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

## TL;DR

The paper reveals a fundamental contradiction in the multi-layer Hele-Shaw model, showing it predicts persistent instability regardless of the number of layers and surface tension effects, challenging its validity.

## Contribution

It demonstrates that the multi-layer Hele-Shaw model inherently predicts instability, contradicting previous assumptions about stabilization with many layers and surface tension.

## Key findings

- The model predicts significant instability even with many layers.
- Surface tension does not fully stabilize the interface.
- Contradicts the classical Saffman-Taylor stability results.

## Abstract

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