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Abstract We consider the mapping properties of the integral operator arising in nonlocal slender body theory (SBT) for the model geometry of a straight, periodic filament. It is well known that the classical singular SBT integral operator suffers from high wavenumber instabilities, making it unsuitable for approximating theslender body inverse problem, where the fiber velocity is prescribed and the integral operator must be inverted to find the force density along the fiber. Regularizations of the integral operator must therefore be used instead. Here, we consider two regularization methods: spectral truncation and the‐regularization of Tornberg and Shelley (2004). We compare the mapping properties of these approximations to the underlying partial differential equation (PDE) solution, which for the inverse problem is simply the Stokes Dirichlet problem with data constrained to be constant on cross sections. For the straight‐but‐periodic fiber with constant radius, we explicitly calculate the spectrum of the operator mapping fiber velocity to force for both the PDE and the approximations. We prove that the spectrum of the original SBT operator agrees closely with the PDE operator at low wavenumbers but differs at high frequencies, allowing us to define a truncated approximation with a wavenumber cutoff. For both the truncated and‐regularized approximations, we obtain rigorous‐based convergence to the PDE solution as: A fiber velocity withregularity givesconvergence, while a fiber velocity with at leastregularity yieldsconvergence. Moreover, we determine the dependence of the‐regularized error estimate on the regularization parameter.
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On the variation of bi‐periodic waves in the transverse direction
Abstract Weakly nonlinear, bi‐periodic patterns of waves that propagate in the‐direction with amplitude variation in the‐direction are generated in a laboratory. The amplitude variation in the‐direction is studied within the framework of the vector (vNLSE) and scalar (sNLSE) nonlinear Schrödinger equations using the uniform‐amplitude, Stokes‐like solution of the vNLSE and the Jacobi elliptic sine function solution of the sNLSE. The wavetrains are generated using the Stokes‐like solution of vNLSE; however, a comparison of both predictions shows that while they both do a reasonably good job of predicting the observed amplitude variation in, the comparison with the elliptic function solution of the sNLSE has significantly less error when the ratio of‐wavenumber to the two‐dimensional wavenumber is less than about 0.25. For ratios between about 0.25 and 0.30 (the limit of the experiments), the two models have comparable errors. When the ratio is less than about 0.17, agreement with the vNLSE solution requires a third‐harmonic term in the‐direction, obtained from a Stokes‐type expansion of interacting, symmetric wavetrains. There is no evidence of instability growth in the‐direction, consistent with the work of Segur and colleagues, who showed that dissipation stabilizes the modulational instability. Finally, there is some extra amplitude variation in, which is examined via a qualitative stability calculation that allows symmetry breaking in that direction.
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- Award ID(s):
- 1716159
- PAR ID:
- 10390241
- Publisher / Repository:
- Wiley-Blackwell
- Date Published:
- Journal Name:
- Studies in Applied Mathematics
- Volume:
- 147
- Issue:
- 4
- ISSN:
- 0022-2526
- Page Range / eLocation ID:
- p. 1388-1408
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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