FIG. 3: The growth rate, ω , of a transverse perturbation to the contact line as a function of the perturbation mode, k , on homogeneous substrates with different wetting properties obtained from lattice-Boltzmann simulations. The growth rate has a maximum that depends on the equilibrium contact angle, θ$_{e}$ . The maximum growth rate is larger if the substrate is hydrophobic. The corresponding most unstable wavelength, Λ$_{max}$( θ$_{e}$ ) = 2 π/k$_{max}$ , is smaller the more hydrophobic is the substrate. The error bars correspond to the standard deviation of the fit performed to estimate ω from the time evolution of the contact line amplitude. librium contact angle, θ$_{e}$ . In order to explore the effect of substrate heterogeneity on the dynamics of the film, we must first explore the dynamics on homogeneous substrates, to characterize ˙ L and Λ$_{max}$. In this section we describe the dynamics of thin films on two separate homogeneous substrates with equilibrium contact angles θ$_{e}$ = 0 ◦ and θ$_{e}$ = 90 ◦ , respectively. Our aim is to characterize the linear instability by measuring the dispersion relation on each substrate, and to characterize the base structures that emerge at long times. A numerical linear stability analysis of the front can be performed by following the evo- lution of an initially flat contact line that is perturbed transversely with a single mode k/k$_{c}$ , where k$_{c}$ = 2 π/x$_{c}$ . At short times, the perturbation either grows or decays exponentially with a growth rate, ω ( k ). We follow the evolution of single mode perturbations of small initial amplitude A$_{0}$/W ≈ 10 − $^{2}$, and focus on timescales where the amplitude shows an ex- A. Linear Stability 14