The role of exponential asymptotics and complex singularities in self-similarity, transitions, and branch merging of nonlinear dynamics

Chapman, S. Jonathan; Dallaston, Michael C.; Kalliadasis, Serafim; Trinh, Philippe H.; Witelski, Thomas P.

doi:10.1016/j.physd.2023.133802

Cited by 3 publications

(1 citation statement)

References 35 publications

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“…However, due to the symmetry of the problem, it is equivalent to consider half the domain and impose symmetry conditions at the origin: q(0) = 0, τ(0) = −π/2. A similar symmetry condition is used in Chapman et al (2023). Enforcing the condition on τ implies…”

Section: Selection Mechanismmentioning

confidence: 99%

On the selection of Saffman–Taylor viscous fingers for divergent flow in a wedge

Andersen,

Lustri,

McCue

et al. 2024

J. Fluid Mech.

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We study self-similar viscous fingering for the case of divergent flow within a wedge-shaped Hele-Shaw cell. Previous authors have conjectured the existence of a countably infinite number of selected solutions, each distinguished by a different value of the relative finger angle. Interestingly, the associated solution branches have been posited to merge and disappear in pairs as the surface tension decreases. For the first time, we demonstrate how the selection mechanism can be derived based on exponential asymptotics. Asymptotic predictions of the finger-to-wedge angle are additionally given for different sized wedges and surface-tension values. The merging of solution branches is explained; this feature is qualitatively different to the case of classic Saffman–Taylor viscous fingering in a parallel channel configuration. Moreover, because the asymptotic framework does not highly depend on specifics of the wedge geometry, the proposed theory for branch merging in our self-similar problem likely relates much more widely to tip-splitting instabilities in time-dependent flows in circular and other geometries, where the viscous fingers destabilise and divide in two.

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Section: Selection Mechanismmentioning

confidence: 99%

On the selection of Saffman–Taylor viscous fingers for divergent flow in a wedge

Andersen,

Lustri,

McCue

et al. 2024

J. Fluid Mech.

Self Cite

View full text Add to dashboard Cite

show abstract

Applied and computational complex analysis in the study of nonlinear phenomena

Protas,

Llewellyn Smith,

Sakajo

2024

Physica D: Nonlinear Phenomena

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Exponential asymptotics and the generation of free-surface flows by submerged line vortices

Shelton

Trinh

2023

J. Fluid Mech.

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There has been significant recent interest in the study of water waves coupled with non-zero vorticity. We derive analytical approximations for the exponentially small free-surface waves generated in two dimensions by one or several submerged point vortices when driven at low Froude numbers. The vortices are fixed in place, and a boundary-integral formulation in the arclength along the surface allows the study of nonlinear waves and strong point vortices. We demonstrate that, for a single point vortex, techniques in exponential asymptotics prescribe the formation of waves in connection with the presence of Stokes lines originating from the vortex. When multiple point vortices are placed within the fluid, trapped waves may occur, which are confined to lie between the vortices. We also demonstrate that, for the two-vortex problem, the phenomenon of trapped waves occurs for a countably infinite set of values of the Froude number. This work will form a basis for other asymptotic investigations of wave–structure interactions where vorticity plays a key role in the formation of surface waves.

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The role of exponential asymptotics and complex singularities in self-similarity, transitions, and branch merging of nonlinear dynamics

Cited by 3 publications

References 35 publications

On the selection of Saffman–Taylor viscous fingers for divergent flow in a wedge

On the selection of Saffman–Taylor viscous fingers for divergent flow in a wedge

Applied and computational complex analysis in the study of nonlinear phenomena

Exponential asymptotics and the generation of free-surface flows by submerged line vortices

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