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We study the effects of some injection policies used in oil recovery process.The Saffman-Taylor instability occurs when a less viscous fluid is displacing amore viscous one, in a rectangular Hele-Shaw cell. The injection of $N$successive intermediate phases with constant viscosities (the multi-layerHele-Shaw model) was studied in some recent papers, where a minimization of theSaffman-Taylor instability was obtained for large enough $N$. However, in thispaper we get a particular eigenfunction of the linear stability system whichleads to eigenvalues which become infinite for large wave numbers. We obtain astrong instability of the multi-layer Hele-Shaw displacement, even if $N$ isvery large.

On the strong instability of the multi-layer Hele-Shaw flows