# On the displacement of two immiscible Oldroyd-B fluids in a Hele-Shaw cell

## TL;DR

This paper analyzes the linear stability of two Oldroyd-B fluids displacing each other in a Hele-Shaw cell, revealing a critical difference in Weissenberg numbers that causes a blow-up in growth rate, aligning with prior experimental findings.

## Contribution

It provides an approximate formula for the growth rate of perturbations in Oldroyd-B fluid displacement, highlighting the effect of Weissenberg number differences and identifying a critical point for instability.

## Key findings

- Growth rate depends on the difference of Weissenberg numbers.
- A blow-up occurs at a critical Weissenberg number difference.
- Results agree with previous numerical and experimental studies.

## Abstract

The Saffman-Taylor instability occurs when { a} Stokes fluid is displaced by a less viscous one { in} a Hele-Shaw cell. { This model is useful to study the secondary oil recovery from a pororus medum.} Since 1960, { polymer solutions} were used as displacing fluids; moreover, the oil in a porous reservoir can often be considered a non-Newtonian fluid. Motivated by this fact, in this paper we study the linear stability of the displacement of two Oldroyd-B fluids in a rectilinear Hele-Shaw cell, even if the direct relevance for the flow through porous media is not so evident. We get an { approximate} formula of the growth rate of perturbations, which depends on the { difference} of the Weissenberg numbers of the two fluids. { A blow-up of the growth rate appears for a critical value of this difference.} This singularity is in agreement with { previous} numerical and experimental results, already reported by several papers concerning the flow of complex fluids in Hele-Shaw cells.