1. Singularities in interfacial fluid dynamics Michael Siegel Dept. of Mathematical Sciences NJIT Supported by National Science Foundation

2. Outline Singularities on interfaces -Motivation and examples -Methods for analyzing singularities -Kelvin-Helmholtz -Rayleigh-Taylor -Hele-Shaw Singularity formation in 3D Euler flow

3. Example 1: Breakup of a viscous drop Shi, Brenner, Nagel ‘94 Similarity solution Eggers ’93 Stone, Lister ’98 (modifications due to exterior fluid)

4. Kelvin-Helmholtz instability Krasny (1986) -u u Evolution of interface at different precision 7 digits 16 digits 29 digits

5. Kelvin-Helmholtz (cont’d) Irregular point vortex motion at later times Krasny ‘86

6. Importance of singularity Mathematical theory (existence of solutions, continuous dependence on data) Numerical computation Physical importance depends on particular problem

7. Roll-up of vortex sheet at edge of circular tube Regularized vortex sheet calculation Krasny 1986 Didden 1979 Singularity removed by regularization

8. Methods for analyzing singularities - Jet pinch-off: Brenner, Eggers, Lister, Papageorgiou -3D Euler: Childress -Vortex Sheets: Pullin Complex Variables Numerical Similarity solutions - Bardos, Hou, Frisch, Sinai, Caflisch, Tanveer, S. Unfolding - Caflisch

9. Complex Variables: Canonical example 1 Laplace equation Initial value problem is ill-posed

10. Complex Variables: Example 2

11. Burger’s equation example, cont’d

12. Numerical analysis of complex singularities Sulem, Sulem, Frisch (1983) Vortex sheets: S., Krasny, Shelley, Baker, Caflisch Taylor-Green: Brachet and collaborators 2D Euler: Frisch and collaborators 3D Euler: Siegel and Caflisch

13. 1-D example

14. Singularity fit for Burger’s equation Initial value problem solved using pseudo-spectral method, u=sin(x) data

15. Singularity fit for Burger near shock

16. Fit to Fourier coefficients in 2D Malakuti, Maung, Vi, Caflisch, Siegel 2007

17. Outline Singularities on interfaces -Motivation and examples -Methods for analyzing singularities -Kelvin-Helmholtz -Rayleigh-Taylor -Hele-Shaw Singularity formation in 3D Euler flow

18. Kelvin-Helmholtz instability Birkhoff-Rott equation:

19. Moore’s analysis (1979)

20. Kelvin-Holmholtz (cont’d) Rigorous construction of singular solution, demonstration of ill-posedness (Duchon & Robert 1988, Caflisch & Orellana 1989) Regularized evolution: vortex blobs (Krasny ’86) Surface tension regularization: Hou, Lowengrub, Shelley (1994), Baker, Nachbin (1997), S. (1995), Ambrose (2004) Numerical validation: Krasny(1986), Shelley(1992), Baker Cowley,Tanveer (1999)

21. Vortex sheet singularity for Rayleigh-Taylor (from Ceniceros and Hou) Baker, S., Caflisch 1993

22. Moore’s construction (Baker, Caflisch, S. ’93 interpretation)

23. Evaluate PV integral by contour integration Equivalence to Moore’s approximation

24. The system of `Moore’s’ equations admit traveling wave solutions (complex wavespeed) with 3/2 singularities The speed of the nonlinear traveling wave is independent of the amplitude and identical to the speed given by a linear analysis This is a general property of upper analytic systems of PDE’s, as long as system of ODE’s resulting from substitution of the traveling wave variable is autonomous ‘Moore’s’ equations for Rayleigh-Taylor (cont’d)

25. Singularity formation: comparison of asymptotics and numerics

26. Two-phase Hele-Shaw, or porous media, flow Boundary Conditions: Hele-Shaw flow: Problem formulation Water Oil

27. Colored water injected into glycerin NJIT Capstone Lab (Kondic)

28. Hele-Shaw flow: one phase problem Exact solutions derived using conformal map (Howison)

29. Singularity Hele-Shaw: Conformal map (e.g., Baker, S., Tanveer ’95)

30. Zero surface tension limit (Tanveer ’93, S., Tanveer, Dai ’96)

31. Channel problem Siegel, Tanveer, Dai ‘96 B=0

32. Hele-Shaw: Radial geometry Comparison of B=0.00025, B=0 evolution B=0.00025 evolution for over Long time

33. Detailed numerical studies suggest cusps can form (Ceniceros, Hou, Si 1999) Singularities in two-phase Hele-Shaw (Muskat) problem Much less is known about two phase Hele-Shaw flow There are no know singular exact solutions Originally proposed as a model for displacement of oil by water in a porous medium

34. Ceniceros, Hou and Si 1999 Numerical solutions

35. Construction of singular solution (S., Caflisch, Howison, CPAM 2004) S., Caflisch, Howison prove a global existence theorem for forward problem with small data. The initial data is allowed to have a curvature (or weaker) singularity, but the solution is analytic for subsequent times Time reversibility implies there are solutions to the backward problem that start smooth but develop a curvature singularity -Not a foregone conclusion: bounded finger velocity in the two fluid case; negative interfacial pressure weakens “runaway” that leads to cusp formation In view of waiting time behavior (King, Lacey, Vasquez ’95), different techniques will be required to show cusp or corner formation Shows backward Muskat problem is ill-posed (on non-linear theory) Apply ideas to 3D Euler Howison (2000) (see also Ambrose (2004), Cordoba et al (2007)

36. Strategy to show existence (stable case) and construct singular solutions Derive preliminary existence result involving class of solutions of the form Remainder terms are estimated using abstract Cauchy-Kowalewski theorem (Caflisch 1990) Approach is similar to that for Kelvin-Helmholtz problem (Duchon, Robert 1988, Caflisch, Orellana 1989) 1. Extend equations to complex 2. Put singularity in initial data 3. Construct solution within class of analytic functions are singular at t=0, e.g., Exact decaying solution of linearized system 0 < p < 1

37. New Challenges presented by Muskat problem Nonlinear term is considerably more complicated Presence of a nonphysical ``reparameterization’’ mode (neutrally stable mode) -Analysis is modified to accommodate this mode by prescribing its data at, i.e., by requiring it to go to 0 as This results in an existence theorem for what appears to be a restricted set of data depends on Introduction of a reparameterization converts to existence for any initial data First (?) global existence result that relies on stable decay rate k to show that solutions become analytic after initial time

38. Euler singularity problem is an outstanding open problem in mathematics & physics Euler singularity connects Navier-Stokes dynamics to Kolmogorov scaling Can a solution to the incompressible Euler equations become singular in finite time, starting from smooth (analytic) initial data?

39. Theoretical Results Beale-Kato-Majda (1984) Constantin-Fefferman-Majda (1996), Deng-Hou-You (2005) (modulo log terms)

40. Numerical Studies Axisymmetric flow with swirl and 2D Boussinesq convection -Grauer & Sideris (1991, 1995), Pumir & Siggia (1992) Meiron & Shelley (1992), E & Shu (1994) Grauer et al (1998), Yin & Tang (2006) High symmetry flows -Kida-Pelz flow: Kida (1985), Pelz & coworkers (1994,1997) -Taylor-Green flow: Brachet & coworkers (1983,2005) Antiparallel vortex tubes -Kerr (1993, 2005) -Hou & Li (2006) Pauls, Frisch et al(2006).: Study of complex space singularities for 2D Euler in short time asymptotic regime

41. Hou and Li (2006) reconsidered Kerr’s (1993,2005) calculation

42. Growth of maximum vorticity from Hou and Li (2006) Rapid growth of vorticity

43. Growth of vorticity is bounded by double exponential No conclusive numerical evidence for singularities e.g., Kerr’s (1993) numerics suggest singularity formation, but higher resolution calculations for same initial data by Hou & Li (2006) show double exponential growth of vorticity From Hou & Li (2006)

45. Complex singularities for axisymmetric flow with swirl Annular geometry Steady background flow chosen to give instability with an unstable eigenmode Caflisch (1993), Caflisch & Siegel (2004) swirl

46. Traveling wave solution Baker, Caflisch & Siegel (1993) Caflisch(1993), Caflisch & Siegel (2004) Exact solution of Euler

47. Motivation for traveling wave form

48. Motivation (cont’d)

49. Numerical method for swirl and axial background velocity

50. Caflisch & Siegel (2004)

53. Perturbation construction of real singular solution Real remainder

54. Difficulties Numerical method is highly unstable, resolved using high precision arithemetic Too numerically intensive for 3D Square root singularity does not satisfy Beale-Kato-Majda theorem

55. 3D Traveling wave solution Pos. wavenumber modes Const. Fourier Coeffs. Exact solution of Euler but for complex velocity Simplify construction -Base flow, instability driven by forcing term No observed numerical instability! Siegel, Caflisch 2007

56. 3D traveling wave (cont’d)

57. Numerical method

58. Fit of singularity parameters BKM satisfied

59. Fit of singularity parameters

61. Singularity amplitude

62. Singular surface Geometry of singular surface is useful for analysis

63. Other singularity types Also find square root and cube root singularities Fit for cube root

64. Square root singularity

65. Conclusions Singularity formation is important in mathematical theory and numerical computation Physical significance depends on particular problem Singularities are often removed by regularization, but are relevant in understanding zero regularization limit New results presented concerning complex singularities for 3D Euler,