Type of Document Dissertation Author Wang, Xiaojun Author's Email Address xjwang08@vt.edu URN etd-05112012-110824 Title Well-posedness results for a class of complex flow problems in the high Weissenberg number limit Degree PhD Department Mathematics Advisory Committee

Advisor Name Title Renardy, Michael J. Committee Chair Borggaard, Jeffrey T. Committee Member Renardy, Yuriko Y. Committee Member Rogers, Robert C. Committee Member Sun, Shu-Ming Committee Member Keywords

- mollifier
- symmetric hyperbolic system
- curvilinear coordinates
- stress boundary layer
- scaling analysis
- multiscale modeling
- Lagrangian coordinates
Date of Defense 2012-04-30 Availability unrestricted 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.

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