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The Quarterly Journal of Mechanics and Applied Mathematics Advance Access originally published online on March 11, 2008
The Quarterly Journal of Mechanics and Applied Mathematics 2008 61(2):161-180; doi:10.1093/qjmam/hbn005
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Q. Jl Mech. Appl. Math, Vol. 61. No. 2 © The author 2008. Published by Oxford University Press; all rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

Exact solutions for the evolution of ellipsoidal inclusions in porous media

Peter Buchak and Darren Crowdy{dagger}

( Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA )

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

Received 16 April 2007. Revise 13 August 2007. Accepted 1 November 2007.


  Abstract

This paper derives exact mathematical solutions for the time-dependent evolution of a single ellipsoidal inclusion in a porous medium when a linear straining flow is active in the far field. This represents a two-phase free boundary problem. It is shown that the dynamics is such that an initially ellipsoidal inclusion remains ellipsoidal under evolution. The theory of ellipsoidal harmonics is used to determine the system of ordinary differential equations governing the geometrical parameters of the ellipsoidal inclusion.


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