3.1 The dynamics of a climbing drop

Figure 2 shows snapshots of a climbing drop at different phases. We can see that the drop motion is accompanied by a capillary wave travelling at the surface of the drop. The capillary wave is more noticeable when the drop moves up than when it slides down. The interaction between the capillary wave and the CL results in complicated CL motion at the upper and lower ends of the drop in one period of acceleration (figure 4 a), and the CL speed $u_{CL}$ reaches its maximum value at roughly $\unicode[STIX]{x1D719}=1/4\unicode[STIX]{x03C0}$ for the upper end and $\unicode[STIX]{x1D719}=\unicode[STIX]{x03C0}$ for the lower end. By contrast, the variation of the volume-averaged $x$ component of the drop velocity, $u_{c}=(1/V)\int _{\unicode[STIX]{x1D6FA}(t)}u\,\text{d}\unicode[STIX]{x1D6FA}$ , is close to a harmonic function with a shift of $U_{m}$ (figure 4 b), where $u$ is the projection of the fluid velocity $\boldsymbol{u}$ on the $x$ -axis and $\unicode[STIX]{x1D6FA}(t)$ is the space occupied by the drop at time $t$ . The oscillation of the drop on the plate has the same frequency as the acceleration but experiences a delay of $\unicode[STIX]{x0394}\unicode[STIX]{x1D719}$ in phase due to the inertia of the drop. Therefore, we can quantitatively divide the drop motion in one period into two stages of equal phase interval: the downhill stage [ $\unicode[STIX]{x0394}\unicode[STIX]{x1D719},\,\unicode[STIX]{x03C0}+\unicode[STIX]{x0394}\unicode[STIX]{x1D719}$ ] and the uphill stage [ $\unicode[STIX]{x03C0}+\unicode[STIX]{x0394}\unicode[STIX]{x1D719},\,2\unicode[STIX]{x03C0}+\unicode[STIX]{x0394}\unicode[STIX]{x1D719}$ ] (figure 4 b). We can see that figures 2(e) and 2(g) correspond to the drop shapes with the maximum average speed of motion downhill and uphill respectively. Their difference indicates the fore–aft asymmetry of drop deformation during its oscillation on the substrate. Figure 4(c) shows the evolution of the wetted area of the drop, $S_{CL}$ . We find that the broken symmetry causes a non-harmonic variation in the wetted area, thereby leading to a hysteresis loop of the wetted area in one period. In particular, the average wetted area in the downhill stage is larger than that in the uphill stage. This will be shown to induce the net uphill displacement of the drop.

Figure 4.

The variation of the flow quantities in one period of acceleration. It should be noted that this is the same case as in figure 2. (a) The variation of the CL speed $u_{CL}$ at the upper end (dash-dotted line) and lower end (solid line) of the CL. (b) The variation of the volume-averaged drop velocity $u_{c}$ (solid line), which is close to a harmonic function (dotted line). The bullets indicate the corresponding phases shown in figure 2. (c) The non-harmonic variation of the wetted area $S_{CL}$ . The dashed line represents an image of the variation of the wetted area in the uphill stage. The region in grey corresponds to the difference in the wetted area between the uphill and downhill stages in one period, i.e. $\unicode[STIX]{x03C0}\cdot \unicode[STIX]{x0394}S$ .

3.2 The forces exerted on the drop

We analyse here the forces acting on the drop, in order to interpret the ratchet mechanism that causes the net uphill movement of the drop. These forces consist of the contributions from the plate (i.e. capillary force at the CL, viscous force and plate vibration), gravity and the surrounding gas. Among them, we can neglect the force exerted by the gas. It is primarily due to the low density and viscosity of the gas and the insignificant difference in velocity between the drop and the surrounding gas. Hence, the variation of drop momentum in a dimensionless form yields

(3.1) $$\begin{eqnarray}\int _{\unicode[STIX]{x1D6FA}}\frac{\text{D}\boldsymbol{u}}{\text{D}t}\,\text{d}V=\frac{1}{Re}\int _{S_{CL}}\boldsymbol{n}_{w}\boldsymbol{\cdot }\unicode[STIX]{x1D64F}\,\text{d}S+\frac{1}{We}\oint _{CL}\unicode[STIX]{x1D749}\,\text{d}l+\int _{\unicode[STIX]{x1D6FA}}\left(\unicode[STIX]{x1D706}_{a}\sin \unicode[STIX]{x1D719}+\frac{1}{Fr^{2}}\right)\boldsymbol{j}\,\text{d}V,\end{eqnarray}$$

where $\boldsymbol{n}_{w}$ denotes the unit vectors normal to the wall and pointing away from the drop, $\unicode[STIX]{x1D64F}$ is the stress tensor and $\unicode[STIX]{x1D749}$ is the unit vector tangential to the interface in the plane spanned by the vectors normal to the substrate and to the CL.

The effect of the gravitational force simply prompts the drop to slide down, and it has been experimentally shown that it does not contribute to the uphill motion of the drop (Brunet et al. Reference Brunet, Eggers and Deegan2007). Therefore, in the following studies, we rule out the gravitational force and mainly focus on the role of the forces exerted by the plate in the uphill net displacement of the drop, or, more specifically, the role of the viscous force, the volumetric force due to plate vibration and the capillary force arising from contact angle hysteresis.

For convenience of analysis, we define the time average of any quantity $q$ over the downhill or uphill stage by

(3.2) $$\begin{eqnarray}\langle q\rangle _{M}=\frac{2}{T}\int _{T/2}q\,\text{d}t,\end{eqnarray}$$

where the subscript $M$ can be up or down, corresponding to the uphill and downhill stages respectively.

3.3 The ratchet mechanism of drop climbing

We first consider cases in the absence of contact angle hysteresis, i.e. $(1/We)\oint _{CL}\unicode[STIX]{x1D749}\,\text{d}l\approx 0$ . Integration of the $x$ component of (3.1) over one period suggests that the time average of the viscous force exerted by the wall should be approximately zero, or $\langle f_{v}\rangle _{up}+\langle f_{v}\rangle _{down}\approx 0$ , where $f_{v}$ is the $x$ component of the viscous force. We tentatively write the average viscous force $\langle f_{v}\rangle$ as $-\unicode[STIX]{x1D6FD}\langle u_{c}\rangle \langle S_{CL}\rangle /Re$ , where $\unicode[STIX]{x1D6FD}$ is a positive coefficient that relates to the distribution of shear strain near the wall. Assuming that the value of $\unicode[STIX]{x1D6FD}$ is roughly the same in the uphill and downhill stages, we can obtain

(3.3) $$\begin{eqnarray}\displaystyle \langle u_{c}\rangle _{up}\langle S_{CL}\rangle _{up}+\langle u_{c}\rangle _{down}\langle S_{CL}\rangle _{down}\approx 0. & & \displaystyle\end{eqnarray}$$

For simplicity, we define $S_{m}=(\langle S_{CL}\rangle _{up}+\langle S_{CL}\rangle _{down})/2$ and $\unicode[STIX]{x0394}S=\langle S_{CL}\rangle _{down}-\langle S_{CL}\rangle _{up}$ (as illustrated in figure 4 b). Then, taking the net velocity of the drop $U_{m}=(\langle u_{c}\rangle _{up}+\langle u_{c}\rangle _{down})/2$ into account and assuming $\unicode[STIX]{x0394}S^{2}\ll S_{m}^{2}$ , (3.3) can be rewritten as

(3.4) $$\begin{eqnarray}\displaystyle -U_{m}\approx \langle f_{v}\rangle _{down}\frac{Re}{\unicode[STIX]{x1D6FD}}\frac{\unicode[STIX]{x0394}S}{S_{m}^{2}}. & & \displaystyle\end{eqnarray}$$

On the other hand, we note that the drop momentum has the same value at the start and end of the downhill (and also uphill) stage (see figure 4 a). This implies that the viscous force should be balanced by the volumetric force arising from the plate acceleration during the downhill (and uphill) stage. Integration of the $x$ component of (3.1) over the period of the downhill stage yields

(3.5) $$\begin{eqnarray}\displaystyle \langle f_{v}\rangle _{down}=\frac{2\unicode[STIX]{x1D706}_{a}}{\unicode[STIX]{x03C0}}\sin \unicode[STIX]{x1D6FC}\cos \unicode[STIX]{x0394}\unicode[STIX]{x1D719}. & & \displaystyle\end{eqnarray}$$

By substituting (3.5) into (3.4), we can obtain a quantitative estimate of the net drop velocity,

(3.6) $$\begin{eqnarray}\displaystyle Ca\propto \unicode[STIX]{x1D706}_{a}\cos \unicode[STIX]{x0394}\unicode[STIX]{x1D719}\,We\,\sin \unicode[STIX]{x1D6FC}\,\frac{\unicode[STIX]{x0394}S}{S_{m}^{2}}, & & \displaystyle\end{eqnarray}$$

where $Ca=-U_{m}\,\unicode[STIX]{x1D707}\sqrt{a_{0}V^{1/3}}/\unicode[STIX]{x1D70E}$ . We can conclude from (3.6) that in the absence of contact angle hysteresis, the drop will always move uphill with $\unicode[STIX]{x0394}S>0$ and $\unicode[STIX]{x0394}\unicode[STIX]{x1D719}<\unicode[STIX]{x03C0}/2$ . The term $\unicode[STIX]{x0394}S/S_{m}^{2}$ represents the non-harmonic response of the drop to the spatial asymmetry and vibration, and thus is directly related to the mobility of the drop. This explains the experimental observations of Sartori et al. (Reference Sartori, Quagliati, Varagnolo, Pierno, Mistura, Magaletti and Casciola2015), namely that it is more likely for a drop to move uphill on a hydrophobic substrate than a hydrophilic one. For a drop of fixed volume, the more hydrophobic the substrate is, the smaller $S_{m}$ is, and thus the higher the drop mobility is.

Figure 5.

Numerical results in the absence of gravity and contact angle hysteresis ( $\unicode[STIX]{x1D703}_{A}=\unicode[STIX]{x1D703}_{R}=60^{\circ }$ ) at $We=6.6$ and $Re=23.6$ . (a) The variation of $\cos \unicode[STIX]{x0394}\unicode[STIX]{x1D719}$ as a function of $St$ . The solid line is a linear fitting of the data. (b) The variation of $\unicode[STIX]{x0394}S/S_{m}$ as a function of $\unicode[STIX]{x1D706}_{a}$ . (c) Comparison of the net drop velocity between numerical results and the predictions of (3.6); the dashed lines represent linear fittings. It should be noted that the same cases are included in (b) and (c), with $St=0.115$ (▪), 0.148 (▵), 0.184 (●), 0.231 (♢) and 0.294 (▸).

Numerical experiments are performed to verify the theoretical prediction of (3.6). Figure 5 shows the numerical results for a variety of values of $\unicode[STIX]{x1D706}_{a}$ and $St$ , and several observations can be made. First, correlation is only seen between $\unicode[STIX]{x0394}\unicode[STIX]{x1D719}$ and $St$ ; this is reasonable since $\unicode[STIX]{x0394}\unicode[STIX]{x1D719}$ reflects the temporal response of the drop inertia to variation of the acceleration. In addition, $\cos \unicode[STIX]{x0394}\unicode[STIX]{x1D719}$ appears to decrease linearly with increase of $St$ in the parameter space considered (figure 5 a). Second, $\unicode[STIX]{x0394}S/S_{m}$ linearly increases with $\unicode[STIX]{x1D706}_{a}$ for a relatively small amplitude of vibration ( $\unicode[STIX]{x1D706}_{a}<1$ ) and the slope is dependent on $St$ : the higher the frequency of the plate vibration is, the gentler the slope is. For sufficiently large $\unicode[STIX]{x1D706}_{a}$ , $\unicode[STIX]{x0394}S/S_{m}$ approaches a maximum value ( ${\sim}20\,\%$ ), which is independent of $St$ . Figure 5(c) shows a comparison of $Ca$ between the numerical results and the theoretical prediction, and good agreement has been obtained. The results at $St=0.148$ , $0.184$ and $0.231$ nearly collapse onto a single line, probably because the frequency of the acceleration in these cases is close to the resonance frequency of the drop, i.e. $f=63$  Hz or $St=0.185$ (Celestini & Kofman Reference Celestini and Kofman2006). For the other two cases ( $St=0.115$ and $0.294$ ), (3.6) remains valid, but the proportion factors are apparently different. This implies that the value of $\unicode[STIX]{x1D6FD}$ may also be a function of the frequency of the plate vibration. Overall, our numerical experiments confirm the prediction of (3.6).

Figure 6.

Numerical results in the absence of gravity at $We=6.6$ , $Re=23.6$ and $\unicode[STIX]{x1D703}_{A}=60^{\circ }$ . (a) The contact angle distribution at $\unicode[STIX]{x1D719}=\unicode[STIX]{x03C0}$ , with $St=0.148$ , $\unicode[STIX]{x1D706}_{a}=1.75$ and $\unicode[STIX]{x1D703}_{R}=30^{\circ }$ . (b,c) The same cases at $St=0.115$ ; the results are denoted by the same symbols. The variation of $\unicode[STIX]{x0394}S/S_{m}$ and the net drop velocity $Ca$ at different values of $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}$ are shown in (b) and (c) respectively. The dashed lines in (c) represent data fittings (quadratic for $\unicode[STIX]{x1D706}_{a}\leqslant 1$ and linear for $\unicode[STIX]{x1D706}_{a}\geqslant 1.25$ ).

3.4 The role of contact angle hysteresis

We further analyse the role of contact angle hysteresis, $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}$ ( $=\unicode[STIX]{x1D703}_{A}-\unicode[STIX]{x1D703}_{R}$ ), in the drop motion. First, contact angle hysteresis gives rise to the capillary force at the CL that is against drop displacement, e.g. subject to the shear force from the surrounding fluid (Ding, Gilani & Spelt Reference Ding, Gilani and Spelt2010) or the gravitational force (Podgorski, Flesselles & Limat Reference Podgorski, Flesselles and Limat2001). The magnitude of the capillary force depends on the distribution of the contact angle along the CL. A typical case is shown in figure 6(a), where the local contact angle on the sides of the CL mainly falls in the range $\unicode[STIX]{x1D703}_{R}<\unicode[STIX]{x1D703}<\unicode[STIX]{x1D703}_{A}$ . Therefore, the capillary force can be approximated by $(\cos \unicode[STIX]{x1D703}_{R}-\cos \unicode[STIX]{x1D703}_{A})W_{CL}\,We^{-1}$ , where $W_{CL}$ is the dimensionless width of the CL. We can expect that the drop stays in the static regime for low acceleration such that $\unicode[STIX]{x1D706}_{a}We\,(\cos \unicode[STIX]{x1D703}_{R}-\cos \unicode[STIX]{x1D703}_{A})^{-1}\sim O(1)$ , in which the vibration is not strong enough to depin the CL. This is confirmed by numerical results in figure 6(c), and the threshold value of $\unicode[STIX]{x1D706}_{a}$ increases with $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}$ ( $\unicode[STIX]{x1D706}_{a}\sim 0.25$ at $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}=10^{\circ }$ and ${\sim}0.5$ at $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}=30^{\circ }$ ).

Second, contact angle hysteresis tends to reduce the drop mobility. Under extreme conditions such as $\unicode[STIX]{x1D703}_{A}=180^{\circ }$ and $\unicode[STIX]{x1D703}_{R}=0^{\circ }$ , in principle, the CL would be completely pinned, regardless of the acceleration amplitude. Numerical results in figure 6(b) show that $\unicode[STIX]{x0394}S/S_{m}$ decreases significantly with increase of $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}$ . We can observe that $\unicode[STIX]{x0394}S/S_{m}$ changes linearly with $\unicode[STIX]{x1D706}_{a}$ for $\unicode[STIX]{x1D706}_{a}\leqslant 1$ and remains at roughly the same value for $\unicode[STIX]{x1D706}_{a}\geqslant 1.25$ ; interestingly, a transition of net drop velocity occurs simultaneously (figure 6 c). For relatively large $\unicode[STIX]{x1D706}_{a}$ , the vibration depins the CL, especially at the upper and lower parts. Since the variation of $W_{CL}$ is insignificant (see figure 2 e–h), it can be assumed that the capillary forces in the uphill and downhill stages cancel each other out. Consequently, the transition of $Ca$ with $\unicode[STIX]{x1D706}_{a}$ in the presence of contact angle hysteresis can be well predicted by (3.6). Specifically, we can obtain $Ca\propto \unicode[STIX]{x1D706}_{a}^{2}$ for $0.5\leqslant \unicode[STIX]{x1D706}_{a}\leqslant 1$ and $Ca\propto \unicode[STIX]{x1D706}_{a}$ for $\unicode[STIX]{x1D706}_{a}\geqslant 1.25$ , in agreement with numerical results (figure 6 c). This transition of $Ca$ with $\unicode[STIX]{x1D706}_{a}$ was also observed in experiments (Brunet et al. Reference Brunet, Eggers and Deegan2007) in a similar range of $\unicode[STIX]{x1D706}_{a}$ ( ${\sim}1\,-\,1.2$ ), accompanied by a change of wetting behaviour at the trailing end of the drop.