Hele-Shaw flow for parity odd three-dimensional fluids

TL;DR

This paper extends Darcy's law to three-dimensional fluids with parity-odd viscosity, predicting observable effects like transverse forces and circulation in Hele-Shaw experiments, aiding the measurement of parity-odd coefficients.

Contribution

It derives a generalized Darcy's law for parity-odd fluids and explores its implications in Hele-Shaw flow setups, revealing new physical effects and stability modifications.

Findings

Transverse forces on walls in channel flow

Circulation development around expanding bubbles

Modified Saffman-Taylor stability conditions

Abstract

A Hele-Shaw cell is a device used to study fluid flow between two parallel plates separated by a small gap. The governing equation of flow within a Hele-Shaw cell is Darcy's law, which also describes flow through a porous medium. In this work, we derive a generalization to Darcy's law starting from a three-dimensional fluid with a parity-broken viscosity tensor with no isotropy. We discuss the observable effects of parity-odd fluids in various physical setups relevant to Hele-Shaw experiments, such as channel flow, flow past an obstacle, bubble dynamics, and the Saffman-Taylor instability. In particular, we show that when such a fluid is pushed through a channel, a transverse force is exerted on the walls, and when a bubble of air expands into a region of such fluid, a circulation develops in the far field, with both effects proportional to the parity-odd viscosity coefficients. The Saffman-Taylor stability condition is also modified, with these terms tending to stabilize the two-fluid interface. Such experiments can in principle facilitate the measurement of parity-odd coefficients in both synthetic and natural active matter systems.