Exponential asymptotics and Stokes lines in a partial differential equation

Chapman, S. Jonathan; Mortimer, David B

doi:10.1098/rspa.2005.1475

Cited by 41 publications

(99 citation statements)

References 11 publications

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“…We find that 10) in the neighbourhood of ζ 0 as n → ∞, where α and λ depend on whether ζ 0 is −1 or −b (through (3.8)-(3.9)), and µ n and ν n are polynomials in α of order n − 1 with a constant term (n − 1)!. Thus, in the neighbourhood of ζ = −1 the late order terms grow in the familiar factorial/power rate when θ I > −π/3 (forcing 3α − 1 < 0), while in the neighbourhood of ζ = −b we encounter the same type of divergence when θ I < π/3 (again forcing 3α − 1 < 0).…”

Section: Late-order Termsmentioning

confidence: 75%

“…It is worth noting, however, that for the (nonlinear) crystal growth model proposed in [6] and studied using beyond all orders techniques by [9,24] for example, linear time-dependent generalisations have been treated successfully by [5,10,21]. The extent to which all these techniques can be applied to fully nonlinear time-dependent free surface flows is as yet untested.…”

Section: Discussionmentioning

confidence: 99%

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Free surface flow past topography: A beyond-all-orders approach

2012

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The problem of steady subcritical free surface flow past a submerged inclined step is considered. The asymptotic limit of small Froude number is treated, with particular emphasis on the effect that changing the angle of the step face has on the surface waves. As demonstrated by Chapman & Vanden-Broeck, (2006) Exponential asymptotics and gravity waves. J. Fluid Mech. 567, 299-326, the divergence of a power series expansion in powers of the square of the Froude number is caused by singularities in the analytic continuation of the free surface; for an inclined step, these singularities may correspond to either the corners or stagnation points of the step, or both, depending on the angle of inclination. Stokes lines emanate from these singularities, and exponentially small waves are switched on at the point the Stokes lines intersect with the free surface. Our results suggest that for a certain range of step angles, two wavetrains are switched on, but the exponentially subdominant one is switched on first, leading to an intermediate wavetrain not previously noted. We extend these ideas to the problem of flow over a submerged bump or trench, again with inclined sides. This time there may be two, three or four active Stokes lines, depending on the inclination angles. We demonstrate how to construct a base topography such that wave contributions from separate Stokes lines are of equal magnitude but opposite phase, thus cancelling out. Our asymptotic results are complemented by numerical solutions to the fully nonlinear equations.

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Section: Late-order Termsmentioning

confidence: 75%

Section: Discussionmentioning

confidence: 99%

Free surface flow past topography: A beyond-all-orders approach

2012

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“…The exponential asymptotic methodology of Olde Daalhuis et al (1995) and Chapman et al (1998) was developed for investigating ordinary differential equations. Because our free surface is two-dimensional, we will require the extension of these techniques to partial differential equations which was developed by Chapman & Mortimer (2005).…”

Section: Methodsmentioning

confidence: 99%

Steady gravity waves due to a submerged source

Lustri

Chapman

2013

J. Fluid Mech.

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In the low-Froude-number limit, free-surface gravity waves caused by flow past a submerged obstacle have amplitude that is exponentially small. Consequently, these cannot be represented using an asymptotic series expansion. Steady linearized flow past a submerged source is considered, and exponential asymptotic methods are applied to determine the behaviour of the free-surface gravity waves. The free surface is found to contain longitudinal and transverse waves that switch on rapidly across curves known as Stokes lines on the free surface. The longitudinal waves are present everywhere downstream of the singularity, while the transverse waves are restricted to two downstream wedges. As the depth of the source approaches the surface, the familiar Kelvin-wedge wave behaviour is recovered.

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“…It is well known that in the case of gravity-capillary flows, the equivalent exponential asymptotics is a great deal more complicated [31,32]. More generally, the study of time-dependent or multi-dimensional problems using exponential asymptotics continues to pose significant challenges [33][34][35][36], particularly for the case of nonlinear phenomena.…”

Section: Discussionmentioning

confidence: 99%

A topological study of gravity free-surface waves generated by bluff bodies using the method of steepest descents

Trinh¹

2016

Proc. R. Soc. A.

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The standard analytical approach for studying steady gravity free-surface waves generated by a moving body often relies upon a linearization of the physical geometry, where the body is considered asymptotically small in one or several of its dimensions. In this paper, a methodology that avoids any such geometrical simplification is presented for the case of steady-state flows at low speeds. The approach is made possible through a reduction of the water-wave equations to a complex-valued integral equation that can be studied using the method of steepest descents. The main result is a theory that establishes a correspondence between different bluff-bodied free-surface flow configurations, with the topology of the Riemann surface formed by the steepest descent paths. Then, when a geometrical feature of the body is modified, a corresponding change to the Riemann surface is observed, and the resultant effects to the water waves can be derived. This visual procedure is demonstrated for the case of two-dimensional free-surface flow past a surface-piercing ship and over an angled step in a channel.

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Exponential asymptotics and Stokes lines in a partial differential equation

Cited by 41 publications

References 11 publications

Free surface flow past topography: A beyond-all-orders approach

Free surface flow past topography: A beyond-all-orders approach

Steady gravity waves due to a submerged source

A topological study of gravity free-surface waves generated by bluff bodies using the method of steepest descents

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