Computation of 2D viscoelastic flows for a differential constitutive equation

Publisher Summary Severe difficulties have been encountered for several years in the numerical simulation of viscoelastic flow for differential constitutive equations. This chapter presents a summary of the numerical problems. The solution of the discretized non linear system of equations is difficult to compute. This is now well known and various adaptations of the Newton method are used with success. Given that numerically computed viscoelastic velocity fields are not very different from the viscous ones it is surprising that sophisticated algorithms are required for convergence. Non-convergence with mesh refinement is reported by various authors. This is possibly connected with the mathematical non existence of the solution for a singular geometry. The chapter proves the existence of solutions in H 3 (Ω) in a smooth geometry. As the Newtonian solution does not satisfy this condition, this obviously fails in a singular geometry. A stabilization of the numerical method can be obtained by introducing diffusion in the numerical upwinding or by a convenient projection appearing for example in the four-field computation.