Unsteady flow over a submerged source with low Froude number

Lustri, Christopher J.; Chapman, S. Jonathan

doi:10.1017/s0956792514000217

Cited by 12 publications

(13 citation statements)

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“…This work also used exponential asymptotics to describe transient ripples that are not present in steady variants of this free-surface wave problem. 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].…”

Section: Introductionmentioning

confidence: 99%

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

Lustri

Koens

Pethiyagoda

2020

Phil. Trans. R. Soc. A.

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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)’.

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Section: Introductionmentioning

confidence: 99%

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

Lustri

Koens

Pethiyagoda

2020

Phil. Trans. R. Soc. A.

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“…These Stokes lines emerge from singularities in the analytical continuation of the leading order term and typically intersect the part of the complex plane corresponding to the physical problem of interest (which is the real axis in many problems, but the unit circle in our problem). Note that this method has been successfully applied to a variety of problems in fluid mechanics, including two-and three-dimensional water waves [13,37,38,39,40,57,58]. Finally, the third key idea is that of selection.…”

Section: Introductionmentioning

confidence: 99%

Selection of a Hele-Shaw Bubble via Exponential Asymptotics

Lustri¹,

Green²,

McCue³

2020

SIAM J. Appl. Math.

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The well-studied selection problems involving Saffman-Taylor fingers or Taylor-Saffman bubbles in a Hele-Shaw channel are prototype examples of pattern selection. Exact solutions to the corresponding zero-surface-tension problems exist for an arbitrary finger or bubble speed, but the addition of surface tension leads to a discrete set of solution branches, all of which approach a single solution in the limit the surface tension vanishes. In this sense, the surface tension selects a single physically meaningful solution from the continuum of zero-surface-tension solutions. Recently, we provided numerical evidence to suggest the selection problem for a bubble propagating in an unbounded Hele-Shaw cell behaves in an analogous way to other finger and bubble problems in a Hele-Shaw channel; however, the selection of the ratio of bubble speeds to background velocity appears to follow a very different surface tension scaling to the channel cases. Here we apply techniques in exponential asymptotics to solve the selection problem analytically, confirming the numerical results, including the predicted surface tension scaling laws. Further, our analysis sheds light on the multiple tips in the shape of the bubbles along solution branches, which appear to be caused by switching on and off exponentially small wavelike contributions across Stokes lines in a conformally mapped plane. These results have ramifications for exotic-shaped Saffman-Taylor fingers as well as for the time-dependent evolution of bubbles propagating in Hele-Shaw cells.

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“…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.…”

Section: Discussionmentioning

confidence: 99%