Wake and wave resistance on viscous thin films

TL;DR

This paper theoretically investigates how an external pressure disturbance moving on a viscous thin film creates a wake and affects wave resistance, revealing how size, speed, and geometry influence these phenomena.

Contribution

It provides a detailed analysis of wake formation and wave resistance in viscous thin films, highlighting the effects of disturbance size, speed, and dimensionality, which were not fully understood before.

Findings

Wave resistance increases with speed up to a transition point

For finite disturbances, wave resistance behavior depends on size and speed

In 2D, wave resistance diverges for a point source

Abstract

The effect of an external pressure disturbance, being displaced with a constant speed along the free surface of a viscous thin film, is studied theoretically in the lubrication approximation in one- and two-dimensional geometries. In the comoving frame, the imposed pressure field creates a stationary deformation of the interface - a wake - that spatially vanishes in the far region. The shape of the wake and the way it vanishes depend on both the speed and size of the external source and the properties of the film. The wave resistance, namely the force that has to be externally furnished in order to maintain the wake, is analysed in details. For finite-size pressure disturbances, it increases with the speed, up to a certain transition value above which a monotonic decrease occurs. The role of the horizontal extent of the pressure field is studied as well, revealing that for a smaller disturbance the latter transition occurs at higher speed. Eventually, for a Dirac pressure source, the wave resistance either saturates in a 1D geometry, or diverges in a 2D geometry.