Well-Posedness of Boundary Layer Equations for Time-Dependent Flow of Non-Newtonian Fluids

Abstract

We consider the flow of an upper convected Maxwell fluid in the limit of high Weissenberg and Reynolds number. In this limit, the no-slip condition cannot be imposed on the solutions. We derive equations for the resulting boundary layer and prove the well-posedness of these equations. A transformation to Lagrangian coordinates is crucial in the argument.

On the dynamic instability of non‐Newtonian fluids driven by gravity

“…Even so, there are many important progress on theoretical analysis for non-Newtonian fluid systems, Bothe-Pruss have given the result of the L p -theory for a class of non-Newtonian fluids [4]; Feireisl-Kwon investigated the long-time behavior of dissipative solutions to models of non-Newtonian compressible fluids [5]; Guo-Tan have given the result of large-time behavior of solutions to a class of non-Newtonian compressible fluids [6]. The results of the well-posedness for non-Newtonian fluids can be refer to [7][8][9][10][11] for details. For more results of non-Newtonian fluids, please refer to [2,6,[10][11][12][13][14][15][16] and the references cited therein.…”

On the small time asymptotics of stochastic non‐Newtonian fluids

“…Zhao et al , studied the upper semicontinuity of the global attractor for a two‐dimensional non‐Newtonian fluid. Renardy and Wang considered the flow of an upper convected Maxwell fluid in the limit of high Weissenberg and Reynolds number. Yuan et al , proved local existence and uniqueness of solutions for a class of non‐Newtonian fluids with vacuum and damping in one‐dimensional bounded intervals.…”

Similarity Solution for High Weissenberg Number Flow of Upper-Convected Maxwell Fluid on a Linearly Stretching Sheet

“…Using an implicit function, the existence of solutions for viscoelastic boundary layer which arises from spatially periodic perturbations of uniform shear flow was addressed [23]. Also, the well-posedness of boundary layer equations for time-dependent flow of UCM fluid in the limit of high Weissenberg and Reynolds numbers was analyzed [24]. Furthermore, a systematic perturbation procedure to solve the initial value problem for creeping flow of the UCM fluid at high Weissenberg number is formulated [25].…”