Regularity of Steady Periodic Capillary Water Waves with Constant Vorticity

Martin, Calin Iulian

doi:10.1142/s1402925112400062

Cited by 9 publications

(6 citation statements)

References 21 publications

(28 reference statements)

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“…Moreover, if the Bernoulli function and the streamline density function are both real analytic functions, then all of the streamlines, including the wave profile, are real analytic for all three physical regimes. These results concur with the recent literature determining the a priori regularity of water waves with vorticity [4,8,15,16,27,28,30,31].…”

Section: The Mathematical Modelsupporting

confidence: 92%

Global bifurcation of capillary–gravity-stratified water waves

Henry

Matioc

2014

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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We study steady periodic water waves with variable vorticity and density, where we allow for stagnation points in the flow, and where we admit the capillarity effects of surface tension. Using global bifurcation theory, we extend a local curve of non-trivial solutions of the governing equations to a global continuum of solutions. Furthermore, we obtain a description of the behaviour of the solutions along this continuum.

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Section: The Mathematical Modelsupporting

confidence: 92%

Global bifurcation of capillary–gravity-stratified water waves

Henry

Matioc

2014

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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“…However, the majority of the literature is focused on the pure-gravity wave scenario (that is, where the governing force is gravitational in nature and the effects of surface tension are neglected) and it is only comparatively recently that the pure-capillary (where gravitational forces are neglected) and capillary-gravity (considering both gravitational and surface tension forces) cases have been studied in great detail. This has yielded results concerning existence of both small-amplitude [14,18,19,28,29] and large-amplitude [22,30] waves of both pure-capillary and capillary-gravity type, as well as other interesting results [20,21].…”

Section: Introductionmentioning

confidence: 93%

“…Focusing specifically on those works that consider constant (non-zero) vorticity [8,9,18,19,20,21], as in this work, the common approach has been to adapt the method of conformal mappings from the irrotational setting. As these many works demonstrate, this reformulation on a fixed domain is an extension of the corresponding irrotational formulation, with extra terms appearing exactly due to the influence of vorticity.…”

Section: Introductionmentioning

confidence: 99%

Global bifurcation for steady finite-depth capillary-gravity waves with constant vorticity

de Boeck

2014

Preprint

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This paper is concerned with two-dimensional, steady, periodic water waves propagating at the free surface of water in a flow of constant vorticity over an impermeable flat bed. The motion of these waves is assumed to be governed both by surface tension and gravitational forces. A new reformulation of this problem is given, which is valid without restriction on the geometry or amplitude of the wave profiles and possesses a structure amenable to global bifurcation. The existence of global curves and continua of solutions bifurcating at certain parameter values from flat, laminar flows is then established through the application of real-analytic global bifurcation in the spirit of Dancer and the degree-theoretic global bifurcation theorem of Rabinowitz, respectively.

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“…Historically, and in more recent years, there have been many studies of this pure-capillary problem, for both the finite-and infinite-depth scenarios (e.g. [3,8,10,11,14]). This particular problem is mainly motivated by the observation that in many situations (such as when the wind begins blowing over the flat, still surface of a body of water [8]) small-amplitude wave trains can arise that are governed principally by capillarity.…”

Section: Introductionmentioning

confidence: 99%

Existence of capillary-gravity waves that are perturbations of Crapper's waves

Boeck¹

2014

Preprint

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This paper is concerned with two-dimensional, steady, periodic water waves propagating at the free surface of water either in a flow of finite depth and constant vorticity over an impermeable flat bed or in an irrotational flow of great depth. In both cases, the motion of these waves is assumed to be governed both by surface tension and gravitational forces. By considering a particular scaling regime, it can be shown that both these schemes approach a limiting form, corresponding to flows of great depth where the effect of gravity is neglected. An application of the Implicit Function Theorem demonstrates that Crapper's explicit solutions to this latter problem can be perturbed by means of the aforementioned scaling to yield capillary-gravity waves of any finite (or infinite) depth and any constant vorticity.

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Regularity of Steady Periodic Capillary Water Waves with Constant Vorticity

Abstract: We prove a regularity result for steady periodic travelling capillary waves of small amplitude at the free surface of water in a flow with constant vorticity over a flat bed.

Cited by 9 publications

References 21 publications

Global bifurcation of capillary–gravity-stratified water waves

Global bifurcation of capillary–gravity-stratified water waves

Global bifurcation for steady finite-depth capillary-gravity waves with constant vorticity

Existence of capillary-gravity waves that are perturbations of Crapper's waves

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