content_image/358825.jpg
## I. INTRODUCTION 

<img src='content_image/358825.jpg'>

Forced liquid thin films appear in many processes, such as solid coating, wetting of bio tissues and microfluidic flow orientation [1–3]. For such driven films advancing on dry substrates, the forcing of the film triggers the destabilization of the contact line [1, 4–7], which gives rise to interfacial structures of different shapes, as depicted in Fig. 1, depending on the driving force, the wettability of the substrate, and, for gravity driven films, on the inclination angle of the substrate [1, 4]. On hydrophilic substrates, the instability typically causes a deformation of the film that has the shape of saturated sawtooth structures, such as the ones shown in Fig. 1 (a), which do not grow in time, and that propagate along the forcing direction at the typical injection velocity U . On the contrary, if the substrate is hydrophobic, the contact line breaks into steadily growing fingers, as in Fig. 1 (b), which propagate in the direction of forcing, growing at an intrinsic rate. Both sawtooth and finger structures have a typical transverse periodicity, which is characterized by the most unstable wavelength in the linear regime, Λ$_{max}$. Again, this intrinsic lengthscale depends on the applied forcing, the inclination angle, and on the wetting properties of the substrate. 

Due to its relevance in technological and biological applications, [1–3], new insight on 

