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

TL;DR

This paper demonstrates that multi-layer Hele-Shaw flows, even with many layers, exhibit strong instability due to eigenvalues becoming infinite at large wave numbers, challenging previous assumptions of stability.

Contribution

It provides a new eigenfunction analysis showing that the multi-layer Hele-Shaw model can have unbounded eigenvalues, indicating persistent instability regardless of the number of layers.

Findings

Eigenvalues become infinite for large wave numbers.

Multi-layer flows are strongly unstable even with many layers.

Previous stability assumptions are challenged by new eigenfunction analysis.

Abstract

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.