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

dc.contributor.affiliationKing Abdullah University of Science and Technology (KAUST)
dc.contributor.authorHelzel, Christiane
dc.contributor.authorTzavaras, Athanasios
dc.contributor.departmentApplied Mathematics and Computational Science Program
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
dc.contributor.institutionInstitute of Mathematics, Heinrich-Heine-University Dusseldorf, Dusseldorf, 40225, , Germany
dc.contributor.institutionInstitute of Applied and Computational Mathematics, FORTH, Heraklion GR, 71409, , Greece
dc.date.accessioned2017-05-01T14:01:23Z
dc.date.available2015-12-02T12:48:50Z
dc.date.available2017-05-01T14:01:23Z
dc.date.issued2017-03-23
dc.date.posted2015-10-12
dc.date.published-online2017-03-23
dc.date.published-print2017-01
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.
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.
dc.eprint.versionPublisher's Version/PDF
dc.identifier.arxivid1510.03235
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.
dc.identifier.doi10.1137/15M1023907
dc.identifier.doi10.1137/15M1023907
dc.identifier.issn1540-3459
dc.identifier.issn1540-3467
dc.identifier.journalMultiscale Modeling & Simulation
dc.identifier.urihttp://hdl.handle.net/10754/583111
dc.language.isoen
dc.publisherSociety for Industrial & Applied Mathematics (SIAM)
dc.relation.urlhttp://epubs.siam.org/doi/10.1137/15M1023907
dc.rightsArchived with thanks to Multiscale Modeling & Simulation
dc.subjectLinear stability
dc.subjectMoment closure
dc.subjectQuasi-dynamic approximation
dc.subjectRod-like particles
dc.subjectSedimentation
dc.titleA Kinetic Model for the Sedimentation of Rod--Like Particles
dc.typeArticle
dc.versionv1
display.details.left<span><h5>Type</h5>Article<br><br><h5>Authors</h5><a href="https://repository.kaust.edu.sa/search?spc.sf=dc.date.issued&spc.sd=DESC&f.author=Helzel, Christiane,equals">Helzel, Christiane</a><br><a href="https://repository.kaust.edu.sa/search?query=orcid.id:0000-0002-1896-2270&spc.sf=dc.date.issued&spc.sd=DESC">Tzavaras, Athanasios</a> <a href="https://orcid.org/0000-0002-1896-2270" target="_blank"><img src="https://repository.kaust.edu.sa/server/api/core/bitstreams/82a625b4-ed4b-40c8-865a-d6a5225a26a4/content" width="16" height="16"/></a><br><br><h5>KAUST Department</h5><a href="https://repository.kaust.edu.sa/search?spc.sf=dc.date.issued&spc.sd=DESC&f.department=Applied Mathematics and Computational Science Program,equals">Applied Mathematics and Computational Science Program</a><br><a href="https://repository.kaust.edu.sa/search?spc.sf=dc.date.issued&spc.sd=DESC&f.department=Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division,equals">Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division</a><br><br><h5>Preprint Posting Date</h5>2015-10-12<br><br><h5>Online Publication Date</h5>2017-03-23<br><br><h5>Print Publication Date</h5>2017-01<br><br><h5>Date</h5>2017-03-23</span>
display.details.right<span><h5>Abstract</h5>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.<br><br><h5>Citation</h5>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.<br><br><h5>Acknowledgements</h5>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.<br><br><h5>Publisher</h5><a href="https://repository.kaust.edu.sa/search?spc.sf=dc.date.issued&spc.sd=DESC&f.publisher=Society for Industrial & Applied Mathematics (SIAM),equals">Society for Industrial & Applied Mathematics (SIAM)</a><br><br><h5>Journal</h5><a href="https://repository.kaust.edu.sa/search?spc.sf=dc.date.issued&spc.sd=DESC&f.journal=Multiscale Modeling & Simulation,equals">Multiscale Modeling & Simulation</a><br><br><h5>DOI</h5><a href="https://doi.org/10.1137/15M1023907">10.1137/15M1023907</a><br><a href="https://doi.org/10.1137/15M1023907">10.1137/15M1023907</a><br><br><h5>arXiv</h5><a href="https://arxiv.org/abs/1510.03235">1510.03235</a><br><br><h5>Additional Links</h5>http://epubs.siam.org/doi/10.1137/15M1023907</span>
kaust.personTzavaras, Athanasios
orcid.authorHelzel, Christiane
orcid.authorTzavaras, Athanasios::0000-0002-1896-2270
orcid.id0000-0002-1896-2270
refterms.dateFOA2018-06-13T16:20:47Z
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