Linear stability of a ridge

We investigate the stability of the three-phase contact line of a thin liquid ridge on a hydrophobic substrate for flow driven by surface tension and van der Waals forces. We study the role of slippage in the emerging instability at the three-phase contact line by comparing the lubrication models for no-slip and slip conditions at the liquid/substrate interface.

For both cases we derive a sharp-interface model via matched asymptotic expansions and derive the eigenvalues from a linear stability analysis of the respective reduced models. We compare our asymptotic results with the eigenvalues obtained numerically for the full lubrication models.