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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:.
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Universitat de Barcelona Nonlinear Saffman-Taylor instability Jordi Ortín Enric Álvarez LacalleJaume Casademunt PASI 2007, Mar del Plata, Argentina
b Darcy’s law: v = -M ÑP M=b2/12m air oil The Saffman-Taylor instability Hele-Shaw cell:
air Unstable oil Stable air oil
Injection Injection + rotation Radial cell: Experiment (2 liquids) Numerical simulation: Eduard Pauné
Formulation (channel geometry): Bulk equations: Boundary conditions: Moving boundary problem
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:
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:
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.
Channel geometry Radial geometry with rotation Balance of capillary and centrifugal forces Balance of capillary and viscous forces
Arbitrary initial condition: Exploiting the sensitivity of capillary pressure to slight gap thickness variations
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
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)