Well-posedness results for a class of complex flow problems in the high Weissenberg number limit
| dc.contributor.author | Wang, Xiaojun | en |
| dc.contributor.committeechair | Renardy, Michael J. | en |
| dc.contributor.committeemember | Borggaard, Jeffrey T. | en |
| dc.contributor.committeemember | Rogers, Robert C. | en |
| dc.contributor.committeemember | Sun, Shu-Ming | en |
| dc.contributor.committeemember | Renardy, Yuriko Y. | en |
| dc.contributor.department | Mathematics | en |
| dc.date.accessioned | 2014-03-14T20:11:54Z | en |
| dc.date.adate | 2012-05-22 | en |
| dc.date.available | 2014-03-14T20:11:54Z | en |
| dc.date.issued | 2012-04-30 | en |
| dc.date.rdate | 2012-05-22 | en |
| dc.date.sdate | 2012-05-11 | en |
| 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. | en |
| dc.description.degree | Ph. D. | en |
| dc.identifier.other | etd-05112012-110824 | en |
| dc.identifier.sourceurl | http://scholar.lib.vt.edu/theses/available/etd-05112012-110824/ | en |
| dc.identifier.uri | http://hdl.handle.net/10919/27669 | en |
| dc.publisher | Virginia Tech | en |
| dc.relation.haspart | Wang_Xiaojun_D_2012.pdf | en |
| dc.rights | In Copyright | en |
| dc.rights.uri | http://rightsstatements.org/vocab/InC/1.0/ | en |
| dc.subject | mollifier | en |
| dc.subject | symmetric hyperbolic system | en |
| dc.subject | curvilinear coordinates | en |
| dc.subject | stress boundary layer | en |
| dc.subject | scaling analysis | en |
| dc.subject | multiscale modeling | en |
| dc.subject | Lagrangian coordinates | en |
| dc.title | Well-posedness results for a class of complex flow problems in the high Weissenberg number limit | en |
| dc.type | Dissertation | en |
| thesis.degree.discipline | Mathematics | en |
| thesis.degree.grantor | Virginia Polytechnic Institute and State University | en |
| thesis.degree.level | doctoral | en |
| thesis.degree.name | Ph. D. | en |
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