A Kinetic Model for the Sedimentation of Rod--Like Particles

Handle URI:
http://hdl.handle.net/10754/583111
Title:
A Kinetic Model for the Sedimentation of Rod--Like Particles
Authors:
Helzel, Christiane; Tzavaras, Athanasios ( 0000-0002-1896-2270 )
Abstract:
We consider a coupled system consisting of a kinetic equation coupled to a macroscopic Stokes (or Navier-Stokes) equation and describing the motion of a suspension of rigid rods in gravity. A reciprocal coupling leads to the formation of clusters: The buoyancy force creates a macroscopic velocity gradient that causes the microscopic particles to align so that their sedimentation reinforces the formation of clusters of higher particle density. We provide a quantitative analysis of cluster formation. We derive a nonlinear moment closure model, which consists of evolution equations for the density and second order moments and that uses the structure of spherical harmonics to suggest a closure strategy. For a rectilinear flow we employ the moment closure together with a quasi-dynamic approximation to derive an effective equation. The effective equation is an advectiondiffusion equation with nonisotropic diffusion coupled to a Poisson equation, and belongs to the class of the so-called flux-limited Keller-Segel models. For shear flows, we provide an argument for the validity of the effective equation and perform numerical comparisons that indicate good agreement between the original system and the effective theory. For rectilinear flow we show numerical results which indicate that the quasi-dynamic provides accurate approximations. Finally, a linear stability analysis on the moment system shows that linear theory predicts a wavelength selection mechanism for the cluster width, provided that the Reynolds number is larger than zero.
KAUST Department:
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Citation:
Helzel C, Tzavaras AE (2017) A Kinetic Model for the Sedimentation of Rod--Like Particles. Multiscale Modeling & Simulation 15: 500–536. Available: http://dx.doi.org/10.1137/15M1023907.
Publisher:
Society for Industrial & Applied Mathematics (SIAM)
Journal:
Multiscale Modeling & Simulation
Issue Date:
12-Oct-2015 ; 23-Mar-2017
DOI:
10.1137/15M1023907; 10.1137/15M1023907
ARXIV:
arXiv:1510.03235
Type:
Article
ISSN:
1540-3459; 1540-3467
Sponsors:
The authors express their thanks to Professor Felix Otto for his valuable comments and his involvement in several discussions concerning this work, and to Professor Jian-Guo Liu for helpful remarks.
Additional Links:
http://epubs.siam.org/doi/10.1137/15M1023907
Appears in Collections:
Articles; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

Full metadata record

DC FieldValue Language
dc.contributor.authorHelzel, Christianeen
dc.contributor.authorTzavaras, Athanasiosen
dc.date.accessioned2017-05-01T14:01:23Z-
dc.date.available2015-12-02T12:48:50Zen
dc.date.available2017-05-01T14:01:23Z-
dc.date.issued2015-10-12-
dc.date.issued2017-03-23en
dc.identifier.citationHelzel C, Tzavaras AE (2017) A Kinetic Model for the Sedimentation of Rod--Like Particles. Multiscale Modeling & Simulation 15: 500–536. Available: http://dx.doi.org/10.1137/15M1023907.en
dc.identifier.issn1540-3459en
dc.identifier.issn1540-3467en
dc.identifier.doi10.1137/15M1023907-
dc.identifier.doi10.1137/15M1023907en
dc.identifier.urihttp://hdl.handle.net/10754/583111-
dc.description.abstractWe consider a coupled system consisting of a kinetic equation coupled to a macroscopic Stokes (or Navier-Stokes) equation and describing the motion of a suspension of rigid rods in gravity. A reciprocal coupling leads to the formation of clusters: The buoyancy force creates a macroscopic velocity gradient that causes the microscopic particles to align so that their sedimentation reinforces the formation of clusters of higher particle density. We provide a quantitative analysis of cluster formation. We derive a nonlinear moment closure model, which consists of evolution equations for the density and second order moments and that uses the structure of spherical harmonics to suggest a closure strategy. For a rectilinear flow we employ the moment closure together with a quasi-dynamic approximation to derive an effective equation. The effective equation is an advectiondiffusion equation with nonisotropic diffusion coupled to a Poisson equation, and belongs to the class of the so-called flux-limited Keller-Segel models. For shear flows, we provide an argument for the validity of the effective equation and perform numerical comparisons that indicate good agreement between the original system and the effective theory. For rectilinear flow we show numerical results which indicate that the quasi-dynamic provides accurate approximations. Finally, a linear stability analysis on the moment system shows that linear theory predicts a wavelength selection mechanism for the cluster width, provided that the Reynolds number is larger than zero.en
dc.description.sponsorshipThe authors express their thanks to Professor Felix Otto for his valuable comments and his involvement in several discussions concerning this work, and to Professor Jian-Guo Liu for helpful remarks.en
dc.language.isoenen
dc.publisherSociety for Industrial & Applied Mathematics (SIAM)en
dc.relation.urlhttp://epubs.siam.org/doi/10.1137/15M1023907en
dc.rightsArchived with thanks to Multiscale Modeling & Simulationen
dc.subjectLinear stabilityen
dc.subjectMoment closureen
dc.subjectQuasi-dynamic approximationen
dc.subjectRod-like particlesen
dc.subjectSedimentationen
dc.titleA Kinetic Model for the Sedimentation of Rod--Like Particlesen
dc.typeArticleen
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
dc.identifier.journalMultiscale Modeling & Simulationen
dc.eprint.versionPublisher's Version/PDFen
dc.contributor.institutionInstitute of Mathematics, Heinrich-Heine-University Dusseldorf, Dusseldorf, 40225, , Germanyen
dc.contributor.institutionInstitute of Applied and Computational Mathematics, FORTH, Heraklion GR, 71409, , Greeceen
dc.contributor.affiliationKing Abdullah University of Science and Technology (KAUST)en
dc.identifier.arxividarXiv:1510.03235-
kaust.authorTzavaras, Athanasiosen

Version History

VersionItem Editor Date Summary
2 10754/583111grenzdm2017-05-01 14:00:44.657Article published. Publisher version allowed to be deposited.
1 10754/583111.1wangh0e2015-12-02 12:48:50.0
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