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  • Cited by 7
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    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Ruiz Álvarez, Juan and Li, Zhilin 2015. The immersed interface method for axis-symmetric problems and application to the Hele–Shaw flow. Applied Mathematics and Computation, Vol. 264, p. 179.

    Ambrose, David M. 2014. The Zero Surface Tension Limit of Two-Dimensional Interfacial Darcy Flow. Journal of Mathematical Fluid Mechanics, Vol. 16, Issue. 1, p. 105.

    Miranda, José A. and Alvarez-Lacalle, Enrique 2005. Viscosity contrast effects on fingering formation in rotating Hele-Shaw flows. Physical Review E, Vol. 72, Issue. 2,

    Casademunt, Jaume 2004. Viscous fingering as a paradigm of interfacial pattern formation: Recent results and new challenges. Chaos: An Interdisciplinary Journal of Nonlinear Science, Vol. 14, Issue. 3, p. 809.

    Alvarez-Lacalle, E. Pauné, E. Casademunt, J. and Ortín, J. 2003. Systematic weakly nonlinear analysis of radial viscous fingering. Physical Review E, Vol. 68, Issue. 2,

    Pauné, E. Siegel, M. and Casademunt, J. 2002. Effects of small surface tension in Hele-Shaw multifinger dynamics: An analytical and numerical study. Physical Review E, Vol. 66, Issue. 4,

    Hou, T.Y. Lowengrub, J.S. and Shelley, M.J. 2001. Boundary Integral Methods for Multicomponent Fluids and Multiphase Materials. Journal of Computational Physics, Vol. 169, Issue. 2, p. 302.

  • Journal of Fluid Mechanics, Volume 409
  • April 2000, pp. 251-272

The singular perturbation of surface tension in Hele-Shaw flows

  • DOI:
  • Published online: 01 April 2000

Morphological instabilities are common to pattern formation problems such as the non-equilibrium growth of crystals and directional solidification. Very small perturbations caused by noise originate convoluted interfacial patterns when surface tension is small. The generic mechanisms in the formation of these complex patterns are present in the simpler problem of a Hele-Shaw interface. Amid this extreme noise sensitivity, what is then the role played by small surface tension in the dynamic formation and selection of these patterns? What is the asymptotic behaviour of the interface in the limit as surface tension tends to zero? The ill-posedness of the zero-surface-tension problem and the singular nature of surface tension pose challenging difficulties in the investigation of these questions. Here, we design a novel numerical method that greatly reduces the impact of noise, and allows us to accurately capture and identify the singular contributions of extremely small surface tensions. The numerical method combines the use of a compact interface parametrization, a rescaling of the governing equations, and very high precision. Our numerical results demonstrate clearly that the zero-surface-tension limit is indeed singular. The impact of a surface-tension-induced complex singularity is revealed in detail. The singular effects of surface tension are first felt at the tip of the interface and subsequently spread around it. The numerical simulations also indicate that surface tension defines a length scale in the fingers developing in a later stage of the interface evolution.

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Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
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