Pressure-driven flow of a rate type fluid with stress threshold in an infinite channel

Abstract

Abstract Abstract In this paper we extend some our previous works on continua with stress threshold. In particular here we propose a mathematical model for a continuum which behaves as a nonlinear upper convected Maxwell fluid if the stress is above a certain threshold and as a Oldroyd-B type fluid if the stress is below such a threshold. We derive the constitutive equations for each phase exploiting the theory of natural configurations (introduced by rajagopal and co-workers) and the criterion of the maximization of the rate of dissipation. We state the mathematical problem for a one-dimensional flow driven by a constant pressure gradient and study two peculiar cases in which the velocity of the inner part of the fluid is spatially homogeneous.

Abstract Highlights This work provides additional insight into the response of rate type fluids with a threshold. The paper extends results previously obtained by the authors on materials with stress threshold (see references in the paper for alist of such works). The one-dimensional formulation, which is presented in the paper, is interesting also from the mathematical point of view, since it turns out to be a very complicated free boundary problem involving a hyperbolic and a parabolic equation. Various simplified cases are studied.