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Nonlinear SaffmanTaylor instability

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PASI 2007, Mar del Plata, Argentina. The Saffman-Taylor instability ... Darcy's law: v = -M P M=b2/12m. Numerical simulation: Eduard Paun . oil. air. oil. air ... – PowerPoint PPT presentation

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Title: Nonlinear SaffmanTaylor instability


1
Nonlinear Saffman-Taylor instability
Jordi Ortín Enric Álvarez Lacalle Jaume
Casademunt
PASI 2007, Mar del Plata, Argentina
2
The Saffman-Taylor instability
Hele-Shaw cell
3
Numerical simulation Eduard Pauné
4
Unstable
oil
Stable
air
oil
5
Radial cell
Experiment (2 liquids)
Numerical simulation Eduard Pauné
6
Formulation (channel geometry)
Bulk equations
Boundary conditions
Moving boundary problem
7
Linear dispersion relation

In a finite system all modes are linearly stable
for B gt 1
Bifurcation diagram
8
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
B1
H maximum-to-minimum distance on the interface
15
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16
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)
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