Unsteady one dimensional motions of a new class of seemingly viscoplastic materials

In this note we study the unsteady rectilinear flow of a fluid whose constitutive equation mimics that of a viscoplastic material. The constitutive relation is non-linear and is such that the stress cannot exceed a certain limit (limit stress fluid). The mathematical problem consists of the mass balance and the linear momentum equations as well as the initial and boundary conditions. We assume that the flow is driven by a horizontal pressure gradient and the movement of the top plate. After rescaling the system, we determine an explicit expression for the solution to the steady-state problem and prove that this solution exists and is unique within a limited range of the model's material parameters. We solve the transient problem (start-up, variation of the Stokes' first problem) using the finite element method and validate the numerical scheme by comparing the numerical and analytical solution. We perform some numerical simulations and show the behavior of the velocity and the stress for different values of the physical parameters. We also consider the case where the top plate oscillates at a given frequency (variation of the Stokes' second problem).