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Über dieses Buch

This book is the first monograph providing an introduction to and an overview of numerical methods for the simulation of two-phase incompressible flows. The Navier-Stokes equations describing the fluid dynamics are examined in combination with models for mass and surfactant transport. The book pursues a comprehensive approach: important modeling issues are treated, appropriate weak formulations are derived, level set and finite element discretization techniques are analyzed, efficient iterative solvers are investigated, implementational aspects are considered and the results of numerical experiments are presented. The book is aimed at M Sc and PhD students and other researchers in the fields of Numerical Analysis and Computational Engineering Science interested in the numerical treatment of two-phase incompressible flows.



1. Introduction

In this introductory chapter we describe the

models of one- and two-phase flow problems

that we consider, namely: 1) Navier-Stokes equations for one-phase flow (NS1), 2) Navier-Stokes equations for two-phase flow (NS2) 3) NS2 combined with transport of a dissolved species (NS2+T) 4)NS2 combined with transport of a surfactant on the interface (NS2+S).

Sven Gross, Arnold Reusken

One-phase incompressible flows


2. Mathematical models

We recall the non-stationary Navier-Stokes equations for modeling a


- phase incompressible flow problem:


$$\rho(\frac{\partial u}{\partial t} + (u. \nabla)u) + \nabla p - \mu\Delta u = \rho g\ in\ \Omega\ div\ u\ = 0\ in\ \Omega,$$

with given constants ρ > 0, µ > 0.

Sven Gross, Arnold Reusken

3. Finite element discretization

In this chapter we treat finite element methods for the discretization of the variational Oseen problem (2.21) and for the


discretization of the variational formulation of the non-stationary Stokes- and Navier-Stokes equations. We restrict ourselves to the class of Hood-Taylor finite elements on tetrahedral grids. In order to perform local grid refinement/coarsening in an efficient way, which is very important in two-phase flow applications, and to be able to use fast multigrid iterative solution methods we will apply such finite element methods not on one grid but on a

hierarchy of nested triangulations

. The construction of such a multilevel grid hierarchy is discussed in Sect. 3.1. In Sect. 3.2 the Hood-Taylor finite element spaces are treated. We present a numerical example in Sect. 3.3, where the approximation order of such a Hood-Taylor finite element space is investigated.

Sven Gross, Arnold Reusken

4. Time integration


$$I:=[0,t_e],f : I\rightarrow\mathbb{R}^N,F : \mathbb{R}^N\rightarrow\mathbb{R}^N$$


$$u_0 \epsilon \mathbb{R}^N$$


Sven Gross, Arnold Reusken

5. Iterative solvers

In this chapter we address the issue of iterative solvers for the large sparse (nonlinear) systems of equations that arise after space and time discretization (using an implicit time integration method) of the non-stationary Stokes and Navier-Stokes equations.

Sven Gross, Arnold Reusken

Two-phase incompressible flows


6. Mathematical model

We recall the Navier-Stokes model (1.19)-(1.21) for two-phase incompressible flows:


$$\left\{\begin{array}{ll} {{\rho i(\frac{\partial u}{\partial t}+(u.\triangledown)u)} = -\triangledown p + \rho i + div(\mu i D(u))}\\ div u=0 \end{array}\right.$$

in Ωi,


$$[\sigma n]=-\tau \kappa n ,\,\,[u] = 0 \,on\,\tau$$


$$[\sigma n]=-\tau \kappa n ,\,\,[u] = 0 \,on \,\tau$$

Sven Gross, Arnold Reusken

7. Finite element discretization of two-phase flow model

In this chapter we treat finite element methods for the two-phase flow model in (6.59).We use a nested family of multilevel triangulations


as explained in Sect. 3.1. In our applications these grids will be locally refined in a (small) neighborhood of the interface. In Sect. 7.2 we discuss a finite element method for discretization of the level set equation.

Sven Gross, Arnold Reusken

8. Time integration

For the two-phase flow problem, the time discretization is based on a generalization of the θ-scheme given in Sect. 4.2 for the one-phase flow Navier-Stokes equations. This generalized method is not found in the literature and therefore we describe its derivation in detail. The need for a generalization has two reasons. Firstly, opposite to the one-phase flow problem the mass matrix


is no longer constant but may vary in time.

Sven Gross, Arnold Reusken

9. Iterative solvers

In this chapter we address the issue of iterative solvers for the coupled nonlinear system of equations that arises in each time step of the implicit time integration methods treated in Chap. 8.

Sven Gross, Arnold Reusken

Mass transport


10. Mathematical model

We consider the model for transport of a dissolved species as given in (1.24). In (1.24) the unknown quantity (concentration) is denoted by





x, t

), the velocity field by


and the diffusion coefficient by


. In this and the next chapter we use a different notation for these quantities: the unknown function (concentration) is denoted by



x, t

) (instead of


), the velocity field by


(instead of


) and the diffusion coefficient by α.

Sven Gross, Arnold Reusken

11. Finite element discretization

In this chapter we discuss a finite element discretization method for the mass transport problem in (10.1). Compared to the usual convection-diffusion prob- lems there are two issues that make this problem more complicated. Firstly, one has to deal with the Henry interface condition in (10.1c) and secondly, due to this condition the solution




across the interface.

Sven Gross, Arnold Reusken

Surfactant transport


12. Mathematical model

We recall the model for transport of surfactants, derived in Sect. 1.1.4. In Sect. 1.1.4 the concentration of the surfactant is denoted by


, the ve- locity field by u and the diffusion coefficient by



Sven Gross, Arnold Reusken

13. Finite element methods for surfactant transport equations

In this chapter we treat different finite element approaches for interfacial trans- port problems as in (12.2). We first present a short overview of important classes of methods and then, in the following sections, treat some of these methods in more detail.

Sven Gross, Arnold Reusken



14. Appendix A: Results from differential geometry

We assume

$$\Gamma \subset \mathbb{R}^d$$

to be an oriented


-hypersurface, i.e., for

$$x^* \epsilon \Gamma$$

there exists an open set

$$U_{x*} \subset \mathbb{R}^d$$


$$x^* \epsilon U_{x*}$$

and a scalar function

$$\psi \epsilon C^2(U_{x*})$$

such that


$$U_{x^*} \cap \Gamma = {x \epsilon U_{x^*} : \psi (x) = 0 }, and \nabla \psi (x) \neq 0 for all x \epsilon U_{x^*} \cap \Gamma.$$

Sven Gross, Arnold Reusken

15. Appendix B: Variational formulations in Hilbert spaces

In this appendix we collect some results on the well-posedness of variational problems in Hilbert spaces. These results are known in the literature. For most of the proofs we refer to the literature.

Sven Gross, Arnold Reusken


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