Computing Through Singularities in Potential Flow with Applications to Electrohydrodynamic Problems

  • Maria Garzon
  • James A. Sethian
  • Len J. Gray
  • August Johansson
Conference paper
Part of the Mathematics in Industry book series (MATHINDUSTRY, volume 26)


Many interesting fluid interface problems, such as wave propagation and breaking, droplet and bubble break-up, electro-jetting, rain drops, etc. can be modeled using the assumption of potential flow. The main challenge, both theoretically and computationally, is due to the presence of singularities in the mathematical models. In all the above mentioned problems, an interface needs to be advanced by a velocity determined by the solution of a surface partial differential equation posed on this moving boundary. By using a level set framework, the two surface equations of the Lagrangian formulation can be implicitly embedded in PDEs posed on one higher dimension fixed domain. The advantage of this approach is that it seamlessly allows breakup or merging of the fluid domain and therefore provide a robust algorithm to compute through these singular events. Numerical results of a solitary wave breaking, the Rayleigh-Taylor instability of a fluid column, droplets and bubbles breaking-up and the electrical droplet distortion and subsequent jet emission can be obtained using this levelset/extended potential model.



This work was supported by the U.S. Department of Energy, under contract Number DE-AC02-05CH11231, the Spanish Ministry of Science and Innovation, Project Number MTM2013-43671-P. The third author was also supported by the Research Council of Norway through a Centers of Excellence grant to the Center for Biomedical Computing at Simula Research Laboratory, Project Number 179578, as well as through the FRIPRO program at Simula Research Laboratory, project number 251237.


  1. 1.
    Burman, E.: Ghost penalty. C. R. Math. 348, 1217–120 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Burman, E., Hansbo, P.: Fictitious domain finite element methods using cut elements: II. A stabilized Nitsche method. Appl. Numer. Math. 62(4), 328–341 (2012)zbMATHGoogle Scholar
  3. 3.
    Eggers, J.: Nonlinear dynamics and breakup of free surface flows. Rev. Mod. Phys. 69(3), 865–929 (1967)CrossRefzbMATHGoogle Scholar
  4. 4.
    Garzon, M., Adalsteinsson, D., Gray, L.J., Sethian, J.A.: A coupled level set-boundary integral method for moving boundary simulations. Interfaces Free Bound. 7, 277–302 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Garzon, M., Gray, L.J., Sethian, J.A.: Numerical simulation of non-viscous liquid pinch-off using a coupled levelset-boundary integral method. J. Comput. Phys. 228, 6079–6106 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Garzon, M., Gray, L.J., Sethian, J.A.: Simulation of the droplet-to-bubble transition in a two-fluid system. Phys. Rev. E. 83, 046318 (2011)CrossRefGoogle Scholar
  7. 7.
    Garzon, M., Gray, L.J., Sethian, J.A.: Axisymmetric boundary integral formulation for a two-fluid system. Int. J. Numer. Methods Fluids 69, 1124–1134 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Garzon, M., Gray, L.J., Sethian, J.A.: Numerical simulations of electrostatically driven jets from nonviscous droplets. Phys. Rev. E. 89, 033011 (2014)CrossRefGoogle Scholar
  9. 9.
    Gray, L.J., Garzon, M., Mantic, V., Graciani, E.: Galerkin boundary integral analysis Iof the axi-symmetric Laplace equation. Int. J. Numer. Methods Eng. 66, 2014–2034 (2006)CrossRefzbMATHGoogle Scholar
  10. 10.
    Grilli, S.T., Guyenne, P., Dias, F.: A fully nonlinear model for three dimensional overturning waves over arbitrary bottom. Int. J. Numer Methods Fluids 35, 828–867 (2001)CrossRefzbMATHGoogle Scholar
  11. 11.
    Grimm, R.L., Beauchamp, J.L.: Dynamics of field induced droplet ionization: time-resolved studies of distortion, jetting and progeny formation from charged and neutral methanol droplets exposed to strong electric fields. J. Phys. Chem. B 109, 8244–8250 (2005)CrossRefGoogle Scholar
  12. 12.
    Johansson, A., Garzon, M., Sethian, J.A.: A three-dimensional coupled Nitsche and level set method for electrohydrodynamic potential flows in moving domains. J. Comput. Phys. 309, 88–111 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Thoroddsen, S.T.: Micro-droplets and micro-bubbles imaging motion at small scales. Nus. Eng. Res. News 22(1) (2007)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  • Maria Garzon
    • 1
  • James A. Sethian
    • 2
  • Len J. Gray
    • 3
  • August Johansson
    • 4
  1. 1.Department of MathematicsUniversidad de OviedoOviedoSpain
  2. 2.University of CaliforniaBerkeleyUSA
  3. 3.Bergen Software Services InternationalBergenNorway
  4. 4.Simula Research LaboratoryFornebuNorway

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