The one-dimensional flow of a fluid with limited strain-rate

We present a model for a continuum in which the strain rate depends linearly on the stress, as long as the latter is below a fixed threshold, but it is frozen to a constant value when the stress exceeds such a threshold. The constitutive equation is given in an implicit form as the stress is a multi-valued function of the strain rate. We derive the model in a general 3D setting and we study the one-dimensional case of a pressure-driven flow between two parallel plates. We prove some analytical results and describe a procedure to determine the main physical parameters (stress threshold and viscosity) by means of a rotational viscometer. Finally we show that the model can be obtained as the limit case of a piecewise linear viscous model