Abstract
We consider a system of equations describing the flows of an incompressible viscoelastic medium with a rheological equation of state containing derivatives of the stress tensor with respect to time. The initial system of equations for two-constant models of the medium is a quasilinear first-order system. Correct formulation of the problem under initial conditions requires the imposition of certain restrictions on the system matrix [ 1]. These restrictions, which are necessary to ensure the evolutionary character of the system, are imposed on the stress tensor in our case. We shall concentrate on one-dimensional motions for which the requirement of evolutionary character renders the system hyperbolic. It is then possible to indicate sufficient conditions which ensure the uniqueness of the continuous solution of the one-dimensional steadystate boundary value problem. Hyperbolic systems of equations of viscoelastic fluid dynamics have discontinuous solutions for certain models (e.g. that of Oldroyd [ 2]). Discontinuous flows of materials with memory in which the stresses are functionals of their “strain history” are discussed in [ 3]. We shall consider the discontinuities in Oldroyd's model when the differential relationship between the stress tensors and straining rates is given. A necessary condition for the existence of discontinuities is formulated. The problem of evolution of a velocity jump in one-dimensional motion is considered.
Published Version
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call