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We compare four finite element implementations of a log-conformation formulation for the numerical solution of viscoelastic fluid flow problems. These formulations essentially differ the treatment of the advective term. The first formulation was introduced in Fattal and Kupferman (2005) [1] while the second one, much simpler to implement, was proposed by Coronado et al. (2007) [2] . We then introduce the two variants of this last formulation. In the first one, a special treatment is done to the computation of the advective term u ⋅ ∇ e s which is more accurate than the one described in Coronado et al. (2007) [2] . In the second variant, we show how e s and ∇ e s can be computed in order to allow a complete linearization of the constitutive equation and its solution by Newton’s method. A comparative study of these four formulations is done on the benchmark problem of the flow of an Oldroyd-B fluid past a circular cylinder in a rectangular channel.

In the last few years, we have developed a fairly general adaptive finite element solution procedure which can be applied to a large variety of problems. In this paper, this strategy is briefly recalled and applied to the solution of two-dimensional viscoelastic fluid flow problems. A log-conformation formulation recently introduced by Fattal and Kupferman [R. Fattal, R. Kupferman, Time-dependent simulation of viscoelastic flows at high Weissenberg number using the log-conformation representation, J. Non-Newtonian Fluid Mech. 126 (2005) 23-37] was implemented in order to improve the convergence properties of the numerical scheme. We confirm some results obtained in Hulsen, Fattal and Kupferman [M. Hulsen, R. Fattal, R. Kupferman, Flow of viscoelastic fluids past a cylinder at high Weissenberg number: stabilized simulations using matrix logarithm, J. Non-Newtonian Fluid Mech. 127 (2005) 27-39] and in some instances, we show that mesh adaptation allows to almost automatically reproduce accurate results obtained on very fine structured meshes.

The numerical computation of viscoelastic fluid flows with differential constitutive equations presents various difficulties. The first one lies in the numerical convergence of the complex numerical scheme solving the non-linear set of equations. Due to the hybrid type of these equations (elliptic and hyperbolic), geometrical singularities such as reentrant corner or die induce stress singularities and hence numerical problems. Another difficulty is the choice of an appropriate constitutive equation and the determination of rheological constants. In this paper, a quasi-Newton method is developed for a fluid obeying a multi-mode Phan-Thien and Tanner constitutive equation. A confined convergent geometry followed by the extrudate swell has been considered. Numerical results obtained for two-dimensional or axisymmetric flows are compared to experimental results (birefringence patterns or extrudate swell) for a linear low density polyethylene (LLDPE) and a low density polyethylene (LDPE).

A numerical method for solving viscoelastic fluid flow problems is presented. Based on the Lesaint-Raviart method, it allows computations at moderate values of the Weissenberg number in a very efficient manner. The components of the stress tensor are computed on an element-by-element basis, reducing substantially the storage requirements with respect to other methods.

A finite-element method for computing viscoelastic fluid flows is presented. The described methodology is based on the introduction of the conformation tensor which is related to the orientation of the polymer macromolecule. Using a model introduced by Grmela [1] and a numerical strategy developed by Fortin and Zine [2], computations are performed for the 4:1 contraction flow problem. No limiting Weissenberg number is observed even though a decoupled numerical method is used for solving the equations. Convergence with mesh size is also addressed.

The numerical simulation of the flow of a viscoelastic fluid by a generalized minimal residual (GMRES) method is investigated and tested to determine its convergence properties. A method based on the Lesaint—Raviart technique is used as a preconditioner for the GMRES method to enhance the convergence rate and robustness of the algorithm. Numerical results are presented for the stick—slip problem.

We present a finite element method for the simulation of viscoelastic flow. A decoupled approach is employed for the computation of the velocity field and of the components of the extra-stress tensor. The high Weissenberg number problem is addressed by using together two different types of upwinding namely the streamline upwinding technique and the Lesaint-Reviart method. Numerical results are presented for the case of the 4 to 1 plane contraction.