shape at long times. Taking into account this periodicity, in the following we will reduce computational costs by considering system sizes whose width is of the order of Λ$_{max}$( θ$_{e}$ ). Figure 4 shows the long time evolution of the most unstable wavelengths for θ$_{e}$ = 0 ◦ and θ$_{e}$ = 90 ◦ . We observe that after a transient, the thin film saturates to a sawtooth shape on the hydrophilic substrate, while on the hydrophobic substrate a finger grows at a steady rate. Given that the growth rate of the finger is constant, the substrate is not covered evenly, and a dry region is left to be covered by the trailing edge. This contrasts with the dynamics in the hydrophilic substrate, on which the sawtooth saturates to a finite amplitude and propagates as a whole with a velocity dictated by the injection rate. Intuitively, one can understand the reason why a finger grows and a sawtooth saturates by examining the structure of the thin film. For unstable fronts, the amplitude of a perturbation grows due to an uncompensated distribution of the thickness of the ridge. As the perturbation grows, the contact line becomes increasingly curved, thus increasing the strength of the restoring capillary force. If the thickness of the capillary ridge is small, as on hydrophilic substrates, the contact line can reach a sufficiently large curvature, and the distribution of the ridge thickness is balanced by surface tension. Thus, the contact line saturates to a sawtooth shape. In contrast, if the thickness of the ridge is large, as on hydrophobic substrates, the curvature of the contact line is always insufficient to balance the driving force, and fingers grow at a rate dictated by the distribution of the ridge thickness along the front. We have demonstrated this effect previously [21], where we have shown that the growth rate of any sawtooth or finger can be collapsed to a universal curve, as a function of the difference of the thickness of the ridge between the leading and trailing edges of the contact line, independently of the capillary number, or the wetting properties or inclination angle of the substrate. Static and dynamic fluid rivulets are known to be unstable to wave lengths which decrease with substrate hydrophobicity, gravity and fluid flow [34–36]. We have never observed the destabilization of a finger for the regimes we have explored, but we cannot rule it out; such a possibility requires a more systematic analysis. To characterize the flow on homogeneous substrates, we compute the local mean velocity by averaging the velocity field, v , in the z direction: 〈 v 〉 = 1 /h ∫ h 0 d z v . Figure 5 shows plots of 〈 v 〉 h , which is a measure of the local mass flow, for both hydrophilic and hydrophobic substrates. The top panels in figure 5 show the x component of the flow. For the sawtooth, we observe only a slight increase of the flow near its tip. This is a signature of the balance 16