Corner and finger formation in Hele-Shaw flow with kinetic undercooling regularisation

Michael C. Dallaston, Scott W. McCue

Research output: Contribution to journalArticlepeer-review

9 Citations (Scopus)
49 Downloads (Pure)


We examine the effect of a kinetic undercooling condition on the evolution of a free boundary in Hele-Shaw flow, in both bubble and channel geometries. We present analytical and numerical evidence that the bubble boundary is unstable and may develop one or more corners in finite time, for both expansion and contraction cases. This loss of regularity is interesting because it occurs regardless of whether the less viscous fluid is displacing the more viscous fluid, or vice versa. We show that small contracting bubbles are described to leading order by a well-studied geometric flow rule. Exact solutions to this asymptotic problem continue past the corner formation until the bubble contracts to a point as a slit in the limit. Lastly, we consider the evolving boundary with kinetic undercooling in a Saffman-Taylor channel geometry. The boundary may either form corners in finite time, or evolve to a single long finger travelling at constant speed, depending on the strength of kinetic undercooling. We demonstrate these two different behaviours numerically. For the travelling finger, we present results of a numerical solution method similar to that used to demonstrate the selection of discrete fingers by surface tension. With kinetic undercooling, a continuum of corner-free travelling fingers exists for any finger width above a critical value, which goes to zero as the kinetic undercooling vanishes. We have not been able to compute the discrete family of analytic solutions, predicted by previous asymptotic analysis, because the numerical scheme cannot distinguish between solutions characterised by analytic fingers and those which are corner-free but non-analytic.

Original languageEnglish
Pages (from-to)707-727
Number of pages21
JournalEuropean Journal of Applied Mathematics
Issue number6
Publication statusPublished - 6 Dec 2014
Externally publishedYes


  • Bubble contraction and expansion
  • Corner formation
  • Hele-Shaw flow
  • Kinetic undercooling
  • Viscous fingering

ASJC Scopus subject areas

  • Applied Mathematics


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