Inertial effects in the Saffman-Taylor instability

TL;DR

This paper investigates how fluid inertia influences the Saffman-Taylor instability, showing that inertia can increase finger width at high Reynolds numbers and proposing a modified control parameter to unify experimental data.

Contribution

It introduces a modified control parameter accounting for inertia effects and demonstrates experimental and numerical analysis of inertial impacts on viscous fingering.

Findings

Inertia causes finger width to increase with speed at high Reynolds numbers.

A critical Weber number determines when inertial effects become significant.

Rescaling data with the modified parameter collapses results onto the classical curve.

Abstract

For the Saffman-Taylor instability, the inertia of the fluid may become important for large Reynolds numbers Re. We investigate the effects of inertia on the width of the viscous fingers experimentally. We find that, due to inertia, the finger width can increase with increasing speed, contrary to what happens at small Re. We find that inertial effects need to be considered above a critical Weber number We. In this case it can be shown that the finger width is governed by a balance between viscous forces and inertia. This allows us to define a modified control parameter 1/B', which takes the corrections due to inertia into account; rescaling the experimental data with 1/B', they all collapse onto the universal curve for the classical Saffman-Taylor instability. Subsequently, we try and rationalize our observations. Numerical simulations taking into account a modification of Darcy law to include inertia, are found to only qualitatively reproduce the experimental findings, pointing to the importance of three-dimensional effects.