Unsteady propagating bubbles in an unbounded Hele-Shaw cell are considered numerically in the case of zero surface tension. The instability of elliptical bubbles and their evolution toward a stable circular boundary, with speed twice that of the fluid speed at infinity, is studied numerically and by stability analysis. Numerical simulations of bubbles demonstrate that the important role played by singularities of the Schwarz function of the bubble boundary in determining the evolution of the bubble. When the singularity lies close to the initial bubble, two types of topological change are observed: (i) bubble splitting into multiple bubbles and (ii) a finite fluid blob pinching off inside the bubble region. C 2015 AIP Publishing LLC.
The behaviour of two-dimensional finite blobs of conducting viscous fluid in a Hele-Shaw cell subject to an electric field is considered. The time-dependent free boundary problem is studied both analytically using the Schwarz function of the free boundary and numerically using a boundary integral method. Various problems are considered, including (i) the behaviour of an initially circular blob of conducting fluid subject to an electric point charge located arbitrarily within the blob, (ii) the delay in cusp formation on the free boundary in sink-driven flow due to a strategically placed electric charge and (iii) the stability of exact steady solutions having both hydrodynamic and electric forcing.
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