A simple method for simulating general viscoelastic fluid flows with an alternate log-conformation formulation

The log-conformation formulation has alleviated the long-standing high Weissenberg number problem associated with the viscoelastic fluid flows [R. Fattal, R. Kupferman, Constitutive laws for the matrix-logarithm of the conformation tensor, J. Non-Newtonian Fluid Mech. 123 (2004) 281–285]. This formulation ensures that solutions of viscoelastic flow problems are physically admissible, and it is able to capture sharp elastic stress layers. However, the implementations presented in literature thus far require changing the evolution equation for the conformation tensor into an equation for its logarithm, and are based on loosely coupled (partitioned) solution procedures [M.A. Hulsen, et al., Flow of viscoelastic fluids past a cylinder at high Weissenberg number: stabilized simulations using matrix logarithms, J. Non-Newtonian Fluid Mech. 127 (2005) 27–39]. A simple alternate form of the log-conformation formulation is presented in this article, and an implementation is demonstrated in the DEVSS-TG/SUPG finite element method. Besides its straightforward implementation, the new log-conformation formulation can be used to solve all the governing equations (continuity, conservation of momentum and constitutive equation) in a strongly coupled way by Newton’s method. The method can be applied to any conformation tensor model. The flows of Oldroyd-B and Larson-type fluids are tested in the benchmark problem of a flow past a cylinder in a channel. The accuracy of the method is assessed by comparing solutions with published results. The benefits of this new implementation and the pending issues are discussed.