Hyperbolic models of unsteady flows of a viscoelastic medium

Unsteady one-dimensional shear flows of a viscoelastic medium are considered. A general approach is formulated for media with several relaxation times, which allows the known models of viscoelastic flows to be presented as evolutionary systems of first-order equations. Conditions of hyperbolicity of flow classes considered are found for the Johnson–Segalman, Giesekus, and Rolie–Poly models. The equations of motion of the viscoelastic medium are presented in the form of a full nonlinear system of conservation laws. A method of calculating unsteady discontinuous flows within the framework of the models under consideration is proposed. The class of unsteady Couette flows in the gap between the cylinders used in rheological tests is studied numerically. The process of shear lamination of its influence on the structure of steady flows are investigated. The numerical results obtained are compared with experimental data.