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

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 a more viscous one, in a rectangular Hele-Shaw cell. The injection of $N$ successive intermediate phases with constant viscosities (the multi-layer Hele-Shaw model) was studied in some recent papers, where a minimization of the Saffman-Taylor instability was obtained for large enough $N$. However, in this paper we get a particular eigenfunction of the linear stability system which leads to eigenvalues which become infinite for large wave numbers. We obtain a strong instability of the multi-layer Hele-Shaw displacement, even if $N$ is very large.