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

Abstract

Steady and unsteady linearised flow past a submerged source are studied in the smallsurface-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.

A note on the Stokes phenomenon in flow under an elastic sheet

“…In [17,18], it was shown that the gravity and capillary waves arise in the linear problem obtained by linearizing around a small step height. This provided an avenue to study these waves in three dimensions using linear theory [26–28]. While we did not consider the linearized elastic sheet problem here, the wave behaviour in this problem can be obtained using a method similar to the corresponding analysis in [18].…”

“…Chapman & Vanden-Broeck subsequently used exponential asymptotics to study two-dimensional nonlinear gravity waves [17] and capillary waves [18]. These two studies led into a number of works that applied related techniques to study other geometries [19–23], gravity–capillary waves [24,25] and waves in three-dimensional domains [26–28]. These studies will be reviewed in more depth in §1a.…”

“…These ideas were extended to three-dimensional unsteady waves in [27], in which the higher-order Stokes phenomenon explains the formation of both transverse and longitudinal gravity waves. More recently, steady and unsteady three-dimensional capillary waves were described in [28].…”

Kelvin wake pattern at small Froude numbers

“…For the present study, the linear problems are for , and , and so the small-Froude-number limits in § 3 are strictly for , or . The details of the corresponding interesting and challenging problem in exponential asymptotics are contained in Lustri & Chapman (2013) and the time-dependent analogue (Lustri & Chapman 2014) which, along with Lustri, Pethiyagoda & Chapman (2019), are the only previous studies of three-dimensional ship waves that use exponential asymptotics in the limit . In terms of future work, one obvious open problem in exponential asymptotics is to perform the equivalent analysis for with , or fixed.…”

Kelvin wake pattern at small Froude numbers

“…For the present study, the linear problems are for 1, µ 1 and δ 1, and so the small-Froude-number limits in §3 are strictly for F 1, µ F 1 or δ F 1. The details of the corresponding interesting and challenging problem in exponential asymptotics are contained in Lustri & Chapman (2013) and the time-dependent analogue (Lustri & Chapman 2014) which, along with Lustri et al (2019), are the only previous studies of three-dimensional ship waves that use exponential asymptotics in the limit F → 0. In terms of future work, one obvious open problem in exponential asymptotics is to perform the equivalent analysis for F → 0 with , µ or δ fixed.…”