1 / 16

Nonlinear Saffman-Taylor instability

Universitat de Barcelona. Nonlinear Saffman-Taylor instability. Jordi Ortín Enric Álvarez Lacalle Jaume Casademunt. PASI 2007, Mar del Plata, Argentina. b. Darcy’s law:. v = -M Ñ P M=b 2 /12 m. air. oil. The Saffman-Taylor instability. Hele-Shaw cell:.

mulan
Download Presentation

Nonlinear Saffman-Taylor instability

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Universitat de Barcelona Nonlinear Saffman-Taylor instability Jordi Ortín Enric Álvarez LacalleJaume Casademunt PASI 2007, Mar del Plata, Argentina

  2. b Darcy’s law: v = -M ÑP M=b2/12m air oil The Saffman-Taylor instability Hele-Shaw cell:

  3. Numerical simulation: Eduard Pauné

  4. air Unstable oil Stable air oil

  5. Injection Injection + rotation Radial cell: Experiment (2 liquids) Numerical simulation: Eduard Pauné

  6. Formulation (channel geometry): Bulk equations: Boundary conditions: Moving boundary problem

  7. w Relación de dispersión lineal: , when fluid 1 displaces fluid 2 Cuando fluido 1 avanza sobre fluido 2 Linearly unstable Linearly stable 1 B Linear dispersion relation: In a finite system all modes are linearly stable for B > 1 Bifurcation diagram:

  8. Linearly unstable Linearly stable 1 B Weakly nonlinear analysis: EAL, JC and JO, Phys. Rev. E 62, 016302 (2001) A weakly nonlinear analysis applied to the situation B = 1 – h, with h small: d1 is the dimensionless amplitude of the mode k = 1 (only unstable mode) New bifurcation diagram:

  9. Gallery of unstable stationary solutions Family of exact elastica solutions: [Nye et al., Eur. J. Phys. 5, 73 (1984)] s arclength q angle between the interface tangent and the x axis The vorticity vanishes identically at the interface: the solutions are stationary. They belong to the unstable branch of the subcritical bifurcation.

  10. Channel geometry Radial geometry with rotation Balance of capillary and centrifugal forces Balance of capillary and viscous forces

  11. The elastica solutions in the bifurcation diagram:

  12. Dynamic relevance of the bifurcation diagram:

  13. Arbitrary initial condition: Exploiting the sensitivity of capillary pressure to slight gap thickness variations

  14. The nonlinear Saffman-Taylor instability: Non-planar stationary solution Pinch-off singularity Planar stationary solution B=1 H: maximum-to-minimum distance on the interface

  15. Conclusions • The elastica solutions of the Saffman-Taylor problem • belong to the unstable branch of the subcritical bifurcation • 2. This branch ends at a topological singularity • (interface pinchoff) • The dynamic relevance of the elastica solutions • can be verified experimentally through the observation • of the nonlinear character of the instability E. Álvarez-Lacalle, JO, J. Casademunt, PRL 92, 054501 (2004)

More Related