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The Saffman-Taylor instability occurs when a less viscous fluid is displacinga more viscous one in a rectangular Hele-Shaw cell. A surface tension on theinterface between the two fluids is improving the stability. The multi-layerHele - Shaw model, consisting of $N$ intermediate fluids with constantviscosities, was studied in some previous papers and very low growth constantswere obtained for large $N$. We prove that this model leads us to a significantinstability, even if $N$ is very large. The maximum value of growth constantscan not decrease under a certain value, not depending on the surface tensionson the interfaces. This contradiction with the Saffman-Taylor result makes ushave some doubts concerning the correctness of multi-layer model.

A strong contradiction in the multi-layer Hele-Shaw model