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IMA Journal of Applied Mathematics Advance Access originally published online on May 22, 2008
IMA Journal of Applied Mathematics 2008 73(5):803-814; doi:10.1093/imamat/hxn014
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© The Author 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Free-surface flow over a trapped bubble

M. G. Blyth{dagger}

J.-M. Vanden-Broeck

School of Mathematics, University of East Anglia, Norwich NR4 7TJ, UK

{dagger} Email: m.blyth{at}uea.ac.uk

Received on May 1, 2007; Revision received April 15, 2008. Accepted on April 15, 2008

We consider the free-surface flow of liquid of finite depth over a bubble trapped on a plane wall. The flow is assumed to be inviscid and irrotational. The problem is formulated using an integral equation and the solution is obtained numerically using a collocation method. The choices for the two contact angles where the bubble is attached to the wall are shown to depend crucially on the bubble drag. For supercritical flow, the bubble drag is zero and the contact angles are required to be equal. For either subcritical or critical flow, the bubble drag is non-zero. In either case, only one of the contact angles may be chosen freely and the second emerges as part of the solution. The effect of the free surface on the shape of the bubble is demonstrated under a variety of flow conditions.

Keywords: bubble; free-surface flow.


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