Linear stability analysis of the Hele-Shaw cell with lifting plates

TL;DR

This paper performs a linear stability analysis of finger formation in a Hele-Shaw cell with lifting plates, revealing how the dominant mode's wavenumber depends on lifting rate and cell parameters, differing from standard cells.

Contribution

It introduces a linear stability framework for the initial finger formation in a Hele-Shaw cell with lifting plates, linking the dominant mode to physical parameters.

Findings

Wavenumber squared is proportional to lifting rate and cell length.

Wavenumber squared is inversely proportional to the cube of the cell gap.

Dependence on parameters differs from standard Hele-Shaw cells.

Abstract

The first stages of finger formation in a Hele-Shaw cell with lifting plates are investigated by means of linear stability analysis. The equation of motion for the pressure field (growth law) results to be that of the directional solidification problem in some unsteady state. At the beginning of lifting the square of the wavenumber of the dominant mode results to be proportional to the lifting rate (in qualitative agreement with the experimental data), to the square of the length of the cell occupied by the more viscous fluid, and inversely proportional to the cube of the cell gap. This dependence on the cell parameters is significantly different of that found in the standard cell.