An ill-posed problem in hydrodynamic stability of multi-layer Hele-Shaw flow

TL;DR

This paper investigates the linear stability of multi-layer Hele-Shaw flow, revealing that the problem is ill-posed due to incompatible interface conditions, which has implications for modeling fluid displacement in porous media.

Contribution

It demonstrates that the linear stability problem for multi-layer Hele-Shaw flow is inherently ill-posed because the necessary compatibility conditions cannot be satisfied.

Findings

Growth rates exist only under specific conditions

Compatibility conditions on interfaces cannot be fulfilled

The problem is fundamentally ill-posed

Abstract

An useful approximation for the displacement of two immiscible fluids in a porous medium is the Hele-Shaw model. We consider several liquids with different constant viscosities, inserted between the displacing fluids. The linear stability analysis of this model leads us to an ill-posed problem. The growth rates (in time) of the perturbations exist iff some compatibility conditions on the interfaces are verified. We prove that these conditions cannot be fulfilled.