content_image/333223.jpg content_image/333222.jpg
leading edge will sweep a hydrophilic domain of length l$_{h}$ with a velocity v h l and will then move across a hydrophobic domain of length l$_{φ}$ with a velocity v φ l . On the other hand, because of the geometry of the checkerboard pattern, the trailing edge will move over a hydrophilic domain of length l$_{φ}$ with a velocity v h t , to later continue over a hydrophobic domain of length l$_{h}$ with a velocity v φ t . 

Accordingly, equation (10) can be written as 

<img src='content_image/333223.jpg'>

in terms of the mean values of the leading an trailing edges, ˆ v$_{l}$ and ˆ v$_{t}$ , respectively. We can finally compute the average growth rate of the contact line as ˙ L = ˆ v$_{l}$ − ˆ v$_{t}$, which within approximations considered reads 

<img src='content_image/333222.jpg'>

Equation (12) gives an estimate of the growth rate of the contact line in terms of the velocities observed in longitudinal striped patterns, the width of the stripes being equal to the width of the domains of the checkerboard arrangement. To compare simulation results with the kinematical prediction, we measure v φ l , v h l , v φ t , and v h t from runs that correspond to a longitudinal stripe configuration for which w$_{h}$ and w$_{φ}$ match with the checkerboard pattern, as displayed in figures 8(a) and 8(b). Figure 12 shows a comparison between the growth rate measured from simulations and the kinematical prediction as a function of the length of the hydrophobic domains. The quantitative agreement shows that the varying wettability of alternating patches determines the growth of the contact line essentially by fixing the local value of the leading and trailing edge velocities, these values being very well approximated by the ones corresponding to long domains. 

## V. DISCUSSION AND CONCLUSIONS 

By means of lattice-Boltzmann simulations and kinematical models, we have studied the dynamics of driven thin films on a variety of heterogeneous hydrophilic-hydrophobic substrates. We have studied the effect of longitudinally stripes and checkerboard patterns. 

We have focused on a scenario where the unstable contact line gives rise to sawtooth and finger structures on homogeneous hydrophilic and hydrophobic substrates, respectively, and 

