Hyperbolic submodels of an incompressible viscoelastic Maxwell medium

V. Yu. Lyapidevskii ; V. V. Pukhnachev

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Modern problems of mechanics, Tome 281 (2013), pp. 84-97

Résumé

A two-dimensional motion of an incompressible viscoelastic Maxwell continuum is considered. The system of quasilinear equations describing this motion has both real and complex characteristics. A class of effectively one-dimensional motions is analyzed for which the original system of equations is decomposed into a hyperbolic subsystem and a quadrature. The properties of the hyperbolic submodels obtained depend on the choice of the invariant derivative in the rheological relation. When one chooses the Jaumann corotational derivative as the invariant derivative, the equations of the submodel remain quasilinear. They can be represented in the form of conservation laws, which allows one to analyze discontinuous solutions to these equations. When one chooses the upper or lower convected derivative, the equations of one-dimensional hyperbolic submodels turn out to be linear. The problem of shear motion between parallel plates and the problem of interaction between the stress field that does not depend on one of the coordinates and a transverse shear flow with initially constant vorticity are studied in detail. It is established that a plane Couette flow in the model with the corotational derivative is unstable in the linear approximation in the class of shear flows if the Weissenberg number is greater than one. The development of small perturbations gives rise to discontinuities in tangential velocities and stresses. The hysteresis phenomenon is observed when the Weissenberg number successively increases and decreases while passing through a critical value. The Couette flow in models with the upper and lower convected derivative remains stable with respect to one-dimensional perturbations.