A new prediction of wavelength selection in radial viscous fingering involving normal and tangential stresses

TL;DR

This paper improves the prediction of wavelength selection in radial viscous fingering by incorporating normal and tangential stresses through Brinkman equations, resulting in better agreement with experimental data.

Contribution

It introduces a modified stability analysis using Brinkman equations to account for in-plane viscous stresses, enhancing the accuracy of wavelength predictions in radial fingering.

Findings

Brinkman equations provide a better fit to experimental data than Darcy's law.

Normal and tangential stresses significantly influence finger wavelength selection.

The dispersion relation aligns more closely with observed instability patterns.

Abstract

We reconsider the radial Saffman-Taylor instability, when a fluid injected from a point source displaces another fluid with a higher viscosity in a Hele-Shaw cell, where the fluids are confined between two neighboring flat plates. The advancing fluid front is unstable and forms fingers along the circumference. The so-called Brinkman equations is used to describe the flow field, which also takes into account viscous stresses in the plane and not only viscous stresses due to the confining plates like the Darcy equation. The dispersion relation agrees better with the experimental results than the classical linear stability analysis of radial fingering in Hele-Shaw cells that uses Darcy's law as a model for the fluid motion.