On the Propulsion of a Rigid Body in a Viscous Liquid Under the Action of a Time-Periodic Force
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912671204376576 |
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| author | Karakouzian, Mher Marc Galdi, Giovanni P. |
| author_facet | Karakouzian, Mher Marc Galdi, Giovanni P. |
| contents | A rigid body $\mathcal{B}$ moves in an otherwise quiescent viscous liquid filling the whole space outside $\mathcal{B}$, under the action of a time-periodic force $\boldsymbol{\mathsf{f}}$ of period $T$ applied to a given point of $\mathcal{B}$ and of fixed direction. We assume that the average of $\boldsymbol{\mathsf{f}}$ over an interval of length $T$ does not not vanish, and that the amplitude, $δ$, of $\boldsymbol{\mathsf{f}}$ is sufficiently small. Our goal is to investigate when $\mathcal{B}$ executes a non-zero net motion; that is, $\mathcal{B}$ is able to cover any prescribed distance in a finite time. We show that, at the order $δ$, this happens if and only if $\boldsymbol{\mathsf{f}}$ and $\mathcal{B}$ satisfy a certain condition. We also show that this is always the case if $\mathcal{B}$ is prevented from spinning. Finally, we provide explicit examples where the condition above is satisfied or not. All our analysis is performed in a general class of weak solutions to the coupled system body-liquid problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_07448 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Propulsion of a Rigid Body in a Viscous Liquid Under the Action of a Time-Periodic Force Karakouzian, Mher Marc Galdi, Giovanni P. Analysis of PDEs Fluid Dynamics A rigid body $\mathcal{B}$ moves in an otherwise quiescent viscous liquid filling the whole space outside $\mathcal{B}$, under the action of a time-periodic force $\boldsymbol{\mathsf{f}}$ of period $T$ applied to a given point of $\mathcal{B}$ and of fixed direction. We assume that the average of $\boldsymbol{\mathsf{f}}$ over an interval of length $T$ does not not vanish, and that the amplitude, $δ$, of $\boldsymbol{\mathsf{f}}$ is sufficiently small. Our goal is to investigate when $\mathcal{B}$ executes a non-zero net motion; that is, $\mathcal{B}$ is able to cover any prescribed distance in a finite time. We show that, at the order $δ$, this happens if and only if $\boldsymbol{\mathsf{f}}$ and $\mathcal{B}$ satisfy a certain condition. We also show that this is always the case if $\mathcal{B}$ is prevented from spinning. Finally, we provide explicit examples where the condition above is satisfied or not. All our analysis is performed in a general class of weak solutions to the coupled system body-liquid problem. |
| title | On the Propulsion of a Rigid Body in a Viscous Liquid Under the Action of a Time-Periodic Force |
| topic | Analysis of PDEs Fluid Dynamics |
| url | https://arxiv.org/abs/2410.07448 |