Extensible membranes in inviscid flow: aerodynamics and singular limits
Abstract
We develop a two-dimensional inviscid model for extensible membranes with continuous vortex shedding from both edges, coupled to a weak spectral Galerkin discretization of the membrane equation that permits direct computation at zero bending rigidity. Viewing the pressure jump as a flux for bound vorticity explains how membranes respond to concentrated fluid loading. For fixed-fixed membranes, compliance draws the body toward the leading-edge vortex, delaying its detachment and enhancing lift primarily during start-up; after periodic shedding develops, compliance mainly amplifies the force oscillations rather than mean lift. Vortex detachment can unload heavy membranes into compression, producing bending-regularized wrinkles. For R2>0, the membrane equation remains well-posed independently of the sign of the tension, provided the membrane remains an immersion. Releasing the trailing edge creates an intrinsic degeneracy: the tension vanishes at the free endpoint, making the R2=0 problem formally underdetermined, while the weak formulation selects its bounded-energy branch. This vanishing-tension endpoint traps pressure disturbances and drives rapid tip-snapping; the dominant frequency and effective wave number approach their zero-rigidity limits with an inverse-logarithmic correction arising from a singular Bessel branch. Finally, decreasing the stretching modulus destabilizes periodic flutter, and the repeated fluid-driven departure and elastic return suggest a homoclinic tangle underlying the transition from periodic to chaotic flutter.
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BibTeX
@misc{loo2026extensible,
title = {{Extensible membranes in inviscid flow: aerodynamics and singular limits}},
author = {Yu Jun Loo and Silas Alben},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16484},
eprint = {2609.16484},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16484},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Extensible membranes in inviscid flow: aerodynamics and singular limits
AU - Yu Jun Loo
AU - Silas Alben
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.16484
UR - https://arxiv.org/abs/2609.16484
AB - We develop a two-dimensional inviscid model for extensible membranes with continuous vortex shedding from both edges, coupled to a weak spectral Galerkin discretization of the membrane equation that permits direct computation at zero bending rigidity. Viewing the pressure jump as a flux for bound vorticity explains how membranes respond to concentrated fluid loading. For fixed-fixed membranes, compliance draws the body toward the leading-edge vortex, delaying its detachment and enhancing lift primarily during start-up; after periodic shedding develops, compliance mainly amplifies the force oscillations rather than mean lift. Vortex detachment can unload heavy membranes into compression, producing bending-regularized wrinkles. For R_2>0, the membrane equation remains well-posed independently of the sign of the tension, provided the membrane remains an immersion. Releasing the trailing edge creates an intrinsic degeneracy: the tension vanishes at the free endpoint, making the R_2=0 problem formally underdetermined, while the weak formulation selects its bounded-energy branch. This vanishing-tension endpoint traps pressure disturbances and drives rapid tip-snapping; the dominant frequency and effective wave number approach their zero-rigidity limits with an inverse-logarithmic correction arising from a singular Bessel branch. Finally, decreasing the stretching modulus destabilizes periodic flutter, and the repeated fluid-driven departure and elastic return suggest a homoclinic tangle underlying the transition from periodic to chaotic flutter.
ER -