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Saffman-Taylor streamer discharges

Saffman-Taylor streamer discharges. Fabian Brau , CWI Amsterdam Alejandro Luque , CWI Amsterdam Ute Ebert , CWI Amsterdam Eindhoven University of Technology. Streamers, sprites, leaders, lightning: from micro- to macroscales Leiden, 11 October 2007. Talk overview.

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Saffman-Taylor streamer discharges

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  1. Saffman-Taylor streamer discharges Fabian Brau,CWI Amsterdam Alejandro Luque, CWI Amsterdam Ute Ebert, CWI Amsterdam Eindhoven University of Technology Streamers, sprites, leaders, lightning: from micro- to macroscales Leiden, 11 October 2007

  2. Talk overview • Minimal PDE model for streamers • Characteristics of single streamers • Interacting streamers : periodic array of streamers in 2D • Numerical solutions + characteristics • FMB explicit solution fits very well the numerical fronts • Conclusions

  3. Minimal PDE model for streamers In dimensionless unit, the minimal PDE model reads • s : Electron density • r: Ion density Solved in the background of a homogeneous field Negative streamer Non attaching gas like nitrogen Normal condition → Electron impact ionization : Townsend’s Approximation → Poisson equation Electric potential Electric field Cf talks of Alajandro Luque and Chao Li

  4. Net charge ( ) Electric Field (kV/cm) z (mm) z (mm) r (mm) r (mm) Characteristics of single streamers(Valid also for interacting streamers) Evolution of some initial condition in the E-field produced by two electrodes: Solutions after the avalanche phase looks like (3D + cylindrical symmetry) Thin charge layer E-field enhancement E-field screening [Montijn, Ebert, Hundsdorfer, J. Comp. Phys. 2006, Phys. Rev. E 2006, J. Phys. D 2006 Luque, Ebert, Montijn, Hundsdorfer, Appl. Phys. Lett. 2007]

  5. Net charge and E-field evolutionfor single streamers NetCharge E-field On branching as a Laplacian instability: [Arrayas, Ebert, Hundsdorfer, Phys. Rev. Lett. 2002, Ebert et al., Plasma Sour. Sci. Techn. 2006]

  6. Interacting streamers as periodic array : previous work G V Naidis, J. Phys. D 29, 779 (1996) • Streamers with fixed radius • Essentially 1D • Charge density along a line • Study how the interaction affects the charge density, the electric field and the velocity

  7. L Interacting streamers : periodic array of streamers in 2D Anode Direction of propagation Cathode Neumann boundary conditions for: Potential and Densities = Symmetry axis

  8. Characteristics of Interacting streamers • If : Streamers do not branch 256

  9. Net charge and E-field evolutionfor interacting streamers NetCharge E-field

  10. Characteristics of Interacting streamers • Uniformly translating streamers • Results are robust against changes in the initial condition

  11. FMB mathematical setup Single streamers x [E. D. Lozansky and O. B. Firsov, J. Phys. D 6, 976 (1973)] Interacting streamers: periodic array of streamers in 2D Ideally conducting streamer Free moving boundary for Hele-Shaw flow Saffman-Taylor solution y L

  12. Colored Water → Hele-Shaw Flow Radial Symmetry Hole Glycerol

  13. Hole Colored Water → Glycerol Hele-Shaw Flow Radial Symmetry

  14. Colored Water → Hele-Shaw Flow Radial Symmetry Hole Glycerol Channel configuration

  15. Saffman-Taylor solution Family of possible solutions

  16. Selection: Boundary condition The boundary condition doesn’t allow any selection mechanism [B. Meulenbroek, U. Ebert and L. Schäfer, PRL 95 (2005) F. Brau, A. Luque, B. Meulenbroek, U. Ebert and L. Schäfer, (2007)]

  17. For small surface tension, only the finger with is selected Saffman-Taylor solution

  18. Saffman-Taylor solution Experiment Theory

  19. Comparison: PDE vs FMB

  20. Comparison: PDE vs FMB The tip of the Saffman – Taylor finger coincide with the maximum of the net charge of the PDE solution No free parameter Saffman-Taylor finger

  21. Comparison: PDE vs FMB Maximum of the net charge along y axis for each value of x

  22. Comparison: PDE vs FMB Saffman-Taylor finger

  23. Various evolutions of the streamer fronts + comparison with Saffman-Taylor Front From the left: Saffman-Taylor finger

  24. Conclusions Interacting streamers viewed as a periodic array of streamers in 2D present remarkable features: • If the streamer spacing is small enough for a given background electric field, streamers do not branch. • When streamers do not branch they reach a steady state which is an attractor of the dynamics. • This steady state is well approximated by a solution from hydrodynamics: the Saffman-Taylor finger.

  25. Predictions If you are in the right part of the phase diagram: • In contrast to single streamers, branchings should be mostly suppressed • After some transient evolution, the velocity should reach a constant value • This value is in 2D. In 3D, we expect that the following linear relation should hold In physical units:

  26. Branching

  27. Electric field along the streamer axis

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