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The Quarterly Journal of Mechanics and Applied Mathematics Advance Access originally published online on August 28, 2006
The Quarterly Journal of Mechanics and Applied Mathematics 2006 59(4):475-485; doi:10.1093/qjmam/hbl012
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Q. Jl Mech. Appl. Math, Vol. 59. No. 4 © The author 2006. Published by Oxford University Press; all rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

Exact solutions to the unsteady two-phase Hele-Shaw problem

Darren G. Crowdy{dagger}

( Department of Mathematics, Imperial College London, 180 Queen's Gate, London SW7 2AZ )

{dagger} d.crowdy{at}imperial.ac.uk


  Abstract

While many explicit solutions to the single-phase Hele-Shaw problem are known, solutions to the two-phase problem (also known as the ‘Muskat problem’) are scarce. This paper presents a new class of exact time-dependent solutions to the two-phase Hele-Shaw problem. It is demonstrated that an elliptical inclusion of one phase remains elliptical under evolution when immersed in any unsteady far-field linear flow of a second ambient phase. On the basis of this solution class, an ‘elliptical inclusion model’ for interactions in inhomogeneous porous media is outlined.


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