Evolution of toroidal free-rim perturbations on an expanding circular liquid sheet

Abstract

The formation of horizontal liquid sheets radially spreading following impact of a droplet onto a second sessile droplet or onto a blunt cylindrical target is studied experimentally. The radius of the spreading lamella and the cross-sectional radius of the rim growing at its moving edge are measured as functions of time. Also, the dynamics of the free-rim perturbations is characterized. Specifically, bending perturbations of the axis of the free rim in the plane of the sheet and the undulation perturbations of its cross section are measured as functions of time. The flow in the rim is described by the quasi-one-dimensional model. A time-dependent flow accounting for the rim stretching along its axis, the deceleration of the axis and the inflow from the sheet are determined as the basic (unperturbed) state, and the instability of the rim is investigated in the linearized approximation for small perturbations. The theoretical predictions are compared with the experimental data. It is revealed that the perturbations increase slower than exponentially in time. The slower growth of the perturbations contradicts the ‘frozen’ assumption, when the basic state is regarded as time independent, as used in previous studies. Moreover, the differences between our results and the instability evolution for time-independent ‘frozen’ basic states in previous studies are elucidated. In comparison with the experiments, it is shown that the present model provides a better prediction of the amplitudes for the wavenumbers corresponding to the largest perturbation amplitudes than the ‘frozen’ models.

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Appendix

Appendix

The coefficients f1(t)–f13(t) in Eqs. (10a, 10b, 10c) are expressed as follows:

$$\begin{aligned} {f_1}(t)=\, & 2\pi {a_0}^{2}{R_0}, \\ {f_2}(t)=\, & \pi {a_0}^{2}{R_0}+h{({R_0} - {a_0})^2}, \\ {f_3}(t)=\, & - i\pi m{a_0}^{2}, \\ {f_4}(t)=\, & h \Bigg[ { - {R_0}{V_0}+{a_0}{V_{\text{S}}}} \\ & {{ - \frac{{{a_0}}}{{{R_0}}}\left\{ {{R_0}\left( {2{V_{\text{S}}} - 3{V_0}} \right) - {a_0}\left( {{V_S} - 2{V_0}} \right)+{m^2}({R_0} - {a_0})\left( {{V_{\text{S}}} - {V_0}} \right)} \right\}} \Bigg]} , \\ {f_5}(t)=\, & - h(2{R_0} - {a_0})\left( {{V_{\text{S}}} - {V_0}} \right)\left( {1 - \frac{{{a_0}}}{{{R_0}}}} \right), \\ {f_6}(t)=\, & \;\rho \pi {a_0}^{2}{R_0}, \\ {f_7}(t)=\, & im\left\{ { - \rho \pi {a_0}^{2}{R_0}{A_0}+\rho h({R_0} - {a_0}){{\left( {{V_{\text{S}}} - {V_0}} \right)}^2}\left( {1 - \frac{{{a_0}}}{{{R_0}}}} \right)+2\alpha {a_0}\left( {1 - \frac{{{a_0}}}{{{R_0}}}} \right)} \right\}, \\ {f_8}(t)=\, & - \rho \pi {a_0}^{2}{V_0} - \rho h({R_0} - {a_0})\left( {{V_{\text{S}}} - {V_0}} \right)\left( {1 - \frac{{{a_0}}}{{{R_0}}}} \right), \\ {f_9}(t)=\, & im\alpha {a_0}\left( {\pi - 2+\frac{{2{a_0}}}{{{R_0}}} - \frac{{\pi {m^2}{a_0}^{2}}}{{{R_0}^{2}}}} \right), \\ {f_{10}}(t)=\, & \rho \pi {a_0}^{2}{R_0}^{2}, \\ {f_{11}}(t)=\, & 2\rho \pi {a_0}^{2}{R_0}{V_0}+2\rho h{({R_0} - {a_0})^2}\left( {{V_S} - {V_0}} \right), \\ {f_{12}}(t)=\, & - 2\rho \pi {a_0}^{2}{R_0}{A_0}+\rho h\left( {{V_{\text{S}}} - {V_0}} \right)\Bigg[ {{R_0}\left( {{V_{\text{S}}} - 3{V_0}} \right)+2{a_0}{V_0}} \\ & {+\frac{{{a_0}}}{{{R_0}}}\left\{ { - {R_0}\left( {3{V_{\text{S}}} - 5{V_0}} \right)+2{a_0}\left( {{V_{\text{S}}} - 2{V_0}} \right) - {m^2}({R_0} - {a_0})\left( {{V_{\text{S}}} - {V_0}} \right)} \right\}} \Bigg] \\ & +\alpha \left[ { - 2{R_0}+\frac{{2{a_0}}}{{{R_0}}}(3{R_0} - 2{a_0}) - {m^2}\left\{ {\pi {a_0} - \frac{{2{a_0}}}{{{R_0}}}({R_0} - {a_0})} \right\}} \right], \\ {f_{13}}(t)=\, & - 2\rho \pi {a_0}^{2}{R_0}{A_0} - \rho h{a_0}{({V_{\text{S}}} - {V_0})^2}\left( {1 - \frac{{{a_0}}}{{{R_0}}}} \right) \\ & - \alpha {a_0}\left( {\pi - 2+\frac{{2{a_0}}}{{{R_0}}} - \frac{{\pi {m^2}{a_0}^{2}}}{{{R_0}^{2}}}} \right). \\ \end{aligned}$$
(21)

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Ogawa, M., Aljedaani, A.B., Li, E.Q. et al. Evolution of toroidal free-rim perturbations on an expanding circular liquid sheet. Exp Fluids 59, 148 (2018). https://doi.org/10.1007/s00348-018-2602-4

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