capacitor and the net motion of the vibrated droplet over a wide range of parameters, to understand the underlying mech- anisms and to discuss transitions in the qualitative system be- haviour. around the maximal velocity. This point needs further investi- gation. We have explored the usage of ratchet mechanisms to trans- port a continuous phase in micro-fluidic settings involvinga free liquid-gas interface and contact lines, i.e., under the in- fluence of capillarity and wettability. In particular, we ha ve studied on the one hand the transport of a dielectric liquid in a capacitor with an asymmetrically spatially modulated elec- trical field that is periodically switched on and off. In this case the ratchet-like potential is imposed externally. On the other hand we have investigated drops on a solid substrate that show net motion under the influence of different types of har- monic substrate vibrations. Analysing the underlying mecha- nism we have found that the component of the harmonic vibra- tion that is orthogonal to the substrate induces a nonlinear (an- harmonic) response in the drop shape. The latter determines the strongly nonlinear drop mobility what results in an asym- metric response of the drop to the vibration component that is parallel to the substrate. The induced symmetry breaking between forward and backward motion during the different phases of the vibration results in the observed net motion of the drop. We have argued that the phenomenon might be seen as a rocked self-ratcheting as the drop itself introduces the local time-reflection asymmetry in the response to the time- periodic driving of the sliding motion. We employ the term “rocked” as the net motion is strongest in the low frequency limit as is typical for rocked particle ratchets [1]. Note as well that in the context of a drop on an incline the component of the vibration parallel to the substrate can be seen as a periodic rocking of the substrate. However, when comparing such lubrication models with particular experiments one always has to keep in mind that the formal range of applicability of the lubrication approxima- tion is limited and does often not overlap with the parameter ranges where experiments work best. The latter are often per- formed with liquid-substrate combinations that lead to equi- librium (and dynamic) contact angles that are not small. In our case (ii) one uses as well large substrate inclinations and large angles between vibration direction and substrate normal. For a direct quantitative comparison all those angles have to be small. Nevertheless, lubrication models are extremely suc- cessful in explaining intriguing effects observed in a wide range of experiments involving capillarity and wettability. A good example for this are studies of morphological changes in sliding drops where some experiments are done for liquids with static contact angles above π/ 4 using substrate inclina- tions varying from zero to π/ 2 [31]. Lubrication theory well explains the effects (see, e.g., [32] and the discussion in [33]). For the vibrated drop in Ref. [15] a liquid is used that has an equilibrium contact angle of about π/ 3 , the substrate incli- nation is π/ 4 , and the vibration is applied at an angle of π/ 4 with the substrate normal, i.e., the experiment is well outside the formal range of applicability of the lubrication approxima- tion. Therefore, a direct quantitative comparison of mean ve- locities could only be done through some ’up-scaling’ proce- dure that lifts drop volume, contact angles and vibration angle from the lubrication theory values to the experimental ones. This might be done (and is sometimes done). However, in our opinion the process has arbitrary elements, and we prefer not to employ it. The theory may, however, be related to the experiment in a semi-quantitative way by comparing, for instance, mean ve- locity in terms of drop size or typical timescales. In Ref. [15] the employed frequencies f range from 25 Hz to 120 Hz. At an acceleration of 15 g and f = 60 Hz one finds mean veloci- ties of about 1mm/s, i.e., the drop moves less than 20 µ m per cycle. For a drop of 3 mm length this implies that it needs about 150 cycles to move by its own length. The situation is very similar in Ref. [17] where for the parameters depicted in their Figs. 1 and 2 it takes about 50 vibration cycles to move the drop by its own length. Note, that their Fig. 1 does not show snapshots taken during a single cycle, but selected pic- tures from 70 cycles. For the vibrated drops our focus has been on two experi- mentally realised cases [15–17]: an oblique substrate vibr a- tion and decoupled normal and parallel substrate vibrations of identical frequency but different amplitudes and/or phase shifts. As in the latter case the use of different amplitudes without phase shift corresponds to the case of the oblique vi- bration we have mainly analysed the dependence of the drop motion on phase shift. The decisive element in both presented systems – the ratchet capacitor and the self-ratcheting vibrated drop – i s the interaction of the external periodic forces with capillar- ity and wettability, i.e., the nonlinearity that is necessary for a net motion results from interface effects. One may call this class of ratchets, “interfacial flow driven ratchets”. As th ey are dominated by interface effects they are very effective on small scales. This implies that they are good candidates for micro- or even nano-fluidic actuators. In terms of the number of cycles needed to move the drop by its own length our results are very close to the experimental findings: At a period of T = 100 we have a scaled velocity of about 0 . 1 × 10 − 3 (our Fig. 9(a)); for a$_{0}$ = 10 and β = 0 . 2 this gives 〈 v 〉 = 2 × 10 − 3 and an advancement per cycle of Δ x = 0 . 2 . For a drop like the one in our Fig. 6 this implies that it takes about 190 cycles to move it by its own length. Employing instead the accelerations of 15 g as in [15] it takes about 90 cycles. This is a fair agreement between experiment and lubrication theory. One may as well discuss the time scale t$_{0}$ employed in the non-dimensionalisation and use it to com- One needs to keep in mind that all models analysed in the present contribution are obtained employing a long-wave or lubrication approximation [18, 19]. That means they are rela- tively simple, but fully dynamical, highly nonlinear models based on a minimal set of ingredients, i.e., Stokes flow in the lubrication approximation, (electro-)capillarity, and wet- tability. They allow to study the net transport in the ratchet VI. CONCLUSIONS 8