Confinement induced control of similarity solutions in premelting dynamics and other thin film problems

Abstract: When films thin the viscous resistance of the confining surfaces controls key aspects of their dynamics. However, those surfaces do more than provide the typical boundary conditions. For example, if one of the boundaries is elastic, there are nonlocal interactions that lead to interesting and challenging mathematical problems and important applications. One unique setting occurs when the free surfaces of most solids approach the bulk melting temperature from below, and they are wet by the melt phase. Mathematically, the fluid dynamics of this so-called “premelting” falls under the rubric of a class of higher-order diffusion equations governing the dynamics of viscous current down an incline, viscous gravity currents between a rigid surface and a deformable elastic sheet, wetting/dewetting, and a spate of other thin-film settings, to name a few. However, the underlying forces driving the flow are uniquely associated with intermolecular forces. We discuss a class of experimentally tested and testable premelting dynamics flows. Finally, through its influence on the viscosity, the confinement effect implicitly introduces a new universal length scale into the volume flux. Thus, there are a host of thin film problems, from droplet breakup to wetting/dewetting dynamics, whose properties (similarity solutions, regularization, and compact support) will change under the action of the confinement effect. Therefore, our study suggests revisiting the mathematical structure and experimental implications of a wide range of problems within the framework of the confinement effect.