0 0.2 0.4 0.6 0.8 1 -0.004 -0.002 0 0.002 0.004 <v> 50 100 200 400 1000 T 0 0.2 0.4 0.6 0.8 1 -0.04 -0.02 0 0.02 0.04 <v>/ 0.05 0.1 0.15 0.2 0.5 1.0 7 FIG. 13: (color online) Shown is the mean velocity as a function of the phase shift δ for various periods as given in the legend. The remaining parameters are V = 192 , G = 0 . 001 , a 0 = 10 , β = 0 . 1 , α = 0 , L = 128 , a 0 = 10 . FIG. 12: (color online) Shown is the mean velocity as a function of the phase shift δ for various amplitude ratios β as given in the legend. Thereby, we use a relatively large period of T = 200 and give the scaled velocity 〈 v 〉 /β . For β ≲ 0 . 2 all curves ’collapse’ onto a master curve. The remaining parameters are V = 192 , G = 0 . 001 , a 0 = 10 , α = 0 , L = 128 , a 0 = 10 . on a single master curve for β ≲ 0 . 2 . This implies that for β ≲ 0 . 2 particularly important values of the phase shift do not depend on the amplitude ratio. In particular, we find that flow reversal occurs at δ = 3 / 4 π and δ = 7 / 4 π . The drop moves fastest to the left [right] at δ = π/ 4 [ δ = 5 / 4 π ]. The master curve shows the symmetry ( δ → δ + π, 〈 v 〉 /β → −〈 v 〉 /β ) . Note that the symmetry holds for all considered amplitude ra- tios and periods (cf. Fig. 12), as can be expected from the set-up of the problem. At T = 70 ( ω = 0 . 090 > ω$_{c}$ ) the drop profiles in Fig. 10 vary much less over time than for the lower frequencies and Fig. 11 shows a rather large phase shift (larger than π/ 3 ) as compared to the other curves. When increasing the frequency, due to the phase shift and nonlinear mobility eventually the average response to the backwards force becomes smaller than the one to the forward force and the drop shows a reversal of the net motion. The net motion to the right becomes maxi- mal at an optimal frequency. The influence of the phase lag still exists at higher frequencies, however, as the changes in the profile shape become smaller with increasing frequency, the mean velocity decreases again and approaches zero in the limit of large frequencies. This is as well the reason why the absolute values of the velocities that can be reached in the flow reversal regime are much smaller than the ones in the low frequency limit. Note, finally, that in the case of larger vibration angles the picture becomes more complicated as the phase lag interacts with the morphological changes. As a re- sult the dependency of mean velocity on frequency might have more than one maximum. However, the scaling found for small β does not hold for larger β (see Fig. 12). For 0 . 2 ≲ β ≲ 1 . 0 the phase shift that results in maximal net motion and the maximal net ve- locity 〈 v 〉 both increase with increasing β . For β ≳ 1 . 0 both decrease again (not shown). This implies that the amplitude ratio that maximises net transport is β = 1 . There the phase shift resulting in maximal net transport to the left is slightly smaller than δ = π/ 2 . The change of the dependency of transport on phase shift with changing period is shown in Fig. 13. In accordance with results in case (ii) one finds that the velocities become mono - tonically smaller (for all phase shifts) with increasing frequen- cies (decreasing periods) as the drops are less able to follow the substrate vibrations. The phase shift where the maximal mean velocity occurs becomes smaller with decreasing fre- quencies. In the limit of low frequency the fastest transport to the left and right are found at δ = 0 and δ = π , respectively. A comparison with the experiments in Ref. [17] (in partic- ular their Fig. 3) shows that there the phase shift value where the maximal mean velocity is found and the value of the max- imal mean velocity both increase with increasing ratio of the amplitudes of parallel and normal vibration. However, as fur- ther explained in the conclusion, a direct quantitative compar- ison is not possible. The experimental results show roughly the symmetry ( δ → δ + π, 〈 v 〉 → −〈 v 〉 ) . Small deviations are explained by the presence of defects on the substrate [17] (see, in particular, the inset of their Fig. 2). A striking differ- ence between the results presented here and Fig. 3(a) of [17] is that in the latter the mean velocity depends in a strongly non-harmonic way on the phase shift. In the simulations we have not seen the experimentally found double peak structure As a second example of vibration-induced transport we dis- cuss case (iii) introduced in Section II – a drop transportedon a horizontal substrate by decoupled normal and parallel sub- strate vibrations as experimentally observed in Ref. [17]. We focus on normal and parallel vibrations of identical frequency that have a phase shift and possibly different amplitudes. Fol- lowing the discussion of the underlyig mechanism in case (ii) it is no surprise that a net motion is found with the present simple long-wave model as well for case (iii). Fig. 12 shows the dependence of the scaled mean velocity on the phase shift for different amplitude ratios β at a moder- ately large vibration period of T = 200 ( ω = 0 . 031 ). Very similar behaviour is found for other periods. Interestingly, the dependencies of the scaled velocity 〈 v 〉 /β for different β fall V. DROP TRANSPORT BY DECOUPLED HORIZONTAL AND VERTICAL SUBSTRATE VIBRATIONS