Spectral element methods for transient viscoelastic flow problems

Spectral element methods are developed for solving transient flows of viscoelastic fluids. The fluids are modeled using the upper-convected Maxwell and Oldroyd B constitutive relationships. Several temporal schemes for dealing with the time dependent nature of the problems are considered. The computation associated with each time step comprises an explicit treatment of the convection and deformation terms and an implicit treatment of the linear terms. The generalized three field Stokes problem generated in the latter is solved using a nested preconditioned conjugate gradient method. The performance of a number of preconditioners is investigated. A modified formulation of the Stokes problem is presented that ensures a unique pressure approximation with zero mean is obtained at each time step. The performance of the schemes and models when applied to the benchmark problem of start-up of plane Poiseuille flow is investigated.