Cuboid drop: A low-dimensional model of drop dynamics on a substrate

Liquid drops placed on substrates may vibrate, slide, take history-dependent shapes, and even detach or break up when subjected to external forces. Although CFD tools can nowadays reproduce these motions, they are affordable for at most a few drops at a time. This work provides a low-dimensional model of such drops, in which the drop shape is approximated by a rectangular cuboid. The model results in three ordinary differential equations per drop. It is sufficiently simple to allow closed-form solutions in a variety of configurations. By systematically comparing the cuboid predictions to experimental, numerical, and theoretical results previously obtained with real drops, we discuss the extent to which the cuboid approximation reproduces the order of magnitude and qualitative dependence on parameters of a series of phenomena. The latter include natural drop vibrations, the detachment of pendant drops, multiple drop shapes allowed by contact angle hysteresis, the nonlinear retraction of drops after spreading, the sliding or climbing of drops on inclined substrates, and drop-induced damping of substrate vibrations. The cuboid model therefore provides a low-cost representation of drop motion on a substrate in response to arbitrarily complex external forcing.

Figure 1

(Left and center) Sessile raindrop residues on the aerial parts of plants: leaf blade of Acer palmatum (left), and flower petals of Senecio speciosus Willd. (center). (Right) A sessile drop is modeled by a rectangular cuboid of length $\ell$, width $w$, and height $h$, placed on a solid substrate. The face on the substrate is centered in $(x,y,z)=(x_{\mathrm{G}},0,0)$. Positions $x=A$ and $x=B$ correspond to the left and right boundaries of the cuboid drop. Inertial forces corresponding to accelerations $a_x$ (along $x$) and $a_z$ (along $z$) are applied at the center of mass of the cuboid.

Figure 2

Velocity profile $\mathbf{u}$ inside the cuboid drop, as a function of normalized coordinates $(x-x_{\mathrm{G}})/\ell \in [-0.5,0.5]$ and $z/h\in [0,1]$, according to Eqs. (4) and (19). The left graph corresponds to the translation of the cuboid drop (shear flow), while the right graph corresponds to the flattening of the cuboid drop (stagnation point flow). The general velocity field inside the cuboid includes a linear combination of both shear flow and stagnation point flow. In both graphs, the function $F$ corresponds to Eq. (19).

Figure 3

Region of existence of equilibrium positions in a $({\mathrm{Bo}}_z,\gamma )$ diagram. The curves represent the maximal Bond number ${\mathrm{Bo}}_{z\mathrm{b}}$ beyond which no equilibrium can be found, for the cuboid drop (thin red line) and for an axisymmetric drop (solution of Young-Laplace equation—thick black line). The shape of the liquid interface is represented for five points (black dots) corresponding to different pairs $(\text{Bo},\gamma )$, for the cuboid drop (light red boxes) and for the axisymmetric drop (black lines). The solid (respectively, dotted) lines correspond to stable (respectively, unstable) solutions of the Young-Laplace equation.

Figure 4

Equilibrium solutions as a function of the Bond number. The red curve represents the equilibrium length $\ell ={\ell }_{\mathrm{e}}$ of the cuboid drop, normalized by the solution ${\ell }_0$ obtained at ${\mathrm{Bo}}_z=0$. The solid (respectively, dashed) part of this curve corresponds to stable (respectively, unstable) solutions. The black dots represent the solutions of the Young-Laplace equation in the full range $\gamma \in [-1,1]$, assuming that the diameter at the drop basis is equivalent to $\ell$ (cf. Supplemental Material [30]).

Figure 5

Variation of the natural frequency $\omega$ of a sessile drop with $\gamma$, for ${\mathrm{Bo}}_z=0$. Red lines correspond to cuboid drops, while dark blue lines correspond to the numerical solution of Zhang et al. [35] for real drops. Thick lines represent the zonal (2,0) mode, while thin lines represent the sectoral (2,2) mode. The dark blue dot corresponds to the experimental data from Mettu and Chaudhury [36].

Figure 6

Variation of the natural frequency $\omega$ of a sessile drop with the Bond number ${\mathrm{Bo}}_z$, relative to the frequency ${\omega }_0$ at ${\mathrm{Bo}}_z=0$. Red lines correspond to cuboid drops, while dark blue lines correspond to the numerical solution of Zhang et al. [35] for real drops. Thick lines represent the zonal (2,0) mode, while thin lines represent the sectoral (2,2) mode. Solid, dashed, and dash-dotted curves correspond to $\gamma =\sqrt{2}/2$, $\gamma =0$ and $\gamma =-\sqrt{2}/2$, respectively.

Figure 7

Region of the phase space ($\ell$, $w$) in which a cuboid drop remains static thanks to contact angle hysteresis, for ${\mathrm{Bo}}_z=0$, $\gamma =0$ and $\zeta =0.1$. Thick lines are exact solutions of Eq. (40) while thin lines correspond to the approximate solutions (41). The dark blue (respectively, red) lines represent variations of ${\lambda }_{\ell }\in [-\zeta ,\zeta ]$ (respectively, ${\lambda }_w\in [-\zeta ,\zeta ]$). Solid blue (respectively, red) lines correspond to ${\lambda }_w=0$ (respectively, ${\lambda }_{\ell }=0$). Dashed blue (respectively, red) lines correspond to ${\lambda }_w=\zeta$ (respectively, ${\lambda }_{\ell }=\zeta$). Dotted blue (respectively, red) lines correspond to ${\lambda }_w=-\zeta$ (respectively, ${\lambda }_{\ell }=-\zeta$).

Figure 8

Time evolution of the length $\ell \left(t\right)$ of a symmetric cuboid drop ($\ell =w$), for different initial conditions ${\ell }_{\mathrm{i}}$, and parameters $\gamma =0$, ${\mathrm{Bo}}_z=0$, $\mathrm{Oh}=0.1$, and $\zeta =0.1$. (Inset) Reached equilibrium position ${\ell }_{\mathrm{e}}$ as a function of ${\ell }_{\mathrm{i}}$. In both the main graph and the inset, the grey shading represents the region in which cuboid drops may remain static owing to hysteresis. In the inset, each color corresponds to trajectories that are qualitatively similar to the trajectory represented in the main graph. The dashed line is the bisector ${\ell }_{\mathrm{e}}={\ell }_{\mathrm{i}}$.

Figure 9

Maximum retraction rate $|\dot{\ell }/{\ell }_{\mathrm{i}}|$ experienced by a sessile drop, as a function of $\mathrm{Oh}$, for ${\mathrm{Bo}}_z=-0.6$, $\gamma =0.17$ and $\zeta =0.01$. The dots correspond to measurements from Bartolo et al. [38] (with ${\mathrm{Bo}}_z\in [-0.72,-0.45]$ and initial normalized spreading diameter between 6 and 10). The thick red line (respectively, thin orange line) corresponds to numerical simulations of cuboid drops with ${\ell }_{\mathrm{i}}=6$ (respectively, ${\ell }_{\mathrm{i}}=10$).

Figure 10

Normalized speed $\mathrm{Ca}$ of a steadily sliding cuboid drop, as a function of the normalized body force ${\mathrm{Bo}}_x$ tangential to the substrate. The parameters are ${\mathrm{Bo}}_z=-2$, $\mathrm{Oh}=0.02$, $\gamma =0.7$, and $\zeta =0.1$. The red shaded area (collapsed in red line at low ${\mathrm{Bo}}_x$) corresponds to different ${\lambda }_w$ in steady state, that can be reached from different initial conditions $({\ell }_{\mathrm{i}},w_{\mathrm{i}})$. The dashed (respectively, dotted) blue line represents equation (42) with ${\lambda }_w=\zeta$ (respectively, ${\lambda }_w=-\zeta$).

Figure 11

Parity plot of the normalized mobility $\partial \mathrm{Ca}/\partial {\mathrm{Bo}}_x$ for the cuboid drop vs experiments from Podgorski et al. [40] and Podgorski [41] (dark blue dots), Le Grand et al. [42] (red dots), Kim et al. [29] (light blue dots) and original measurements reported in the Supplemental Material [30] (orange dots). The error bars correspond to ${\lambda }_w\in [-\zeta ,\zeta ]$ in the cuboid model [Eqs. (40) and (42)] and the dots are in ${\lambda }_w=0$. The black line is the bisector of the first quadrant.

Figure 12

Parity plot of the critical Bond number ${\mathrm{Bo}}_{xc}$ for the cuboid drop vs experiments from Podgorski et al. [40] and Podgorski [41] (dark blue dots), Le Grand et al. [42] (red dots), and original measurements reported in Supplemental Material [30] (orange dots). The error bars correspond to ${\lambda }_w\in [-\zeta ,\zeta ]$ in the cuboid model [Eqs. (40) and (42)] and the dots are in ${\lambda }_w=0$. The black line is the bisector of the first quadrant.

Figure 13

Time evolution of the position $x_{\mathrm{G}}$ of the cuboid center of mass for parameters $\gamma =0.47$, $\zeta =0.25$, $\mathrm{Oh}=0.1$, ${\omega }_{\mathrm{s}}=1.79$, $\mathrm{Bo}=0.517$, $a_{\mathrm{s}}=30$, and ${\mathrm{Bo}}_x={\mathrm{Bo}}_z=(\mathrm{Bo}/\sqrt{2})[a_{\mathrm{s}}sin\left({\omega }_{\mathrm{s}}t\right)-1]$. The thick line represents $x_{\mathrm{G}}$ vs time normalized by the vibration period, ${\omega }_{\mathrm{s}}t/\left(2\pi \right)$. The dashed line passes through the maxima of $x_{\mathrm{G}}$ (starting from the second) and corresponds to climbing at the time-averaged velocity $\langle {\dot{x}}_{\mathrm{G}}\rangle$ of the cuboid drop. (inset) Time evolution of the contact line A (thick orange line), the contact line B (thick light blue line), the cuboid length $\ell$ (dashed dark blue line) and its width $w$ (dashed red line) over one forcing period. The black line corresponds to $sin\left({\omega }_{\mathrm{s}}t\right)$, which varies linearly with the forcing acceleration.

Figure 14

Lumped-element model of the coupled vibration of a drop on a cantilever beam. (Left) The configuration investigated by Erfanul Alam and Dickerson [45]. (Right) The associated lumped-element model. The real drop is replaced by a cuboid drop of height $h\left(t\right)$. The beam is replaced by a rigid board (proof mass) of mass $\chi$ at vertical position $Z\left(t\right)$ supported by a spring of stiffness $\kappa$. All variables in the lumped-element model are dimensionless.

Figure 15

Decay rates ${\beta }_1$ (dark blue) and ${\beta }_2$ (red) of the vibration modes associated to the cuboid drop on mass-spring system, as a function of the Ohnesorge number $\mathrm{Oh}$, for $\chi =2.1$ and $\kappa =160$. The solid lines correspond to the cuboid drop while the dark blue dots correspond to values inferred from the data of Erfanul Alam and Dickerson [45], for their beam of intermediate thickness. The corresponding asymptotic regimes are given in Table 2.