Unbounded eigenfunctions in the stability problem for a three-layer flow in porous media

TL;DR

This paper investigates the linear stability of a three-layer fluid displacement in porous media using the Hele-Shaw model, revealing unbounded eigenfunctions that challenge the physical interpretation of the stability analysis.

Contribution

It demonstrates that the eigenfunctions in the stability problem are unbounded, indicating the problem's lack of physical meaningfulness in this configuration.

Findings

Eigenfunctions are unbounded in the stability analysis.

The stability problem lacks physical sense due to unbounded solutions.

The study questions the applicability of linear stability in this context.

Abstract

We study the linear stability of the displacement of three Stokes fluids with constant viscosity in a porous medium when the middle fluid is contained in a bounded region. We use the Hele-Shaw model. The eigenfunctions of the stability system are the amplitudes of the linear perturbations. These amplitudes must be small. We get unbounded eigenfunctions. So the stability problem has no physical sense.