A TAYLOR-PETROV-GALERKIN ALGORITHM FOR VISCOELASTIC FLOW

A viscoelastic flow is solved using a generalised Taylor-Galerkin/pressure correction scheme that incorporates consistent Petrov-Galerkin streamline upwinding within the discretisation of the constitutive equations. The numerical approach is indirect, in the sense that fractional equation solution stages are introduced within the framework of a time-stepping scheme. The Oldroyd-B and Phan-Thien-Tanner constitutive models are considered and the proposed Taylor-streamline upwind/Petrov-Galerkin algorithm is used to simulate flow through a 4:1 planar contraction. For the Oldroyd-B model, the algorithm indicates the onset of instability as elasticity is increased and a limiting Weissenberg number is observed, with no lip vortices apparent for values less than five. For a particular class of Phan-Thien-Tanner models, it is found that the algorithm is stable at high Weissenberg numbers. In this case the solutions exhibit a lip-vortex mechanism for the establishment of recirculating regions as has been observed in some experiments. Results presented for various combinations of fluid parameters suggest that the extensional behaviour of the viscoelastic model is the single most important factor governing the stability and convergence of the algorithms for highly elastic fluids.