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On finite element modelling of surface tension Variational formulation and applications – Part I: Quasistatic problems

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Abstract

This paper describes variational formulation and finite element discretization of surface tension. The finite element formulation is cast in the Lagrangian framework, which describes explicitly the interface evolution. In this context surface tension formulation emerges naturally through the weak form of the Laplace–Young equation.

The constitutive equations describing the behaviour of Newtonian fluids are approximated over a finite time step, leaving the governing equations for the free surface flow function of geometry change rather than velocities. These nonlinear equations are then solved by using Newton-Raphson procedure.

Numerical examples have been executed and verified against the solution of the ordinary differential equation resulting from a parameterization of the Laplace-Young equation for equilibrium shapes of drops and liquid bridges under the influence of gravity and for various contact angle boundary conditions.

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References

  1. Landau LD, Lifshitz EM (1987) Fluid Mechanics. Pergamon Press, 2nd edition

  2. Pozrikidis C (1997) Introduction to Theoretical and Computational Fluid Mechanics. Oxford University Press

  3. Daly BJ (1969) A technique for including surface tension effects in hydrodynamics calculations. J. Comp. Physics 4:97–117

    MATH  Google Scholar 

  4. Tezduyar TE (2001) Finite element methods for flow problems with moving boundaries and interfaces. Arch. Comput. Meth. Engng. 8:83–130

    Google Scholar 

  5. Esmaeeli A, Juric D, Al-Rawahi N, Tauber W, Han J, Nas S, Tryggvason G, Bunner B, Jan YJ (2001) Front tracking method for the computations of multiphase flow. J. Comp. Physics 169:708–759

    MATH  Google Scholar 

  6. Harlow FH, Welch JE 1965 Numerical calculation of time-depen dent viscous incomressible flow of fluid with free surface. Phys. Fluids 8:2182–2189

    Google Scholar 

  7. Hirt CW, Nichols BD (1981) Volume of fluid (vof) method for the dynamics of an incompressible fluid with free surfaces. J. Comp. Physics 39:201–225

    MATH  Google Scholar 

  8. Sethian JA (1985) Curvature and the evolution of fronts. Commun. Math. Physics 101:487–499

    MATH  MathSciNet  Google Scholar 

  9. Bach P, Hassager O (1985) An algorithm for the use of the Lagrangian specification in Newtonian fluid mechanics with applications to free surface flow. J. Fluid Mech. 152:173–190

    Google Scholar 

  10. Ramaswamy B, Kawahara M, Nakayama T (1986) Lagrangian finite element method for the analysis of two dimensional sloshing problems. Int. J. Num. Meth. Fluids 6:659–670

    MATH  MathSciNet  Google Scholar 

  11. Ramaswamy B, Kawahara M (1987) Lagrangian finite element analysis applied to viscous free surface fluid flow. Int. J. Num. Meth. Fluids 7:953–984

    MATH  Google Scholar 

  12. Radovitzky R, Ortiz M (1998) Lagrangian finite element analysis of newtonian fluid flows. Int. J. Num. Meth. Engng. 43:607–619

    Google Scholar 

  13. Bonet J (1994) The incremental flow formulation for the numerical analysis of plane stress and thin sheet viscous forming processes. Comput. Methods Appl. Mech. Engrg. 114:103–122

    Google Scholar 

  14. Bonet J, Bhargava P (1995) The incremental flow formulation for the numerical analysis of 3-dimensional viscous deformation processes: continuum formulation and computational aspects. Comput. Methods Appl. Mech. Engrg. 122:51–68

    Google Scholar 

  15. Bonet J (1998) Recent developments in the incremental flow formulation for the numerical simulation of metal forming processes. Engng. Comp. 15:345–356

    Google Scholar 

  16. Daly BJ, Pracht WE (1968) Numerical study of density-current surges. Phys. Fluids 11:15–30

    Article  MATH  Google Scholar 

  17. Daly BJ (1969) Numerical study of the effects of surface tension on interface stability. Phys. Fluids 12:1340–1354

    Article  MATH  Google Scholar 

  18. Brackbill JU, Kothe DB, Zemach C (1992) A continuum method for modelling surface tension. J. Comp. Physics 100:335–354

    MATH  MathSciNet  Google Scholar 

  19. Jamet D, Torres D, Brackbill JU (2002) On the theory and computations of surface tensions: The elimination of parasitic currents through energy conservation in the second-gradient method. J. Comp. Physics 182:262–276

    MATH  Google Scholar 

  20. Kothe DB, Mjolsness RC (1992) Ripple: a new model for incompressible flows with free surfaces. AIAA J. 30:2694–2700

    Google Scholar 

  21. Lafaurie B, Nardone C, Scardovelli R, Zalesky S, Zanetti G (1994) Modelling merging and fragmentation in multiphase flows with SURFFER. J. Comp. Physics 113:134–147

    MathSciNet  MATH  Google Scholar 

  22. Wu J, Yu ST, Jiang BN (1998) Simulation of two-fluid flows by the least-square finite element method using a continuum surface tension model. Int. J. Num. Meth. Engng. 42:583–600

    Google Scholar 

  23. Ruschak, KJ (1980) A method for incorporating free surface boundaries with surface tension in finite element fluid-flow simulators. Int. J. Num. Meth. Engng. 15:639–648

    Google Scholar 

  24. Christodoulou, KN, Scriven LE (1989) The fluid mechanics of slide coating. J. Fluid Mech. 208:321–354

    Google Scholar 

  25. Yildirim, OE, Basaran OA (2001) Deformation and breakup of stretching bridges of Newtonian and shear-thinning liquids: comparison of one- and two-dimensional models. Chem. Eng. Sci. 56:211–233

    Google Scholar 

  26. Dettmer W, Saksono PH, Perić D (2003) On a finite element formulation for incompressible Newtonian fluid flows on moving domains in the presence of surface tension. Commun. Numer. Meth. Engng. 19:659–668

    Google Scholar 

  27. Dettmer W, Perić D (2005) A computational framework for free surface fluid flows accounting for surface tension. Comput. Methods Appl. Mech. Engrg. in press

  28. Cairncross RA, Schunk PR, Baer TA, Rao RR, Sackinger PA (2000) Finite element method for free surface flows of incompressible fluids in three dimensions. Part I. Boundary fitted mesh motion. Int. J. Numer. Meth. Fluids 33:375–403

    MATH  Google Scholar 

  29. Bellet M (2001) Implementation of surface tension with wall adhesion effects in a three-dimensional finite element model for fluid flow. Commun. Numer. Meth. Engng. 17:563–579

    Google Scholar 

  30. Saksono PH, Perić D (2005) On finite element modelling of surface tension: variational formulation and applications – Part II: dynamic problems. Comp. Mech. DOI 10.1007/s00466-005-0745-7

  31. Simo JC, Taylor RL (1982) Penalty function formulations for incompressible nonlinear elastostatics. Comput. Methods Appl. Mech. Engrg. 35:107–118

    Google Scholar 

  32. Sussman T, Bathe KJ (1987) A finite element formulation for nonlinear incompressible elastic and inelastic analysis. Comp. Struct. 26:357–409

    Google Scholar 

  33. Liu CH, Hofstetter G, Mang HA (1994) 3D finite element analysis of rubber-like materials at finite strains. Engng. Comp. 11:111–128

    Google Scholar 

  34. Doll S, Schweizerhof K (2000) On the development of volumetric strain energy functions. J. Appl. Mech. 67:17–21

    Google Scholar 

  35. Weatherburn CE (1930) Differential Geometry and Three Dimensions. Cambridge University Press

  36. Hughes TJR (1987) The Finite Element Method. Linear Static and Dynamic Finite Element Analysis. Prentice-Hall

  37. Zienkiewicz OC, Taylor RL (1989) The Finite Element Method volume 1, Wiley

  38. Bathe KJ (1996) Finite Element Procedures. Prentice-Hall

  39. Pitts E (1974) The stability of pendent drops – Part 2. Axial symmetry. J. Fluid Mech. 63:487–508

    Google Scholar 

  40. Michael DH, Williams PG (1976) The equilibrium and stability of axisymmetric pendent drops. Proc. R. Soc. Lond. A 351:117–127

    Google Scholar 

  41. Finn R (1986) Equilibrium Capillary Surfaces. Springer-Verlag

  42. Padday JF (1971) The profiles of axially symmetric menisci. Phil. Trans. R. Soc. Lond. A 269:265–293

    Google Scholar 

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  1. Civil and Computational Engineering Centre, School of Engineering, University of Wales Swansea, Swansea, SA2 8PP, U.K

    P. H. Saksono & D. Perić

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  1. P. H. Saksono
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  2. D. Perić
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Correspondence to D. Perić.

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Saksono, P., Perić, D. On finite element modelling of surface tension Variational formulation and applications – Part I: Quasistatic problems. Comput Mech 38, 265–281 (2006). https://doi.org/10.1007/s00466-005-0747-5

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