Singularities in interfacial fluid dynamics

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) • Evolution of interface at different precision 7 digits -u u 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. Singularity removed by regularization Roll-up of vortex sheet at edge of circular tube Regularized vortex sheet calculation Didden 1979 Krasny 1986

8. Methods for analyzing singularities • Numerical • Similarity solutions -Jet pinch-off: Brenner, Eggers, Lister, Papageorgiou -3D Euler: Childress -Vortex Sheets: Pullin • Complex Variables -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) • Numerical validation: Krasny(1986), Shelley(1992), Baker • Cowley,Tanveer (1999) • 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)

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

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

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

24. ‘Moore’s’ equations for Rayleigh-Taylor (cont’d) • 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

25. Singularity formation: comparison of asymptotics and numerics

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

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

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

29. Hele-Shaw: Conformal map Singularity (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 B=0.00025 evolution for over Long time Comparison of B=0.00025, B=0 evolution

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

34. Numerical solutions Ceniceros, Hou and Si 1999

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 (see also Ambrose (2004), Cordoba et al (2007) Howison (2000)

36. Strategy to show existence (stable case) and construct singular solutions • 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 • Derive preliminary existence result involving class of solutions of the form are singular at t=0, e.g., Exact decaying solution of linearized system 0 < p < 1 • Remainder terms are estimated using abstract Cauchy-Kowalewski • theorem (Caflisch 1990)

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) (modulo log terms) • Constantin-Fefferman-Majda (1996), • Deng-Hou-You (2005)

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. 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 Growth of vorticity is bounded by double exponential From Hou & Li (2006)

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

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)