Well-posedness results for a class of complex flow problems in the high Weissenberg number limit

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dc.contributor.authorWang, Xiaojunen
dc.contributor.committeechairRenardy, Michael J.en
dc.contributor.committeememberBorggaard, Jeffrey T.en
dc.contributor.committeememberRogers, Robert C.en
dc.contributor.committeememberSun, Shu-Mingen
dc.contributor.committeememberRenardy, Yuriko Y.en
dc.contributor.departmentMathematicsen
dc.date.accessioned2014-03-14T20:11:54Zen
dc.date.adate2012-05-22en
dc.date.available2014-03-14T20:11:54Zen
dc.date.issued2012-04-30en
dc.date.rdate2012-05-22en
dc.date.sdate2012-05-11en
dc.description.abstractFor simple fluids, or Newtonian fluids, the study of the Navier-Stokes equations in the high Reynolds number limit brings about two fundamental research subjects, the Euler equations and the Prandtl's system. The consideration of infinite Reynolds number reduces the Navier-Stokes equations to the Euler equations, both of which are dealing with the entire flow region. Prandtl's system consists of the governing equations of the boundary layer, a thin layer formed at the wall boundary where viscosity cannot be neglected. In this dissertation, we investigate the upper convected Maxwell(UCM) model for complex fluids, or non-Newtonian fluids, in the high Weissenberg number limit. This is analogous to the Newtonian fluids in the high Reynolds number limit. We present two well-posedness results. The first result is on an initial-boundary value problem for incompressible hypoelastic materials which arise as a high Weissenberg number limit of viscoelastic fluids. We first assume the stress tensor is rank-one and develop energy estimates to show the problem is locally well-posed. Then we show the more general case can be handled in the same spirit. This problem is closely related to the incompressible ideal magneto-hydrodynamics (MHD) system. The second result addresses the formulation of a time-dependent elastic boundary layer through scaling analysis. We show the well-posedness of this boundary layer by transforming to Lagrangian coordinates. In contrast to the possible ill-posedness of Prandtl's system in Newtonian fluids, we prove that in non-Newtonian fluids the stress boundary layer problem is well-posed.en
dc.description.degreePh. D.en
dc.identifier.otheretd-05112012-110824en
dc.identifier.sourceurlhttp://scholar.lib.vt.edu/theses/available/etd-05112012-110824/en
dc.identifier.urihttp://hdl.handle.net/10919/27669en
dc.publisherVirginia Techen
dc.relation.haspartWang_Xiaojun_D_2012.pdfen
dc.rightsIn Copyrighten
dc.rights.urihttp://rightsstatements.org/vocab/InC/1.0/en
dc.subjectmollifieren
dc.subjectsymmetric hyperbolic systemen
dc.subjectcurvilinear coordinatesen
dc.subjectstress boundary layeren
dc.subjectscaling analysisen
dc.subjectmultiscale modelingen
dc.subjectLagrangian coordinatesen
dc.titleWell-posedness results for a class of complex flow problems in the high Weissenberg number limiten
dc.typeDissertationen
thesis.degree.disciplineMathematicsen
thesis.degree.grantorVirginia Polytechnic Institute and State Universityen
thesis.degree.leveldoctoralen
thesis.degree.namePh. D.en

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