0  32 0 1000 2000 3000 0 1 x t h on min. static droplet coarsening film instability gradient driven potential off on min. max. spreading droplet 10 -4 10 -3 10 -2 10 -1 10 -5 10 -4 10 -3 <j> 0.3 0.5 0.7 h 0.1 1 10 10 -6 10 -5 10 -4 10 -3 <j> 0.3 0.5 0.7 h FIG. 5: (Color online) Shown is the net transport as measured by the mean flux 〈 j 〉 in its dependence on the flashing ratio χ for various film thicknesses as indicated in the legend. T = 500 and the remain- ing parameters are as in Fig. 3. The dotted straight line corresponds to a power law of exponent -1.3. FIG. 4: (Color online) Shown is the net transport as measured by the mean flux 〈 j 〉 in its dependence on the flashing frequency ω for various film thicknesses as indicated in the legend. The rema ining parameters are as in Fig. 3. The dotted straight line corresponds to a power law of exponent one. FIG. 3: Shown is a space-time plot of the typical evolution of the film thickness profile of a film of dielectric liquid in a hetero geneous capacitor (one spatial period shown) during one flashing per iod. For capacitor geometry and flashing cycle see Fig. 2. Parametersare ¯ h = 0 . 5 , Ψ = 0 . 5 , L = 32 , φ = 5 , T = 5000 , χ = 1 , A = 0 . 001 , ε r = 2 . 5 . The starting time is well after initial transients have decayed. 4 frequency is small enough. Each of the ’steps’ in the curves of Fig. 4 is related to a coarsening step that does not take place above the frequency at the step. At smaller mean film thick- nesses the maximum resembles a plateau with a flux that does nearly not change between ω = 10 − 3 and 10 − $^{2}$. The maxi- mal flux increases with increasing mean film thickness. The particularly interesting non-monotonic behaviour close to the $_{flow maximum (here well visible for}$¯ h = 0 . 5 ) results from the interaction of the dynamics of the ’last’ coarsening step and the switching cycle. Second, Fig. 5 shows the dependence of the mean flux on the flashing ratio. When it is low, i.e., the ratchet is most ofthe time in the off-phase, the flux is very low. This is because the liquid remains nearly homogeneously spread out as it has no time to assemble at the points of maximal voltage. By increas- ing the time of the on-phase the flux increases in a non-trivia l manner until it reaches a maximum at χ ≈ 1 . Depending on the film thickness one may find almost a plateau, i.e. a range of flashing ratios with nearly constant flux. For high flashing ratios the flux decreases again, since the ratchet is most of t he time in the on-phase. That implies electrically formed drops have not enough time to spread out. The decrease follows a power law with an exponent of about − 4 / 3 . Further calculations (not shown) indicate that the mean flux increases monotonically with increasing spatial asymmetry ratio φ and the amplitude Ψ of the ratchet potential. Note that as expected the net transport is zero at φ = 1 , i.e., for a symmetric potential. For further results for the presented one- layer geometry see Refs. [13, 14]. Exchanging the air layer by a second dielectric liquid one is able to transport the two liquids into opposite direction. There, depending on the ratios of the viscosities and relative dielectric constants one may as well find a flux reversal. This is further discussed in Ref. [14 ]. With this we close our present discussion of case (i), i.e., the study of the transport of a dielectric liquid by a ratchet ca- pacitor. Our results are typical for many externally imposed ratchets. The two cases discussed next are both related to net transport through substrate vibrations. Note that they can be geometry the flux is always positive and approaches zero in the zero frequency limit as well as in the high frequency limit. For large frequencies the fluid does not have enough time to dewet in the on-phase or to spread in the off-phase. At small frequencies both processes reach the respective equilibrium well before the next switching, i.e. most time is spend waiting. As a result the mean flow increases proportionally with fre- quency when increasing the frequency from zero (cf. Fig. 4). The flux reaches a maximum at ω = 10 − 3 . . . 10 − 2 before it decreases again. The decrease seems to follow a power law as well. It is, however, not universal: the exponent increases with increasing mean film thickness. The decaying part shows non-monotonic step-like behaviour that is more pronounced for smaller film thicknesses. This results from that fact tha t films of smaller thicknesses dewet in a homogeneous electric field with a wavelength well below the spatial period of the ratchet. The more wavelengths fit into a ratchet period the more coarsening steps can take place in the on-phase if the