ABSTRACT
As observed in recent experiments of Lim and Schowalter, the piston-driven channel flow of a highly elastic and very viscous non-Newtonian fluid exhibits persistent oscillations. We suggest an explanation for these oscillations in terms of a constitutive model that has a non-monotonic relationship between steady shear stress and strain rate. We describe the reduction of the three-dimensional equations of motion and stress to approximating systems that can be solved by a combination of numerical and analytic methods. One such approximation results in a quadratic system of first-order functional differential equations in which the volumetric flow rate is imposed by a feedback control. Numerical solution of this system exhibits a transition to a regime with persistent oscillations that compare favorably to the Lim–Schowalter observations. To better understand this behavior, we make a further approximation suggested by the numerical solution of the feedback system. The resulting quadratic system of four ordinary differential equations provides precise predictions, through eigenvalue analysis of its linearization, of the transitions of the feedback system.
