Unsteady Couette flow of a class of fluids described by non-monotone models

In this paper we study the unsteady Couette motion of an incompressible non-linear fluid driven by a prescribed shear stress or a prescribed velocity imposed on the top surface. The fluid under consideration is described by a non-monotone relation between stress and strain rate. We consider the two-dimensional unsteady problem and solve it numerically by means of Crank–Nicolson scheme combined with spectral collocation method. We investigate the time behaviour of the velocity field, in order to find out, starting from suitable initial conditions, whether the numerical solutions tend to the corresponding stationary one or not. Our results confirm both the stability of the two solutions belonging to the ascending branches of the constitutive law and the linear instability of the one on the descending branch.