Nonlinear Dynamics of Elastic Filaments Conveying a Fluid and Numerical Applications to the Static Kirchhoff Equations

Persistent Link:
http://hdl.handle.net/10150/194164
Title:
Nonlinear Dynamics of Elastic Filaments Conveying a Fluid and Numerical Applications to the Static Kirchhoff Equations
Author:
Beauregard, Matthew Alan
Issue Date:
2008
Publisher:
The University of Arizona.
Rights:
Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.
Abstract:
Two problems in the study of elastic filaments are considered.First, a reliable numerical algorithm is developed that candetermine the shape of a static elastic rod under a variety ofconditions. In this algorithm the governing equations are writtenentirely in terms of local coordinates and are discretized usingfinite differences. The algorithm has two significant advantages:firstly, it can be implemented for a wide variety of the boundaryconditions and, secondly, it enables the user to work with generalconstitutive relationships with only minor changes to thealgorithm. In the second problem a model is presented describingthe dynamics of an elastic tube conveying a fluid. First weanalyze instabilities that are present in a straight rod or tubeunder tension subject to increasing twist in the absence of afluid. As the twist is increased beyond a critical value, thefilament undergoes a twist-to-writhe bifurcation. A multiplescales expansion is used to derive nonlinear amplitude equationsto examine the dynamics of the elastic rod beyond the bifurcationthreshold. This problem is then reinvestigated for an elastic tubeconveying a fluid to study the effect of fluid flow on thetwist-to-writhe instability. A linear stability analysisdemonstrates that for an infinite rod the twist-to-writhethreshold is lowered by the presence of a fluid flow. Amplitudeequations are then derived from which the delay of bifurcation dueto finite tube length is determined. It is shown that the delayedbifurcation threshold depends delicately on the length of the tubeand that it can be either raised or lowered relative to thefluid-free case. The amplitude equations derived for the case of aconstant average fluid flux are compared to the case where theflux depends on the curvature. In this latter case it is shownthat inclusion of curvature results in small changes in some ofthe coefficients in the amplitude equations and has only a smalleffect on the post-bifurcation dynamics.
Type:
text; Electronic Dissertation
Keywords:
Continuum mechanics; Elastic filaments; Elasticity theory; Flow through tubes; Fluid mechanics
Degree Name:
Ph.D.
Degree Level:
doctoral
Degree Program:
Applied Mathematics; Graduate College
Degree Grantor:
University of Arizona
Committee Chair:
Tabor, Michael; Goriely, Alain

Full metadata record

DC FieldValue Language
dc.language.isoENen_US
dc.titleNonlinear Dynamics of Elastic Filaments Conveying a Fluid and Numerical Applications to the Static Kirchhoff Equationsen_US
dc.creatorBeauregard, Matthew Alanen_US
dc.contributor.authorBeauregard, Matthew Alanen_US
dc.date.issued2008en_US
dc.publisherThe University of Arizona.en_US
dc.rightsCopyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.en_US
dc.description.abstractTwo problems in the study of elastic filaments are considered.First, a reliable numerical algorithm is developed that candetermine the shape of a static elastic rod under a variety ofconditions. In this algorithm the governing equations are writtenentirely in terms of local coordinates and are discretized usingfinite differences. The algorithm has two significant advantages:firstly, it can be implemented for a wide variety of the boundaryconditions and, secondly, it enables the user to work with generalconstitutive relationships with only minor changes to thealgorithm. In the second problem a model is presented describingthe dynamics of an elastic tube conveying a fluid. First weanalyze instabilities that are present in a straight rod or tubeunder tension subject to increasing twist in the absence of afluid. As the twist is increased beyond a critical value, thefilament undergoes a twist-to-writhe bifurcation. A multiplescales expansion is used to derive nonlinear amplitude equationsto examine the dynamics of the elastic rod beyond the bifurcationthreshold. This problem is then reinvestigated for an elastic tubeconveying a fluid to study the effect of fluid flow on thetwist-to-writhe instability. A linear stability analysisdemonstrates that for an infinite rod the twist-to-writhethreshold is lowered by the presence of a fluid flow. Amplitudeequations are then derived from which the delay of bifurcation dueto finite tube length is determined. It is shown that the delayedbifurcation threshold depends delicately on the length of the tubeand that it can be either raised or lowered relative to thefluid-free case. The amplitude equations derived for the case of aconstant average fluid flux are compared to the case where theflux depends on the curvature. In this latter case it is shownthat inclusion of curvature results in small changes in some ofthe coefficients in the amplitude equations and has only a smalleffect on the post-bifurcation dynamics.en_US
dc.typetexten_US
dc.typeElectronic Dissertationen_US
dc.subjectContinuum mechanicsen_US
dc.subjectElastic filamentsen_US
dc.subjectElasticity theoryen_US
dc.subjectFlow through tubesen_US
dc.subjectFluid mechanicsen_US
thesis.degree.namePh.D.en_US
thesis.degree.leveldoctoralen_US
thesis.degree.disciplineApplied Mathematicsen_US
thesis.degree.disciplineGraduate Collegeen_US
thesis.degree.grantorUniversity of Arizonaen_US
dc.contributor.chairTabor, Michaelen_US
dc.contributor.chairGoriely, Alainen_US
dc.contributor.committeememberTumin, Anatolien_US
dc.contributor.committeememberBrio, Moyseyen_US
dc.identifier.proquest10127en_US
dc.identifier.oclc659750671en_US
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