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

In this paper we extend some of our previous works on continua with stress threshold. In particular here we propose a mathematical model for a continuum which behaves as a non-linear 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.