A paradox in Hele-Shaw displacements

TL;DR

This paper investigates Hele-Shaw displacements with zero surface tension, revealing that variable viscosity can nearly suppress the Saffman-Taylor instability, unlike constant viscosity layers which lead to unbounded growth rates.

Contribution

It demonstrates that an intermediate variable viscosity liquid can almost suppress the Saffman-Taylor instability in Hele-Shaw flows, contrasting with the destabilizing effect of multiple constant viscosity layers.

Findings

Variable viscosity can suppress instability

Constant viscosity layers cause unbounded growth rates

Intermediate liquid use is consistent in both cases

Abstract

We study the Hele-Shaw immiscible displacements when all surfaces tensions on the interfaces are zero. The Saffman-Taylor instability occurs when a less viscous fluid is displacing a more viscous one, in a rectangular Hele-Shaw cell. We prove that an intermediate liquid with a variable viscosity can almost suppress this instability. On the contrary, a large number of constant viscosity liquid-layers inserted between the initial fluids gives us boundless growth rates with respect to the wave numbers of perturbations. The same amount of intermediate liquid is used in both cases.