Abstract
An elastic sheet that deforms near a solid substrate in a viscous fluid is a situation relevant to various dynamical processes in biology, geophysics and engineering. Here, we study the relaxation dynamics of an elastic plate resting on a thin viscous film that is supported by a solid substrate. By combining scaling analysis, numerical simulations and experiments, we identify asymptotic regimes for the elastohydrodynamic leveling of a surface perturbation of the form of a bump, when the flow is driven by either the elastic bending of the plate or thermal fluctuations. In both cases, two distinct regimes are identified when the bump height is either much larger or much smaller than the thickness of the pre-wetted viscous film. Our analysis reveals a distinct crossover between the similarity exponents with the ratio of the perturbation height to the film height.
Highlights
Thin viscous films have been of great interest to mankind for centuries with a notable early application related to the invention of the wheel and axle and its need for lubricating films [1]
We study the dynamic wetting of a self-propelled viscous droplet using the time-dependent lubrication equation on a conical-shaped substrate for different cone radii, cone angles and slip lengths
Our results show that manipulating the droplet size, the cone angle and the slip length provides different schemes for guiding droplet motion and coating the substrate with a film
Summary
Thin viscous films have been of great interest to mankind for centuries with a notable early application related to the invention of the wheel and axle and its need for lubricating films [1]. The motion of an elastic sheet supported by a thin layer of viscous fluid is a phenomenon that manifests itself in processes spanning wide ranges of time and length scales, from, e.g., magmatic intrusion in the Earth’s crust [1,2], to fracturing and crack formation in glaciers [3], to pumping in the digestive and arterial systems [4–6], or the construction of two-dimensional (2D) crystals for electronic engineering [7]. Elastohydrodynamic flows have been studied in model geometries in order to understand their generic features and the inherent coupling between the driving force from the elastic deformations of the material and the viscous friction force resisting motion [8–16]. If the supporting film is instead of nanoscopic thickness, elastic bending generates a restoring force trying to oppose the van der Waals force that pulls the plate towards the wall and can lead to an elastohydrodynamic touchdown [24] similar to the dewetting of a liquid film [25]. Not much is known about how elastohydrodynamic flows are affected by the ratio between the geometric parameters that characterize the system as it undergoes large changes while the driving force remain the same
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