longitudinal pattern for different stripe widths. In this section we analyze the growth of the contact line on longitudinal patterns of hydrophilic-hydrophobic properties. 

We first impose a pattern of alternating hydrophilic-hydrophobic stripes of equal width w , oriented in the longitudinal direction. Given that the periodicity of the front is determined by Λ$_{max}$ /x$_{c}$ ≈ 8, we consider the effect of increasing the stripe width from Λ$_{max}$, choosing the widths w/x$_{c}$ = 8, w/x$_{c}$ = 11, and w/x$_{c}$ = 13. Due to the periodicity of the front, larger widths corresponding to w > 2Λ$_{max}$ are not explored. As in the homogeneous case, we explore the evolution of only one mode, for which the corresponding wavelength is fixed as Λ = w . 

Figure 6 shows the evolution of the front for stripes of width w/x$_{c}$ = 8. Given that for θ$_{e}$ = 90 ◦ , Λ$_{max}$ /x$_{c}$ ≈ 8, the front destabilizes as it moves over the hydrophobic stripe and grows steadily. In contrast, when in contact with the hydrophilic stripe, the front bends and saturates to a sawtooth shape. The last three plots in figure 6 show the contact line for the heterogeneous and homogeneous substrates at the same time. By comparing these plots we notice that for the same injection rate the length of the contact line on the hydrophobic stripe is smaller than the length of the finger that would have grown on a homogeneous substrate. In addition, its tip is located at a less advanced position. In contrast, the sawtooth advances more rapidly on the heterogeneous substrate than on the homogeneous one. 

To examine this partial inhibition of the growth on the hydrophobic stripe caused by the longitudinal pattern, we compute the y component of the flow, which we show in figure 7(a) for w/x$_{c}$ = 8. The grey scale in this plot matches that of figure 5, which corresponds to a homogeneous substrate. The flow field exhibits essentially the same regions of transverse flow as in the homogeneous case. These regions are stationary and only translate forward as time proceeds. Therefore, the inflow to each stripe is constant in time. Nonetheless, by comparing the intensity of the regions between figure 5 and figure 7(a) one can appreciate that the inflow to the hydrophobic stripe is smaller for the heterogeneous substrate, while there is a clear increase in the flow to the hydrophilic stripe, which makes the sawtooth propagate with a larger velocity compared to the homogeneous case. 

This effect is what causes the contact line on the hydrophobic stripe to have a smaller growth rate compared to the homogeneous situation. Nonetheless, given that transverse flows originate in the vicinity of the boundary between adjacent stripes, the effect should be observable only when the stripe width, w , is small enough. Figures 7(b), 7(c), and 7(d) 

