On the characteristics and compatibility equations for the UCM model fluid

The quasilinear system of partial differential equations governing the flow of a UCM fluid is known to be of mixed elliptic-hyperbolic type. The compatibility equations associated with the hyperbolic part of the system are derived in this paper. There are two characteristic variables that are transported along the characteristics. These are both associated with the conditional well-posedness of the system. The compatibility equations in this case are derived from the constitutive equation alone. There are two additional characteristics that are always present for unsteady inertial flow and which may be present for steady inertial flow. The compatibility equations associated with these characteristics involve a coupling between the momentum and constitutive equations. At an inflow boundary, it is shown that all stress components can be prescribed, despite the fact that there are only two characteristic variables that enter the domain along the streamlines. The remaining stress component is provided by fixing the incoming elastic shear waves. (c) 2008 WILEY-VCH Verlag GmbH & Co. KGaA. Weinheim.