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arXiv · 1704.00452

Navier-Stokes flow past a rigid body: attainability of steady solutions as limits of unsteady weak solutions, starting and landing cases

Toshiaki Hishida·Paolo Maremonti
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Abstract

Consider the Navier-Stokes flow in 3-dimensional exterior domains, where a rigid body is translating with prescribed translational velocity $-h(t)u_\infty$ with constant vector $u_\infty\in \mathbb R^3\setminus\{0\}$. Finn raised the question whether his steady slutions are attainable as limits for $t\to\infty$ of unsteady solutions starting from motionless state when $h(t)=1$ after some finite time and $h(0)=0$ (starting problem). This was affirmatively solved by Galdi, Heywood and Shibata for small $u_\infty$. We study some generalized situation in which unsteady solutions start from large motions being in $L^3$. We then conclude that the steady solutions for small $u_\infty$ are still attainable as limits of evolution of those fluid motions which are found as a sort of weak solutions. The opposite situation, in which $h(t)=0$ after some finite time and $h(0)=1$ (landing problem), is also discussed. In this latter case, the rest state is attainable no matter how large $u_\infty$ is.

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Toshiaki Hishida, Paolo Maremonti. 2017-04-03. Navier-Stokes flow past a rigid body: attainability of steady solutions as limits of unsteady weak solutions, starting and landing cases. https://doi.org/10.1007/s00021-017-0344-3

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