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

Abstract

The Stokes phenomenon is a class of asymptotic behaviour that was first discovered by Stokes in his study of the Airy function. It has since been shown that the Stokes phenomenon plays a significant role in the behaviour of surface waves on flows past submerged obstacles. A detailed review of recent research in this area is presented, which outlines the role that the Stokes phenomenon plays in a wide range of free surface flow geometries. The problem of inviscid, irrotational, incompressible flow past a submerged step under a thin elastic sheet is then considered. It is shown that the method for computing this wave behaviour is extremely similar to previous work on computing the behaviour of capillary waves. Exponential asymptotics are used to show that free-surface waves appear on the surface of the flow, caused by singular fluid behaviour in the neighbourhood of the base and top of the step. The amplitude of these waves is computed and compared to numerical simulations, showing excellent agreements between the asymptotic theory and computational solutions. This article is part of the theme issue ‘Stokes at 200 (part 2)’.

Elastic and elastic-gravity waves on flow past submerged obstacles with small bending stiffness

“…Even in linear geometries, these studies demonstrated that there exist interactions between gravitational and capillary effects which affect the surface wave behaviour. Subsequent studies on elastic waves in the absence of gravity [33] found that elastic wave behaviour in the absence of gravity has similar behaviour to the capillary waves studied in [10]. This formulation will allow for direct comparison with the methods of [62], which demonstrated interaction effects between gravity and capillary waves in a similar scaling limit (small surface tension and small Froude number); they found that the wave behaviour changed depending on the parameter regime.…”

“…These results were extended to linearized and nonlinear gravity-capillary waves in two dimensions [62,63], as well as gravity and capillary waves in linearized three-dimensional geometries [31,32,34]. Recently, flexural waves through an elastic sheet in the absence of gravitational effects was studies in [33], using the full nonlinear boundary condition. The present study extends directly on this body of work, exploring elastic wave effects in more detail.…”

“…In the other geime, the two waves decay exponentially in space away from the submerged obstacle. The purpose of this study is to determine whether a similar variety of wave behaviour is obtained by the inclusion of gravity effects into the elastic sheet geometry studied in [33] -we will identify a similar bifurcation structure in flexuralgravity waves. Finally, we will discuss the challenges required to extend this analysis to the nonlinear case, as in [63], and obtain some preliminary asymptotic results.…”

“…In order to study the behaviour of hydroelastic waves, we will adapt the methodology of [62] for the problem of two-dimensional flexural-gravity waves, and [33] for the study of purely elastic waves in a three-dimensional setting. These studies considered exponentially small waves on a free surface due to gravity or capillary effects; in each study, it was found that the waves on the surface existed in regions of the domain bounded by curves known as Stokes curves.…”

Ideal Planar Fluid Flow Over a Submerged Obstacle: Review and Extension

“…At early times, a train of solitons also moves ahead of the obstacle, as apparently first reported by Ertekin et al [11], but moves infinitely far upstream leaving a time-independent region of almost uniform flow ahead of the bump. This advancing soliton packet is believed not to alter conditions far ahead of the obstacle, although this has not been established definitively, and may perhaps be resolved using the technique of exponential asymptotics, as reviewed by Lustri et al [36]. An interesting feature of steady-state subcritical flows is that a bottom bump of appropriate height and length may give a drag-free flow in which the downstream waves have cancelled precisely, even in the nonlinear case, as discussed for the linearized solution in Figure 3(b).…”