{"id":{"repo_id":"vt","oai_identifier":"oai:vtechworks.lib.vt.edu:10919/27669"},"canonical_url":"https://search.dev.ndltd.org/etd/vt/oai:vtechworks.lib.vt.edu:10919/27669","repository":{"repo_id":"vt","name":"Virginia Tech","base_url":"https://vtechworks.lib.vt.edu/oai/request"},"display":{"title":"Well-posedness results for a class of complex flow problems in the high Weissenberg number limit","abstract":"For 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.","abstract_html":"For 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.","abstract_has_math":false,"creators":["Wang, Xiaojun"],"institution":"Virginia Tech","degree_name":"Ph. 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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."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph. D."]},{"key":"dc:title","label":"Title","values":["Well-posedness results for a class of complex flow problems in the high Weissenberg number limit"]}]}],"canonical_facts":{"dc:contributor.committeechair":["Renardy, Michael J."],"dc:contributor.committeemember":["Borggaard, Jeffrey T.","Rogers, Robert C.","Sun, Shu-Ming","Renardy, Yuriko Y."],"dc:contributor.department":["Mathematics"],"dc:creator":["Wang, Xiaojun"],"dc:date.accessioned":["2014-03-14T20:11:54Z"],"dc:date.available":["2014-03-14T20:11:54Z","2012-05-22"],"dc:date.issued":["2012-04-30"],"dc:description.abstract":["For 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."],"dc:description.degree":["Ph. D."],"dc:identifier.other":["etd-05112012-110824"],"dc:identifier.uri":["http://hdl.handle.net/10919/27669"],"dc:publisher":["Virginia Tech"],"dc:rights":["In Copyright"],"dc:rights.uri":["http://rightsstatements.org/vocab/InC/1.0/"],"dc:subject":["mollifier","symmetric hyperbolic system","curvilinear coordinates","stress boundary layer","scaling analysis","multiscale modeling","Lagrangian coordinates"],"dc:title":["Well-posedness results for a class of complex flow problems in the high Weissenberg number limit"],"dc:type":["Dissertation"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["doctoral"],"thesis:degree_name":["Ph. 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