Computation of viscoelastic fluid flows using continuation methods

Abstract

a b s t r a c tThe numerical simulation of viscoelastic fluid flow becomes more difficult as a physical parameter, the Weissenberg number, increases. Specifically, at a Weissenberg number larger than a critical value, the iterative nonlinear solver fails to converge, a phenomenon known as the high Weissenberg number problem. In this work we describe the application and implementation of continuation methods to the nonlinear Johnson-Segalman model for steady-state viscoelastic flows. Simple, natural, and pseudo-arclength continuation approaches in Weissenberg number are investigated for a discontinuous Galerkin finite element discretization of the equations. Computations are performed for a benchmark contraction flow and, several aspects of the performance of the continuation methods including high Weissenberg number limits, are discussed.

The DESC Stellarator Code Suite Part II: Perturbation and continuation methods

“…The perturbation techniques described previously can also be adapted for optimization, where instead of choosing a particular change in the parameters ∆c we instead let ∆c be a free parameter that is chosen to minimize a cost function g(x, c), such as quasisymmetry error 2,25 , coil complexity [26][27][28] , or fast particle confinement 29,30 : ∆c * = arg min ∆c g(x+∆x, c+∆c) s.t. f (x+∆x, c+∆c) = 0 (15) Where as before ∆x is an implicit function of ∆c.…”

“…a) wconlin@princeton.edu b) ddudt@princeton.edu c) dpanici@princeton.edu d) ekolemen@princeton.edu Continuation methods have received less attention in the fusion community, though they have seen extensive use in other fields such as 15,16 . For the purposes of the present work, continuation methods can be used to solve parameterized equations of the form F (x, η) = 0, where we identify x as the solution vector, and η as a continuation parameter.…”

The DESC stellarator code suite. Part 2. Perturbation and continuation methods

“…Continuation methods have received less attention in the fusion community, though they have seen extensive use in other fields (e.g. Richter & DeCarlo 1983; Howell 2009). For the purposes of the present work, continuation methods can be used to solve parameterized equations of the form , where we identify as the solution vector and as a continuation parameter.…”

Approximation of the two-fluid flow problem for viscoelastic fluids using the level set method and pressure enriched finite element shape functions

“…The log-conformation formulation for the elastic stress tensor proposed by Fattal and Kupfermann [28] is a common approach to solve highly elastic uids and has been used for the free surface evolution problem successfully with dierent numerical methods [9,30]. Continuation methods are another numerical tool to increase the Weissenberg number limit that can be solved by a standard formulation, as shown in the work of Howell [31]. For the treatment of local oscillations, discontinuity-capturing techniques have proved to produce good results in [32,27].…”