Khalid, AH;
(2015)
Free boundary problems in a Hele-Shaw cell.
Doctoral thesis , UCL (University College London).
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Abstract
The motion of a free boundary separating two immiscible fluids in an unbounded Hele-Shaw cell is considered. In the one-phase problem, a viscous fluid is separated from an inviscid fluid by a simple closed boundary. Preliminaries for a complex variable technique are presented by which the one-phase problem can be solved explicitly via conformal mappings. The Schwarz function of the boundary plays a major role giving rise to the so called Schwarz function equation which governs the evolution of exact solutions. The Schwarz function approach is used to study the stability of a translating elliptical bubble due to a uniform background flow, and the stability of a blob (or bubble) subject to an external electric field. The one-phase problem of a translating free boundary and of a free boundary subject to an external field are studied numerically. A boundary integral method is formulated in the complex plane by considering the Cauchy integral formula and the complex velocity of a fluid particle on the free boundary. In the case of a free boundary subject to an external electric field due to a point charge, it is demonstrated that a stable steady state is achieved for appropriate charge strength. The method is also employed to study breakup of a single translating bubble in which the Schwarz function singularities (shown to be stationary) of the initial boundary play an important role. The two-phase problem is also considered, where the free boundary now separates two viscous fluids, and the construction of exact solutions is studied. The one-phase numerical model is enhanced, where a boundary integral method is formulated to accommodate the variable pressure in both viscous phases. Some numerical experiments are presented with a comparison to analytical results, in particular for the case where the free boundary is driven by a uniform background flow.
| Type: | Thesis (Doctoral) |
|---|---|
| Title: | Free boundary problems in a Hele-Shaw cell |
| Open access status: | An open access version is available from UCL Discovery |
| Language: | English |
| Keywords: | Hele-Shaw, Free boundary problems, Complex variable techniques, Fluid mechanics, Applied mathematics |
| UCL classification: | UCL > Provost and Vice Provost Offices UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences |
| URI: | https://discovery.ucl.ac.uk/id/eprint/1463159 |
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