Scaling law for a buckled elastic filament in a shear flow
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866912523838554112 |
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| author | Sznajder, Pawel Zdybel, Piotr Liu, Lujia Ekiel-Jezewska, Maria L. |
| author_facet | Sznajder, Pawel Zdybel, Piotr Liu, Lujia Ekiel-Jezewska, Maria L. |
| contents | We analyze the three-dimensional buckling of an elastic filament in a shear flow of a viscous fluid at low Reynolds number and high Peclet number. We apply the Euler-Bernoulli beam (elastica) theoretical model. We show the universal character of the full 3D spectral problem for the small perturbation of the thin filament from a straight position of arbitrary orientation. We use the eigenvalues and eigenfunctions for the linearized elastica equation in the shear plane, found earlier by [Liu et al., 2024] with the Chebyshev spectral collocation method, to solve the full 3D eigenproblem. We provide a simple analytic approximation to the eigenfunctions, represented as Gaussian wavepackets. As the main result of the paper, we derive square-root dependence of the eigenfunction wavenumber on the parameter $\tildeχ=-η\sin 2ϕ\sin^2θ$, where $η$ is the elastoviscous number, and the filament orientation is determined by the zenith angle $θ$ with respect to the vorticity direction and the azimuthal angle $ϕ$ relative to the flow direction. We also compare the eigenfunctions with shapes of slightly buckled elastic filaments with a non-negligible thickness with the same Young's modulus, using the bead model and performing numerical simulations with the precise Hydromultipole numerical codes. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2307_07215 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Scaling law for a buckled elastic filament in a shear flow Sznajder, Pawel Zdybel, Piotr Liu, Lujia Ekiel-Jezewska, Maria L. Fluid Dynamics Soft Condensed Matter We analyze the three-dimensional buckling of an elastic filament in a shear flow of a viscous fluid at low Reynolds number and high Peclet number. We apply the Euler-Bernoulli beam (elastica) theoretical model. We show the universal character of the full 3D spectral problem for the small perturbation of the thin filament from a straight position of arbitrary orientation. We use the eigenvalues and eigenfunctions for the linearized elastica equation in the shear plane, found earlier by [Liu et al., 2024] with the Chebyshev spectral collocation method, to solve the full 3D eigenproblem. We provide a simple analytic approximation to the eigenfunctions, represented as Gaussian wavepackets. As the main result of the paper, we derive square-root dependence of the eigenfunction wavenumber on the parameter $\tildeχ=-η\sin 2ϕ\sin^2θ$, where $η$ is the elastoviscous number, and the filament orientation is determined by the zenith angle $θ$ with respect to the vorticity direction and the azimuthal angle $ϕ$ relative to the flow direction. We also compare the eigenfunctions with shapes of slightly buckled elastic filaments with a non-negligible thickness with the same Young's modulus, using the bead model and performing numerical simulations with the precise Hydromultipole numerical codes. |
| title | Scaling law for a buckled elastic filament in a shear flow |
| topic | Fluid Dynamics Soft Condensed Matter |
| url | https://arxiv.org/abs/2307.07215 |