{\displaystyle {\zeta}_{\mathrm{I}\mathrm{I}}=\frac{{c}_{\mathrm{I}\mathrm{I}}}{2\sqrt{m{k}_{\mathrm{I}\mathrm{I}}}}}\end{array}$ is the damping ratio, *ω*_{II} =
$\begin{array}{}{\displaystyle \sqrt{\frac{{k}_{\mathrm{I}\mathrm{I}}}{m}}}\end{array}$ is the undamped angular frequency of the spring,
$\begin{array}{}{\displaystyle {\omega}_{\mathrm{I}\mathrm{d},\mathrm{I}}=\sqrt{1-{\zeta}_{\mathrm{I}\mathrm{I}}^{2}}{\omega}_{\mathrm{I}\mathrm{I}}}\end{array}$ is the under-damped harmonic oscillator, and *ψ*_{II} is the phase at lift-off.

The coefficients were obtained according to the description given in Appendix A. The damping coefficient for regime I, *c*_{I}, was determined to be 0.7 × 10^{–3} kgs^{–1} using Eq. (A.5) with the result for *We* = 7 and was reasonably assigned for all Weber numbers in this study, while that for regime II, *c*_{II}, was determined to be half the value of *c*_{I}. This difference of the damping coefficient indicates that the mechanism for energy loss is different between regimes I and II. The stiffness *k*, determined by Eq. (A.6), tends to decrease with an increase of the Weber number.

The sum of the kinetic, potential, and elastic energies as the surface energy of the spring model can be calculated for each regime:

$$\begin{array}{}{\displaystyle {E}_{\mathrm{m}\mathrm{s},\mathrm{I}}=\frac{1}{4}m{\left(\frac{\mathrm{d}{\u03f5}_{\mathrm{I}}}{\mathrm{d}t}\right)}^{2}+\frac{1}{2}mg({D}_{0}+{\u03f5}_{\mathrm{I}})}\\ {\displaystyle \phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}+\frac{1}{4}{k}_{\mathrm{I}}{\u03f5}_{\mathrm{I}}^{2}+\sigma \pi {D}_{0}^{2},}\end{array}$$(26)

$$\begin{array}{}{\displaystyle {E}_{\mathrm{m}\mathrm{s},\mathrm{I}\mathrm{I}}=\frac{1}{2}m{\left(\frac{\mathrm{d}{x}_{\mathrm{c},\mathrm{I}\mathrm{I}}}{\mathrm{d}t}\right)}^{2}+\frac{1}{8}m{\left(\frac{\mathrm{d}{\u03f5}_{\mathrm{I}\mathrm{I}}}{\mathrm{d}{t}_{\mathrm{I}\mathrm{I}}}\right)}^{2}+mg{x}_{\mathrm{c},\mathrm{I}\mathrm{I}}}\\ {\displaystyle \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}+\frac{1}{8}{k}_{\mathrm{I}\mathrm{I}}{\u03f5}_{\mathrm{I}\mathrm{I}}^{2}+\sigma \pi {D}_{0}^{2},}\end{array}$$(27)

where *E*_{ms,I} and *E*_{ms,II} are sums of the mechanical energy and the surface energy in regime I and regime II, respectively. These energies are shown in Figure 10 with *We* = 7, 15, 23.

The overall energy loss rate in regime I is expressed as

$$\begin{array}{}{\displaystyle {\lambda}_{\mathrm{I}}=1-\frac{{E}_{\mathrm{m}\mathrm{s},\mathrm{I}}({t}_{\mathrm{I}\mathrm{I}}=0)}{{E}_{\mathrm{m}\mathrm{s},\mathrm{I}}({t}_{\mathrm{I}}=0)},}\end{array}$$(28)

and the energy loss rate over 1 cycle of oscillation in regime II,

$$\begin{array}{}{\displaystyle {\lambda}_{\mathrm{I}\mathrm{I}}=1-\frac{{E}_{\mathrm{m}\mathrm{s},\mathrm{I}\mathrm{I}}({t}_{\mathrm{I}\mathrm{I}}={T}_{\mathrm{I}\mathrm{d},\mathrm{I}})}{{E}_{\mathrm{m}\mathrm{s},\mathrm{I}\mathrm{I}}({t}_{\mathrm{I}\mathrm{I}}=0)},}\end{array}$$(29)

corresponds well for both the simulation and the spring model (Figure 11).

While the vertical strain and energy loss rates of the drop were well explained by the spring model (Figures 9 and 11), the second decrease of the spring model lagged that of the simulated drop (Figure 10). This time lag indicates that true damping factor has an other period than the damping term of the spring model.

Figure 9 Time series of drop’s vertical strain for the experiment, simulation, and the spring model with *We* = 7.

Figure 10 Time series for the sum of kinetic, potential, and surface energies from the simulation and spring model results. Each dashed line represents the transitional time from regime I to regime II. Major energy losses were observed at two moments, 2 ms and 12 ms after the impact as indicated by the “*” marks. The dash-dotted line shows *t* = *D*_{0}/*U*_{impact}, which predicts the start time of the second energy decay of the drop.

Figure 11 Energy loss rates over regime I *λ*_{I}, and the energy loss rate over 1 cycle of oscillation in regime II *λ*_{II}, with the simulated drop and the spring model.

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