Finite Element Discretization Error Analysis of a Surface Tension Force in Two-Phase Incompressible Flows

We consider a standard model for a stationary two-phase incompressible flow with surface tension. In the variational formulation of the model a linear functional which describes the surface tension force occurs. This functional depends on the location and the curvature of the interface. In a finite element discretization method the functional has to be approximated. For an approximation method based on a Laplace-Beltrami representation of the curvature we derive sharp bounds for the approximation error. A new modified approximation method with a significantly smaller error is introduced.

[1]  S. Osher,et al.  Level set methods: an overview and some recent results , 2001 .

[2]  Arnold Reusken,et al.  Parallel Multilevel Tetrahedral Grid Refinement , 2005, SIAM J. Sci. Comput..

[3]  L. Tobiska,et al.  FINITE ELEMENT SIMULATION OF A DROPLET IMPINGING A HORIZONTAL SURFACE , 2005 .

[4]  S. Osher,et al.  A level set approach for computing solutions to incompressible two-phase flow , 1994 .

[5]  Vivette Girault,et al.  Finite Element Methods for Navier-Stokes Equations - Theory and Algorithms , 1986, Springer Series in Computational Mathematics.

[6]  Eberhard Bänsch,et al.  Finite element discretization of the Navier–Stokes equations with a free capillary surface , 2001, Numerische Mathematik.

[7]  P. Wesseling,et al.  A mass‐conserving Level‐Set method for modelling of multi‐phase flows , 2005 .

[8]  Xiaofeng Yang,et al.  An adaptive coupled level-set/volume-of-fluid interface capturing method for unstructured triangular grids , 2006, J. Comput. Phys..

[9]  Maxim A. Olshanskii,et al.  A STOKES INTERFACE PROBLEM: STABILITY, FINITE ELEMENT ANALYSIS AND A ROBUST SOLVER , 2004 .

[10]  J. Brackbill,et al.  A continuum method for modeling surface tension , 1992 .

[11]  Maxim A. Olshanskii,et al.  Analysis of a Stokes interface problem , 2006, Numerische Mathematik.

[12]  V. A. Solonnikov,et al.  Smooth interface in a two-component Stokes flow , 2001, ANNALI DELL UNIVERSITA DI FERRARA.

[13]  S. Pillapakkam,et al.  A level-set method for computing solutions to viscoelastic two-phase flow , 2001 .

[14]  Alan Demlow,et al.  An Adaptive Finite Element Method for the Laplace-Beltrami Operator on Implicitly Defined Surfaces , 2007, SIAM J. Numer. Anal..

[15]  G. Dziuk,et al.  An algorithm for evolutionary surfaces , 1990 .

[16]  V. Cristini,et al.  Adaptive unstructured volume remeshing - II: Application to two- and three-dimensional level-set simulations of multiphase flow , 2005 .

[17]  李幼升,et al.  Ph , 1989 .

[18]  P. Colella,et al.  An Adaptive Level Set Approach for Incompressible Two-Phase Flows , 1997 .

[19]  James A. Sethian,et al.  Level Set Methods and Fast Marching Methods , 1999 .

[20]  S. Osher,et al.  A Level Set Formulation of Eulerian Interface Capturing Methods for Incompressible Fluid Flows , 1996 .

[21]  A. Reusken,et al.  A finite element based level set method for two-phase incompressible flows , 2006 .

[22]  S. Hysing,et al.  A new implicit surface tension implementation for interfacial flows , 2006 .

[23]  Arnold Reusken,et al.  An extended pressure finite element space for two-phase incompressible flows with surface tension , 2007, J. Comput. Phys..

[24]  L. Tobiska,et al.  On spurious velocities in incompressible flow problems with interfaces , 2007 .

[25]  V. Yurinsky,et al.  A free-boundary problem for Stokes equations: classical solutions , 2000 .

[26]  Anna-Karin Tornberg,et al.  Interface tracking methods with application to multiphase flows , 2000 .

[27]  Maxim A. Olshanskii,et al.  Uniform preconditioners for a parameter dependent saddle point problem with application to generalized Stokes interface equations , 2006, Numerische Mathematik.

[28]  Björn Engquist,et al.  A finite element based level-set method for multiphase flow applications , 2000 .