Well-posedness of the upper convected Maxwell fluid in the limit of infinite Weissenberg number

Abstract

An iteration scheme is used to show the well-posedness of the initial-boundary value problem for incompressible hypoelastic materials, which arise as a high Weissenberg number limit of viscoelastic fluids. We first assume that the stress is a rank-one matrix T = qq T ,q∈ R n , and develop energy estimates to show that the problem is locally wellposed. This problem is related to incompressible ideal magnetohydrodynamics (MHD). We show that the general case T = CC T ,C ∈ R n×n can be handled by a generalization of the method we developed.

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

“…In [22], we proved the well-posedness of this system. We considered the initialboundary value problem in a smooth domain Ω, subject to initial conditions for v 0 and T 0 , and the boundary condition v 0 · n = 0.…”

“…In the limit of high elasticity, a limiting equation can be derived which is similar to the system of ideal magnetohydrodynamics and, like the Euler equations, does not allow the imposition of a no-slip boundary condition. The well-posedness of this system has been established in [22]. The goal of this manuscript is to supplement this analysis with a study of the well-posedness of the accompanying boundary layer equations.…”

The initial value problem for creeping flow of the upper convected Maxwell fluid at high Weissenberg number

“…The analog of the combined Euler-Prandtl system for the upper convected Maxwell fluid has recently been analyzed in [4,5]. It was assumed there that both the Reynolds and Weissenberg numbers are large.…”

“…In , the well‐posedness of this system is analyzed in the limit where both the Reynolds number R and Weissenberg number W are large. The stress is assumed only positive semidefinite, and at leading order T · n vanishes at the boundary.…”