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 numerique d'ecoulements de fluides viscoelastiques par eRments finis incompressibles et une methode de directions alternees; applications, These de l'Institut National Polytechnique de Grenoble, 1990 and P. Saramito, Numerical simulation of viscoelastic fluid flows using incompressible finite element method and a 0-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.

Channel flow of a tensorial shear-thinning Maxwell model: Lattice Boltzmann simulations

“…[63][64][65], or for the simpler generalized-Newtonian fluids 66 . Also within these schemes, the inclusion of non-trivial constitutive equations poses subtleties.…”

Understanding and predicting viscous, elastic, plastic flows

“…As in 1D, the stationary solution is obtained by solving the time-dependent problem with a second-order time-splitting algorithm, already used for VE [45], generalized here to VEP; it allows to treat separatly the two main non-linearities of the equations, namely the plasticity term in eq. (3) and the stress transport term in the objective derivative (eq.…”

Mixed finite element methods for viscoelastic flow analysis: a review

“…Saratimo et al [28], [63] and [64] have applied the so-called Raviart-Thomas element where the diagonal components of the extra stress tensor are interpolated discontinuous piece-wise constant while the off-diagonal components are interpolated with a continuous bi-linear polynomial. Clearly, the piece-wise constant interpolation limits the accuracy to first order only.…”

“…For instance Singh and Leal [29] first applied the three step operator splitting methodology developed by Glowinksi and Pironneau [74] to viscoelastic flows, later followed by Saratimo et al [28], [63] and [64] and Luo [52]. In the first and third step a generalized Stokes problem is obtained, while in the second step a convection-diffusion type of problem needs to be solved.…”