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seen as ’intrinsic’ or self-inflicted ratchets. 

## IV. DROP TRANSPORT BY OBLIQUE SUBSTRATE VIBRATION 

As the first case of vibration-induced transport we discuss case (ii) introduced in Section II – a drop transported by an oblique substrate vibration as sketched in Fig. 1(b) and ex- perimentally observed in Ref. [15]. The typical behaviour of a drop during one vibration cycle as obtained in a time sim- ulation of Eq. (1) is given in Fig. 6. One observes that the drop undergoes changes of shape and moves back and forth. Such simulations in time may be employed to analyse the net transport over a wide frequency range. In addition one may use continuation techniques [27–29] to study the behaviourin the low frequency limit, i.e., for a slowly vibrating substrate. In this limit the intrinsic timescale of the drop dynamics t$_{0}$ is much smaller than the vibration period T and the drop moves in a quasi-stationary manner. This means that drop shape and velocity at each instant during the vibration cycle correspond to the ones of a stationary moving drop at the corresponding constant force. They are parametrized by a ( t ) . Averaging sta- tionary drop velocities v ( a ( t )) over one vibration period gives the low frequency limit of 〈 v 〉 . 

Fig. 7 presents results for the low frequency limit. Panel (a) shows profiles of stationary moving drops at various phases of the cycle on a horizontal substrate. Panels (b) and (c) give drop velocity and the dynamic mesoscopic contact angles, re- spectively, over one vibration cycle. The angles are shown for the left ( θ$_{l}$ ) and right ( θ$_{r}$ ) contact line. In the given setup, temporal modulations of the right contact angle are larger than those of the left one. The deviation of the velocity of the slid- ing drop from a harmonic modulation [dotted line in Fig. 7(b)] is a measure of the net motion of the drop. 

The overall behaviour of the drop over one vibration cycle is very similar in Fig. 6 (finite, but small frequency) and Fig . 7 

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(low frequency limit): during the first half of the cycle ( t < T/ 2 ) the drop is flattened and moves to the right. Note that this happens when the substrate is accelerated upwards and to the left. In the second half of the cycle the drop becomes taller and less wide while it slides to the left ( t > T/ 2 , substrate ac- celeration is downward and to the right). After one period the drop has moved a small net distance to the left. For the param- eters of Fig. 6 it takes about 100 vibration cycles to move the drop by its own length. As detailed below in the conclusion, this does well correspond to the available experimental results [15]. 

Based on the described findings one is able to understand the mechanism that leads to the net motion of the drop: The component of the oblique vibration that is normal to the sub- strate strongly modulates the hydrostatic pressure and there- fore provokes a nonlinear response in the drop shape, i.e., when the substrate accellerates upwards [downwards] it com- presses [decompresses] the drop. That in turn determines the strongly nonlinear drop mobility Q ( h ) in Eq. (1) and is there- fore responsible for an asymmetric response to the back and forth forcing that results from the parallel vibration compo- nent. As a result of this nonlinear coupling of the effects of normal and parallel vibration component one obtains an an- harmonic response of the drop to the harmonic but oblique vibration. 

The examples given in Figs. 6 and 7 where for a rather small vibration angle β . Therefore the change in drop shape was mainly due to the modulation of the hydrostatic pressure. For larger β the drop additionally changes its shape in re- sponse to the parallel vibration component. An examples is given in Fig. 8. The characteristic backwards shoulders are known from strongly driven drops on homogeneous substrate [29, 30]. For even larger β the process may become very com- plicated as the drop can during a single cycle undergo several 

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