3 of the drop are completely periodic in time. We quantify the transport by the mean drop velocity 〈 v 〉 = Δ x/T where Δ x denotes the distance the drop moves within one period T . (iii) The final example is closely related to the second one. In the experiments of Ref. [17] the normal and lateral substrate vibrations are mechanically decoupled and may have different amplitudes, frequencies and phases, i.e., the normal vibration in Eq. (5) is a ( t ) = a$_{0}$ sin( ω$_{a}$t + δ ) , and the lateral one in Eq. (6) is βb ( t ) = βa$_{0}$ sin( ω$_{b}$t ) . In the particular case of [17], a horizontal substrate is used, i.e., α = 0 and ω$_{b}$ = ω$_{a}$ = ω . The parameter β takes the role of the ratio of the vibration amplitudes in the directions parallel and normal to the substrate. To be consistent with the long-wave approach taken, the physical amplitude ratio has to be small. However, as in (ii) β is the scaled ratio and therefore of O (1) . with the dimensionless Hamaker constants A$_{l}$ > 0 and A$_{u}$ < 0 . The dielectric film is subject to an electrical ’disjoining ’ pressure [22, 23] that varies in space as described by Φ( x ) and is periodically switched in time as Ω( t ) . In Eq. (4), ε$_{r}$ denotes the ratio of the relative dielectric constants of the liquid, ε$_{liq}$ , and the gas phase, ε$_{air}$ . The function Φ( x ) models an electric field that is periodic in x , but with an asymmetric profile, i.e., there exists no reflection symmetry in x . The employed spatial variation Φ( x ) and the temporal modulation Ω( t ) are defined by the sketches in Figs 2(d) and 2(e), respectively. The introduced scales are 3 γη/dκ 2 $_{el}$, √ γd/κ$_{el}$ , and d for time t , position x , and film thickness h , respectively. Thereby, we have defined an electrostatic ’spreading coefficient’ κ$_{el}$ = ε$_{0}$ε$_{liq}$U $^{2}$/ 2 d 2 where ε$_{0}$ is the absolute dielectric constant and U is the applied voltage. γ and η denote surface tension and dynamic viscosity of the liquid, respectively. For simplicity we assume A$_{l}$ = − A$_{u}$ = A > 0 . The di- mensionless Hamaker constant A is related to the dimensional one by A = A$_{dim}$/ 6 πd $^{3}$κ$_{el}$ . To characterise the ratchet we in- troduce the flashing ratio χ = τ/ ( T − τ ) and the asymmetry ratio φ = λ/ ( L − λ ) . The flashing frequency is ω = 2 π/T . The net transport along the substrate is measured by the mean flow 〈 j 〉 . For more details see [13, 14]. (ii) In the second example we look at drop transport by an oblique vibration that determines both, P and F . The pressure Fig. 2 sketches an idealised electrical wettability ratchet. The working principle is as follows. The flat free surface of a film of a dielectric liquid that wets the lower wall of the capa c- itor is stable when the electric field is switched off (Fig. 2 ( a)). When switching on the spatially inhomogeneous electric fiel d at t = 0 (see spatial profile and time dependence in Figs. 2 (d) and 2 (e), respectively) the film dewets the substrate and de- cays into a set of drops (we call this the ’on-phase’). This is due to a destabilisation of the surface by the overall elec- tric field and its gradients parallel to the substrate (Fig. 2 (d)). The latter interfere with the wavelength selection in the linear phase of the surface instability. They do as well accelerate the coarsening process. The different processes during the on-phase can be well appreciated in Fig. 3. The qualitative behaviour in the on-phase of the cycle resembles dewetting of a liquid film on a substrate with a chemical wettability pat- tern [25, 26]. If the on-phase is long enough (as is the case in Fig. 3) all the liquid collects in drops close to the posi- tions of the maximal voltage (Fig. 2 (b)). Then the field is switched off (’off-phase’) at t = τ (Fig. 2 (e)) and the drops spread (Fig. 2 (c)), merge and become a homogeneous film again (Fig. 2 (a)). The next cycle starts at t = T . If the pe- riod of the cycle is not large enough, the surface may not be- come entirely flat, a small modulation remains as indicated i n Fig. 2(c). contains the disjoining pressure Π( h ) = − 1 /h 3 + 1 /h 6 and the hydrostatic pressure where the time-dependence results from the vibration component normal to the substrate. Π con- tains long-range destabilising and short-range stabilising van der Waals interactions [24]. The lateral force contains a constant part (force down the incline) and a time- modulated part (vibration component parallel to the sub- strate). For a harmonic oblique vibration, the substrate accel- eration is a ( t ) = b ( t ) = a$_{0}$ sin( ωt ) . Note, that the physical vibration angle is O ( θ$_{e}$β ) where θ$_{e}$ is the mesoscopic equilib- rium contact angle, i.e., a scaled angle β of order one corre- sponds to a small physical angle. In this case, the introduced scales are 3 γη/h$_{0}$κ $^{2}$, √ γh$_{0}$/κ , and h$_{0}$ = ( B/ | A | ) 1 / 3 for time t , position x , and thickness h , respectively. A < 0 and B > 0 are the Hamaker constants for the long- and the short-range part of the disjoining pressure, respectively. Furthermore, κ = | A | / 6 πh 3 $_{0}$, and G = ρgh$_{0}$/κ , where ρ denotes the density of the liquid and g the gravita- tional acceleration. The non-dimensional vibration period is T = 2 π/ω . The fixed drop volume V = L ( ¯ h − h$_{p}$ ) is deter- mined by the domain size L , the mean film thickness ¯ h and the dimensionless precursor film thickness h$_{p}$ = 1 . The re- sulting transport along the substrate is measured after all tran- sients have decayed and the vibration-induced shape changes When initially switching on the device there are several cy- cles (typically ∼ 5 − 50 , the number varies with the employed parameters) that show transient behaviour. However, the ini- tial transients die out and the evolution of the film profile du r- ing one cycle is exactly time-periodic. Note, that there exist small parameter ranges where this is not the case due to reso- nance phenomena and/or multistability of various film state s. However, this shall not concern us here. Although the evolution of the film profile is exactly time- periodic, the process re-distributes the liquid within the film. This results in a net transport that depends in a non-trivial way on the various control parameters. As an example we give in Fig. 4 a log-log plot of the dependence of the mean flux on the flashing frequency. For the used flashing ratio and ratche t F = G [ α + βb ( t )] (6) P ( h, x, t ) = Π( h ) − Gh [1 + a ( t )] (5) III. FLOW IN A RATCHET CAPACITOR Π$_{el}$ = ( ε$_{r}$ − 1) [ ε$_{r}$ + (1 − ε$_{r}$ ) h ] 2 (4)