g Local Heating

Thin liquid films driven up an inclined plane by temperature-induced sur-face 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 differen-tial 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 evi-dence 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 numer-ical 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. The Thin Film Experiment Figure 1 contains a schematic of a physical experiment described in [5], in which a thin film is driven up an inclined substrate by a surface tension gradient known as a Marangoni force. The substrate is inclined at an angl