Efficient simulation of nonlinear viscoelastic fluid flows

Abstract

This paper presents a new and efficient method for computing the flow of a non-Newtonian fluid. The approach is based on two independent concepts: Time-dependent simulation of viscoelastic flow: A new decoupled algorithm, presented in P. Saramito, Simulation numérique d'ecoulements de fluides viscoélastiques par éléments finis incompressibles et une méthode de directions alternées; applications, Thèse de l'Institut National Polytechnique de Grenoble, 1990 and P. Saramito, Numerical simulation of viscoelastic fluid flows using incompressible finite element method and a θ-method, Math. Modelling Num. Anal., 35 (1994) 1–35, enables us to split the major difficulties of this problem, and to solve it more efficiently. Moreover, this scheme is of order two in time, and can be used to obtain stationary flows in an efficient way. Conservative finite element method: this method combines the incompressible Raviart Thomas element, the discontinuous Lesaint-Raviart element, and a finite volume element method. It satisfies exactly the mass conservation law, and leads to an optimal size for the nonlinear system in terms of the total degree of freedom versus the mesh size. We apply our numerical procedure to the Phan-Thien-Tanner model with a classical benchmark: the four to one abrupt contraction. The numerical solutions exhibit good behavior, especially near the singularity, in the vicinity of the re-entrant corner. The numerical experiments present the main features of such flows: vortex development and overshooting of the velocity profile along the axis of symmetry in the entry region.