Low-Mach-number--slenderness limit for elastic Cosserat rods
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2015
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| _version_ | 1866929499688402944 |
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| author | Baus, Franziska Klar, Axel Marheineke, Nicole Wegener, Raimund |
| author_facet | Baus, Franziska Klar, Axel Marheineke, Nicole Wegener, Raimund |
| contents | This paper deals with the relation of the dynamic elastic Cosserat rod model and the Kirchhoff beam equations. We show that the Kirchhoff beam without angular inertia is the asymptotic limit of the Cosserat rod, as the slenderness parameter (ratio between rod diameter and length) and the Mach number (ratio between rod velocity and typical speed of sound) approach zero, i.e. low-Mach-number--slenderness limit. The asymptotic framework is exact up to fourth order in the small parameter and reveals a mathematical structure that allows a uniform handling of the transition regime between the models. To investigate this regime numerically, we apply a scheme that is based on a spatial Gauss-Legendre collocation and an $α$-method in time. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1507_03432 |
| institution | arXiv |
| publishDate | 2015 |
| record_format | arxiv |
| spellingShingle | Low-Mach-number--slenderness limit for elastic Cosserat rods Baus, Franziska Klar, Axel Marheineke, Nicole Wegener, Raimund Analysis of PDEs 74K10, 35Q74, 35B40, 35C20, 65Mxx This paper deals with the relation of the dynamic elastic Cosserat rod model and the Kirchhoff beam equations. We show that the Kirchhoff beam without angular inertia is the asymptotic limit of the Cosserat rod, as the slenderness parameter (ratio between rod diameter and length) and the Mach number (ratio between rod velocity and typical speed of sound) approach zero, i.e. low-Mach-number--slenderness limit. The asymptotic framework is exact up to fourth order in the small parameter and reveals a mathematical structure that allows a uniform handling of the transition regime between the models. To investigate this regime numerically, we apply a scheme that is based on a spatial Gauss-Legendre collocation and an $α$-method in time. |
| title | Low-Mach-number--slenderness limit for elastic Cosserat rods |
| topic | Analysis of PDEs 74K10, 35Q74, 35B40, 35C20, 65Mxx |
| url | https://arxiv.org/abs/1507.03432 |