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Existence and Spatial Decay of Periodic Navier-Stokes Flows in Exterior Domains

Eiter, Thomas (2020):
Existence and Spatial Decay of Periodic Navier-Stokes Flows in Exterior Domains.
Darmstadt, Logos Verlag Berlin, ISBN 978-3-8325-5108-7,
DOI: 10.25534/tuprints-00011629,
[Book]

Abstract

A classical problem in the field of mathematical fluid mechanics is the flow of a viscous incompressible fluid past a rigid body. In his doctoral thesis, Thomas Walter Eiter investigates time-periodic solutions to the associated Navier--Stokes equations when the body performs a non-trivial translation. The first part of the thesis is concerned with the question of existence of time-periodic solutions in the case of a non-rotating and of a rotating obstacle. Based on an investigation of the corresponding Oseen linearizations, new existence results in suitable function spaces are established. The second part deals with the study of spatially asymptotic properties of time-periodic solutions. For this purpose, time-periodic fundamental solutions to the Stokes and Oseen linearizations are introduced and investigated, and the concept of a time-periodic fundamental solution for the vorticity field is developed. With these results, new pointwise estimates of the velocity and the vorticity field associated to a time-periodic fluid flow are derived.

Item Type: Book
Erschienen: 2020
Creators: Eiter, Thomas
Title: Existence and Spatial Decay of Periodic Navier-Stokes Flows in Exterior Domains
Language: English
Abstract:

A classical problem in the field of mathematical fluid mechanics is the flow of a viscous incompressible fluid past a rigid body. In his doctoral thesis, Thomas Walter Eiter investigates time-periodic solutions to the associated Navier--Stokes equations when the body performs a non-trivial translation. The first part of the thesis is concerned with the question of existence of time-periodic solutions in the case of a non-rotating and of a rotating obstacle. Based on an investigation of the corresponding Oseen linearizations, new existence results in suitable function spaces are established. The second part deals with the study of spatially asymptotic properties of time-periodic solutions. For this purpose, time-periodic fundamental solutions to the Stokes and Oseen linearizations are introduced and investigated, and the concept of a time-periodic fundamental solution for the vorticity field is developed. With these results, new pointwise estimates of the velocity and the vorticity field associated to a time-periodic fluid flow are derived.

Place of Publication: Darmstadt
Publisher: Logos Verlag Berlin
ISBN: 978-3-8325-5108-7
Divisions: 04 Department of Mathematics
04 Department of Mathematics > Analysis
04 Department of Mathematics > Analysis > Partial Differential Equations and Applications
Date Deposited: 15 Jul 2020 10:44
DOI: 10.25534/tuprints-00011629
URL / URN: https://tuprints.ulb.tu-darmstadt.de/11629
URN: urn:nbn:de:tuda-tuprints-116294
PPN:
Referees: Farwig, Prof. Dr. Reinhard ; Kyed, Prof. Dr. Mads ; Galdi, Prof. Dr. Giovanni Paolo
Refereed / Verteidigung / mdl. Prüfung: 27 February 2020
Alternative Abstract:
Alternative abstract Language

Ein klassisches Problem im Bereich der mathematischen Strömungsmechanik ist der Fluss eines viskosen inkompressiblen Fluids entlang eines starren Körpers. In seiner Doktorarbeit untersucht Thomas Walter Eiter zeitperiodische Lösungen der zugehörigen Navier-Stokes-Gleichungen, wenn der Körper eine nichttriviale Translation vollzieht. Der erste Teil der Arbeit beschäftigt sich mit der Frage der Existenz von zeitperiodischen Lösungen im Fall eines nichtrotierenden und eines rotierenden Körpers. Basierend auf einer Untersuchung der zugehörigen Oseen-Linearisierungen werden neue Existenzresultate in geeigneten Funktionenräumen bewiesen. Der zweite Teil behandelt die Untersuchung von räumlich-asymptotischen Eigenschaften von zeitperiodischen Lösungen. Hierzu werden zeitperiodische Fundamentallösungen der Stokes- und Oseen-Linearisierungen eingeführt und untersucht sowie das Konzept einer zeitperiodischen Fundamentallösung für das Wirbelfeld entwickelt. Mit diesen Ergebnissen werden neue punktweise Abschätzungen von Geschwindigkeits- und Wirbelfeld einer zeitperiodischen Strömung hergeleitet.

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