Improving Eulerian two-phase flow finite element approximation with discontinuous gradient pressure shape functions

2005 ◽  
Vol 49 (12) ◽  
pp. 1287-1304 ◽  
Author(s):  
A. H. Coppola-Owen ◽  
R. Codina
Keyword(s):  
Finite Element ◽  
Two Phase Flow ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
Phase Flow ◽  
Shape Functions ◽  
Two Phase

On convergent schemes for a two-phase Oldroyd-B type model with variable polymer density

2020 ◽  
Vol 28 (2) ◽  
pp. 99-129
Author(s):  
Oliver Sieber
Keyword(s):  
Finite Element ◽  
Stress Tensor ◽  
Two Phase Flow ◽  
Particle Density ◽  
Type Equation ◽  
Diffuse Interface ◽  
Type Model ◽  
Element Approximation ◽  
Phase Flow ◽  
Two Phase

AbstractThe paper is concerned with a diffuse-interface model that describes two-phase flow of dilute polymeric solutions with a variable particle density. The additional stresses, which arise by elongations of the polymers caused by deformations of the fluid, are described by Kramers stress tensor. The evolution of Kramers stress tensor is modeled by an Oldroyd-B type equation that is coupled to a Navier–Stokes type equation, a Cahn–Hilliard type equation, and a parabolic equation for the particle density. We present a regularized finite element approximation of this model, prove that our scheme is energy stable and that there exist discrete solutions to it. Furthermore, in the case of equal mass densities and two space dimensions, we are able to pass to the limit rigorously as the regularization parameters and the spatial and temporal discretization parameters tend towards zero and prove that a subsequence of discrete solutions converges to a global-in-time weak solution to the unregularized coupled system. To the best of our knowledge, this is the first existence result for a two-phase flow model of viscoelastic fluids with an Oldroyd-B type equation. Additionally, we show that our finite element scheme is fully practical and we present numerical simulations.

Finite element implementation of an improved conservative level set method for two-phase flow

2014 ◽  
Vol 100 ◽  
pp. 138-154 ◽  
Author(s):  
Lanhao Zhao ◽  
Jia Mao ◽  
Xin Bai ◽  
Xiaoqing Liu ◽  
Tongchun Li ◽  
...  
Keyword(s):  
Finite Element ◽  
Level Set Method ◽  
Level Set ◽  
Two Phase Flow ◽  
Phase Flow ◽  
Two Phase ◽  
Finite Element Implementation ◽  
Conservative Level Set

On convergent schemes for two-phase flow of dilute polymeric solutions

2018 ◽  
Vol 52 (6) ◽  
pp. 2357-2408 ◽  
Author(s):  
Stefan Metzger
Keyword(s):  
Finite Element ◽  
Two Phase Flow ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Phase Flow ◽  
Two Phase ◽  
Extra Stress Tensor ◽  
Spring Force ◽  
Polymeric Solutions ◽  
Fokker Planck

We construct a Galerkin finite element method for the numerical approximation of weak solutions to a recent micro-macro bead-spring model for two-phase flow of dilute polymeric solutions derived by methods from nonequilibrium thermodynamics ([Grün, Metzger, M3AS 26 (2016) 823–866]). The model consists of Cahn-Hilliard type equations describing the evolution of the fluids and the unsteady incompressible Navier-Stokes equations in a bounded domain in two or three spatial dimensions for the velocity and the pressure of the fluids with an elastic extra-stress tensor on the right-hand side in the momentum equation which originates from the presence of dissolved polymer chains. The polymers are modeled by dumbbells subjected to a finitely extensible, nonlinear elastic (FENE) spring-force potential. Their density and orientation are described by a Fokker-Planck type parabolic equation with a center-of-mass diffusion term. We perform a rigorous passage to the limit as the spatial and temporal discretization parameters simultaneously tend to zero, and show that a subsequence of these finite element approximations converges towards a weak solution of the coupled Cahn-Hilliard-Navier-Stokes-Fokker-Planck system. To underline the practicality of the presented scheme, we provide simulations of oscillating dilute polymeric droplets and compare their oscillatory behaviour to the one of Newtonian droplets.

Variational multi-scale finite element method for the two-phase flow of polymer melt filling process

2019 ◽  
Vol 30 (3) ◽  
pp. 1407-1426 ◽  
Author(s):  
Xuejuan Li ◽  
Ji-Huan He
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Numerical Method ◽  
Two Phase Flow ◽  
Polymer Melt ◽  
Phase Flow ◽  
Filling Process ◽  
Two Phase ◽  
Content Type ◽  
Element Method

Purpose The purpose of this paper is to develop an effective numerical algorithm for a gas-melt two-phase flow and use it to simulate a polymer melt filling process. Moreover, the suggested algorithm can deal with the moving interface and discontinuities of unknowns across the interface. Design/methodology/approach The algebraic sub-grid scales-variational multi-scale (ASGS-VMS) finite element method is used to solve the polymer melt filling process. Meanwhile, the time is discretized using the Crank–Nicolson-based split fractional step algorithm to reduce the computational time. The improved level set method is used to capture the melt front interface, and the related equations are discretized by the second-order Taylor–Galerkin scheme in space and the third-order total variation diminishing Runge–Kutta scheme in time. Findings The numerical method is validated by the benchmark problem. Moreover, the viscoelastic polymer melt filling process is investigated in a rectangular cavity. The front interface, pressure field and flow-induced stresses of polymer melt during the filling process are predicted. Overall, this paper presents a VMS method for polymer injection molding. The present numerical method is extremely suitable for two free surface problems. Originality/value For the first time ever, the ASGS-VMS finite element method is performed for the two-phase flow of polymer melt filling process, and an effective numerical method is designed to catch the moving surface.

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