Three-dimensional capillary waves due to a submerged source with small surface tension

TL;DR

This paper investigates the generation and evolution of three-dimensional capillary waves caused by a submerged source in a small-surface-tension regime, using exponential asymptotics to analyze steady and unsteady flows without gravity.

Contribution

It applies exponential asymptotics to characterize capillary waves in submerged source flows, including steady and transient behaviors, in the small-surface-tension limit.

Findings

Capillary waves extend upstream from the source, switching on across Stokes lines.

Asymptotic predictions match computational solutions for free surface position.

Transient effects lead to complex wave behaviors, including higher-order Stokes lines.

Abstract

Steady and unsteady linearised flow past a submerged source are studied in the small-surface-tension limit, in the absence of gravitational effects. The free-surface capillary waves generated are exponentially small in the surface tension, and are determined using the theory of exponential asymptotics. In the steady problem, capillary waves are found to extend upstream from the source, switching on across curves on the free surface known as Stokes lines. Asymptotic predictions and compared with computational solutions for the position of the free surface. In the unsteady problem, transient effects cause the solution to display more complicated asymptotic behaviour, such as higher-order Stokes lines. The theory of exponential asymptotics is applied to show how the capillary waves evolve over time, and eventually tend to the steady solution.