Fitted finite element discretization of two‐phase Stokes flow

Agnese, Marco; Nürnberg, Robert

doi:10.1002/fld.4237

Cited by 8 publications

(15 citation statements)

References 35 publications

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“…Similar work for the two-phase Stokes flow without contact lines has been done in Ref. [52]; there the method was shown to be unconditionally stable.…”

Section: The Finite Element Methodssupporting

confidence: 56%

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An Energy-stable Finite Element Method for the Simulation of Moving Contact Lines in Two-phase Flows

Zhao,

Ren

2020

Preprint

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We consider the dynamics of two-phase fluids, in particular the moving contact line, on a solid substrate. The dynamics are governed by the sharp-interface model consisting of the incompressible Navier-Stokes/ Stokes equations with the classical interface conditions, the Navier boundary condition for the slip velocity along the wall and a contact line condition which relates the dynamic contact angle of the interface to the contact line velocity. We propose an efficient numerical method for the model. The method combines a finite element method for the Navier-Stokes/Stokes equations on a moving mesh with a parametric finite element method for the dynamics of the fluid interface. The contact line condition is formulated as a timedependent Robin-type of boundary condition for the interface so it is naturally imposed in the weak form of the contact line model. For the Navier-Stokes equations, the numerical scheme obeys a similar energy law as in the continuum model but up to an error due to the interpolation of numerical solutions on the moving mesh. In contrast, for Stokes flows, the interpolation is not needed so we can prove the global unconditional stability of the numerical method in terms of the energy. Numerical examples are presented to demonstrate the convergence and accuracy of the numerical methods.

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“…Similar work for the two-phase Stokes flow without contact lines has been done in Ref. [52]; there the method was shown to be unconditionally stable.…”

Section: The Finite Element Methodssupporting

confidence: 56%

“…where U and P are defined in (2.16) and (2.17), respectively. These two choices satisfy the inf-sup stability condition [50,52],…”

Section: The Finite Element Methodsmentioning

confidence: 99%

An Energy-stable Finite Element Method for the Simulation of Moving Contact Lines in Two-phase Flows

Zhao,

Ren

2020

Preprint

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“…Hence there is no need to employ an ALE method. In fact, in the case of two-phase Stokes flow, all our proposed numerical methods collapse the approximation considered by the authors in [3].…”

Section: Introductionmentioning

confidence: 69%

“…Following the authors' approach in [3], see also [12,10], for the solution of (4.2) we use a Schur complement approach that eliminates (κ m+1 , δ X m+1 ) from (4.2), and then use an iterative solver for the remaining system in ( U m+1 , P m+1 ). This approach has the advantage that for the reduced system well-known solution methods for finite element discretizations for standard Navier-Stokes discretizations may be employed.…”

Section: Solvers For the Linear (Sub)problemsmentioning

confidence: 99%

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Fitted front tracking methods for two-phase incompressible Navier--Stokes flow: Eulerian and ALE finite element discretizations

Agnese¹,

Nürnberg²

2019

Preprint

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We investigate novel fitted finite element approximations for two-phase Navier-Stokes flow. In particular, we consider both Eulerian and Arbitrary Lagrangian-Eulerian (ALE) finite element formulations. The moving interface is approximated with the help of parametric piecewise linear finite element functions. The bulk mesh is fitted to the interface approximation, so that standard bulk finite element spaces can be used throughout. The meshes describing the discrete interface in general do not deteriorate in time, which means that in numerical simulations a smoothing or a remeshing of the interface mesh is not necessary. We present several numerical experiments, including convergence experiments and benchmark computations, for the introduced numerical methods, which demonstrate the accuracy and robustness of the proposed algorithms. We also compare the accuracy and efficiency of the Eulerian and ALE formulations.

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A Stable Parametric Finite Element Discretization of Two-Phase Navier–Stokes Flow

2014

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We present a parametric finite element approximation of two-phase flow. This free boundary problem is given by the Navier-Stokes equations in the two phases, which are coupled via jump conditions across the interface. Using a novel variational formulation for the interface evolution gives rise to a natural discretization of the mean curvature of the interface. The parametric finite element approximation of the evolving interface is then coupled to a standard finite element approximation of the two-phase Navier-Stokes equations in the bulk. Here enriching the pressure approximation space with the help of an XFEM function ensures good volume conservation properties for the two phase regions. In addition, the mesh quality of the parametric approximation of the interface in general does not deteriorate over time, and an equidistribution property can be shown for a semidiscrete continuousin-time variant of our scheme in two space dimensions. Moreover, our finite element approximation can be shown to be unconditionally stable. We demonstrate the applicability of our method with some numerical results in two and three space dimensions.

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Fitted finite element discretization of two‐phase Stokes flow

Cited by 8 publications

References 35 publications

An Energy-stable Finite Element Method for the Simulation of Moving Contact Lines in Two-phase Flows

An Energy-stable Finite Element Method for the Simulation of Moving Contact Lines in Two-phase Flows

Fitted front tracking methods for two-phase incompressible Navier--Stokes flow: Eulerian and ALE finite element discretizations

A Stable Parametric Finite Element Discretization of Two-Phase Navier–Stokes Flow

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