New exact solutions for Hele-Shaw flow in doubly connected regions
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.
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|Item Type:||Journal Article|
|Keywords:||Hele-Shaw flow, Doubly-connected, Saffman-Taylor instability, finite-time blow-up, bubble extinction, viscous fingering, complex variable theory, ill-posedness, Polubarinova-Galin equation, Villat's integral formula, loxodromic functions|
|Subjects:||Australian and New Zealand Standard Research Classification > MATHEMATICAL SCIENCES (010000) > APPLIED MATHEMATICS (010200) > Theoretical and Applied Mechanics (010207)|
Australian and New Zealand Standard Research Classification > PHYSICAL SCIENCES (020000) > CLASSICAL PHYSICS (020300) > Fluid Physics (020303)
|Divisions:||Current > Schools > School of Mathematical Sciences|
Current > QUT Faculties and Divisions > Science & Engineering Faculty
|Copyright Owner:||Copyright 2012 American Institute of Physics|
|Deposited On:||23 Apr 2012 16:05|
|Last Modified:||09 May 2012 22:10|
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