Effects of Localized Forcing on Driven Thin Liquid Films

Thin liquid films driven up an inclined plane by temperature-induced surface tension gradients have been the focus of extensive mathematical and experimental research. Using the lubrication approximation, the motion of a thin liquid film is described by a single fourth-order partial differential equation (PDE) that models the evolution of the height of the film. Such films are now known to exhibit both classical and non-classical wave structures. Bertozzi, Munch, and Shearer [2] first proved the existence of non-classical undercompressive waves in such films, and experimental evidence was provided by Cazabat, Heslot, Troian, and Carles [3] and later by Sur, Bertozzi, and Behringer [8]. Levy and Shearer [7] applied theory developed by Shearer and LeFloch [6] to classify types of wave structures emerging from a Marangoni-driven film. In a recent analytical and numerical study, Haskett, Witelski, and Sur [5] use localized Marangoni forcing to produce a “microfluidic valve” that provides control of a thin flow of Marangoni and gravity-driven viscous fluid. Two long-time solutions are classified, one determined by the upstream thickness (weak forcing) and the other controlled by the forcing amplitude (strong forcing). For a given initial upstream height, there is a bifurcation between the two types of solutions at a critical level of forcing. This work employs the analysis of [5] and [7] to further explore the development of wave structures in the film. The effect of the “microfluidic valve” on both classical and non-classical wave structures is explored, and for early times, a classical N-wave is discovered in the PDE simulations.