On a modified non-singular log-conformation formulation for Johnson–Segalman viscoelastic fluids

Abstract

A modified log-conformation formulation of viscoelastic fluid flows is presented in this paper. This new formulation is non-singular for vanishing Weissenberg numbers and allows a direct steady numerical resolution by a Newton method. Moreover, an exact computation of all the terms of the linearized problem is provided. The use of an exact divergence-free finite element method for velocity-pressure approximation and a discontinuous Galerkin upwinding treatment for stresses leads to a robust discretization. A demonstration is provided by the computation of steady solutions at high Weissenberg numbers for the difficult benchmark case of the lid driven cavity flow. Numerical results are in good agreement, qualitatively with experiment measurements on real viscoelastic flows, and quantitatively with computations performed by others authors. The numerical algorithm is both robust and very efficient, as it requires a low mesh-invariant number of linear systems resolution to obtain solutions at high Weissenberg number. An adaptive mesh procedure is also presented: it alows representing accurately both boundary layers and the main and secondary vorties.

The fully-implicit log-conformation formulation and its application to three-dimensional flows

“…Furthermore, we especially sought a description that could be applied in three dimensions with the same ease as the previously published two-dimensional approaches [13,14].…”

“…As a continuation of these earlier works, this paper is devoted to a generalization to arbitrary dimensions, along which we will also bridge the gap between the two expositions in [13] and [14].…”

“…Generalizations of the subsuming methods to the Johnson-Segalman model, as, e.g., done in [14], or other models are in principle possible, but omitted here for the sake of brevity.…”

“…Nonetheless, it falls short in many applications when it comes to study perturbations of Ψ h , as it is the case for the variational derivative needed in the Newton-Raphson algorithm. Existing implementations, as, e.g., in [13,14], were limited to the two-dimensional case, since for a 2 × 2 matrix it is still feasible to derive an algebraic closed expression for eigenvalues and eigenvectors in dependence of Ψ h . Another approach would be general perturbation theory [23], which directly applies to eigenvalues λ i and their projection operators P i , but this theory is prone to singularities in the vicinity of degenerate eigenvalues.…”

“…Eq. (14). Nonetheless, the Stokesian drag formula can be generalized to ellipsoids in principle [36].…”