FIG. 5: Grey scale maps of the average flow within the film for a steady sawtooth ( θ$_{e}$ = 0 ◦ ) and a finger ( θ$_{e}$ = 90 ◦ ) at t/t$_{c}$ = 134, starting from a slightly perturbed front at t/t$_{c}$ = 0. The width of the system is fixed to W/x$_{c}$ = 13 for the hydrophilic substrate, and W/x$_{c}$ = 8 for the hydrophobic substrate. Top panels: the flow in the x direction, 〈 v 〉 $_{x}$h/ ( Uh$_{c}$ ), is homogeneous far from the contact line, and increases sharply on the body of the finger. The sawtooth shows a small increase, due to the variation of the film thickness. Bottom panels: the lateral flow, 〈 v 〉 $_{y}$h/ ( Uh$_{c}$ ), shows clearly that both the finger and the sawtooth are fed by transverse flows. For the finger, streamlines are computed close to the trailing edge and show that particles flow to the body of the finger as they approach this region. between the driving force caused by the variation of mass along the contact line, and the restoring surface tension. For a finger, this balance cannot be sustained, and a sharp flow increase in the body of the finger is observed. As shown in the bottom panels in figure 5, there exist two stationary regions that feed both the sawtooth and the finger. For the sawtooth, these regions extend along the whole contact line structure sustaining its stationary shape. For a finger, they are located at the trailing edge of the contact line and fix a constant flow of mass that causes the finger to grow steadily (see, e.g. , the streamlines depicted at the lowermost panel of the figure). In the previous section we presented results for sawtooth and finger growth on homo- geneous substrates. In both hydrophilic and hydrophobic substrates, the lengthscale of instability is characterized by the most unstable wavelength of the linear regime, Λ$_{max}$( θ$_{e}$ ), which sets the typical spacing of both sawtooth and finger structures. 17 IV. HETEROGENEOUS SUBSTRATES