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Carnegie Mellon NSF logoCenter for Nonlinear Analysis
Xiaojun Wang
Afilliation: Virginia Tech

Title: High Weissenberg number limit and boundary layer phenomena in complex fluids

Abstract: In an earlier paper, we have proved well-posedness for the high Weissenberg number limit of the UCM model for non-Newtonian fluids. The limiting equations do not admit the full boundary conditions for the finite Weissenberg number case, like the Euler equations do not do for Navier-Stoke equations in Newtonian case. In Newtonian fluids the nonslip boundary condition causes velocity boundary layers. While in non-Newtonian, stress boundary layers form due to different mechanism. Under finite Weissenberg number, the convected derivatives in the constitutive relation vanish at the wall. However at a short distance from the wall, the convected derivative terms can not be neglected. When the Weissenberg number becomes large, a large stress gradient leads the formation of a sharp stress layer. The problem in passing to the high Weiessenberg number limit is that nonslip boundary conditions cannot be enforced. In this work we derive boundary layers equations to describe the behavior near the boundary and discuss the mathematical difficulties which arise in trying to establish the well-posedness of these boundary layer equations.