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)'.
Highlights
Stokes phenomenon was discovered by George Gabriel Stokes in his analysis of the Airy function [42, 43]
The switching associated with Stokes phenomenon in singularly perturbed problems is exponentially small in the asymptotic limit, and new asymptotic techniques known were required in order to fully capture this switching behaviour
The authors performed an exponential asymptotic analysis of these Stokes curves, and were able to compute the amplitude of these exponentially small waves in the far field. These results clearly showed that exponential asymptotics would be a valuable tool for computing water wave behaviour in various limits, and inspired a number of studies which aimed to extend the picture laid out in [16, 15]
Summary
Stokes phenomenon was discovered by George Gabriel Stokes in his analysis of the Airy function [42, 43]. The switching associated with Stokes phenomenon in singularly perturbed problems is exponentially small in the asymptotic limit, and new asymptotic techniques known were required in order to fully capture this switching behaviour. Rescaling the problem to examine this truncation error allows for direct asymptotic study of these exponentially small terms This idea was used to develop new asymptotic methods for studying exponentially small behaviour, known as exponential asymptotics, asymptotics-beyond-all-orders, or hyperasymptotics. Berry used this idea to study Stokes curves in the Airy function [6]. We suggest a number of interesting future directions arising in the study of flow past elastic sheets
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