On Self-Propulsion by Oscillations in a Viscous Liquid
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866916613946605568 |
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| author | Galdi, Giovanni P. Muha, Boris Radošević, Ana |
| author_facet | Galdi, Giovanni P. Muha, Boris Radošević, Ana |
| contents | Suppose that a body $\mathscr B$ can move by translatory motion with velocity $\boldsymbolγ$ in an otherwise quiescent Navier-Stokes liquid, $\mathscr L$, filling the entire space outside $\mathscr B$. Denote by $Ω= Ω(t)$, $t\in\mathbb{R}$, the one-parameter family of bounded, sufficiently smooth domains of $\mathbb{R}^3$, each one representing the configuration of $\mathscr B$ at time $t$ with respect to a frame with the origin at the center of mass $G$ and axes parallel to those of an inertial frame. We assume that there are no external forces acting on the coupled system $\mathscr S := \mathscr B +\mathscr L$ and that the only driving mechanism is a prescribed change in shape of $Ω$ with time. The self-propulsion problem that we would like to address can be thus qualitatively formulated as follows. Suppose that $\mathscr B$ changes its shape in a given time-periodic fashion, namely, $Ω(t+T) = Ω(t)$, for some $T > 0$ and all $t \in \mathbb{R}$. Then, find necessary and sufficient conditions on the map $t\mapsto Ω(t)$ securing that $\mathscr B$ self-propels, that is, $G$ covers any given finite distance in a finite time. We show that this problem is solvable, in a suitable function class, provided the amplitude of the oscillations is below a given constant. Moreover, we provide examples where the propelling velocity of $\mathscr B$ is explicitly evaluated in terms of the physical parameters and the frequency of oscillations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_10009 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Self-Propulsion by Oscillations in a Viscous Liquid Galdi, Giovanni P. Muha, Boris Radošević, Ana Analysis of PDEs 76D05, 35A01, 35B10, 35Q70, 35Q74 Suppose that a body $\mathscr B$ can move by translatory motion with velocity $\boldsymbolγ$ in an otherwise quiescent Navier-Stokes liquid, $\mathscr L$, filling the entire space outside $\mathscr B$. Denote by $Ω= Ω(t)$, $t\in\mathbb{R}$, the one-parameter family of bounded, sufficiently smooth domains of $\mathbb{R}^3$, each one representing the configuration of $\mathscr B$ at time $t$ with respect to a frame with the origin at the center of mass $G$ and axes parallel to those of an inertial frame. We assume that there are no external forces acting on the coupled system $\mathscr S := \mathscr B +\mathscr L$ and that the only driving mechanism is a prescribed change in shape of $Ω$ with time. The self-propulsion problem that we would like to address can be thus qualitatively formulated as follows. Suppose that $\mathscr B$ changes its shape in a given time-periodic fashion, namely, $Ω(t+T) = Ω(t)$, for some $T > 0$ and all $t \in \mathbb{R}$. Then, find necessary and sufficient conditions on the map $t\mapsto Ω(t)$ securing that $\mathscr B$ self-propels, that is, $G$ covers any given finite distance in a finite time. We show that this problem is solvable, in a suitable function class, provided the amplitude of the oscillations is below a given constant. Moreover, we provide examples where the propelling velocity of $\mathscr B$ is explicitly evaluated in terms of the physical parameters and the frequency of oscillations. |
| title | On Self-Propulsion by Oscillations in a Viscous Liquid |
| topic | Analysis of PDEs 76D05, 35A01, 35B10, 35Q70, 35Q74 |
| url | https://arxiv.org/abs/2502.10009 |