New exact solutions for Hele-Shaw flow in doubly connected regions

Michael C. Dallaston, Scott W. McCue

Research output: Contribution to journalArticle

10 Citations (Scopus)
14 Downloads (Pure)

Abstract

Radial Hele-Shaw flows are treated analytically using conformal mapping techniques. The geometry of interest has a doubly connected annular region of viscous fluid surrounding an inviscid bubble that is either expanding or contracting due to a pressure difference caused by injection or suction of the inviscid fluid. The zero-surface-tension problem is ill-posed for both bubble expansion and contraction, as both scenarios involve viscous fluid displacing inviscid fluid. Exact solutions are derived by tracking the location of singularities and critical points in the analytic continuation of the mapping function. We show that by treating the critical points, it is easy to observe finite-time blow-up, and the evolution equations may be written in exact form using complex residues. We present solutions that start with cusps on one interface and end with cusps on the other, as well as solutions that have the bubble contracting to a point. For the latter solutions, the bubble approaches an ellipse in shape at extinction.

Original languageEnglish
Article number052101
JournalPhysics of Fluids
Volume24
Issue number5
DOIs
Publication statusPublished - 1 May 2012
Externally publishedYes

Fingerprint

bubbles
viscous fluids
cusps
critical point
conformal mapping
fluids
suction
ellipses
contraction
interfacial tension
extinction
injection
expansion
geometry

Keywords

  • CRITICAL POINT
  • PHENOMENA
  • VISCOSITY
  • EXACT SOLUTIONS
  • WAVE ATTENUATION
  • BERNOULLI'S PRINCIPLE

ASJC Scopus subject areas

  • Condensed Matter Physics

Cite this

New exact solutions for Hele-Shaw flow in doubly connected regions. / Dallaston, Michael C.; McCue, Scott W.

In: Physics of Fluids, Vol. 24, No. 5, 052101, 01.05.2012.

Research output: Contribution to journalArticle

Dallaston, Michael C. ; McCue, Scott W. / New exact solutions for Hele-Shaw flow in doubly connected regions. In: Physics of Fluids. 2012 ; Vol. 24, No. 5.
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