Numerical study on viscous fingering using electric fields in a Hele-Shaw cell

TL;DR

This study explores how electric fields influence viscous fingering in a Hele-Shaw cell, developing a computational method to analyze interface dynamics and revealing how electric currents can stabilize or destabilize pattern formation.

Contribution

A novel spectral boundary integral method with a rescaling scheme is introduced to efficiently simulate electric field effects on viscous fingering in Hele-Shaw cells.

Findings

Positive currents stabilize finger patterns.

Negative currents induce interface instability and tail formation.

The pinch-off time follows an algebraic law.

Abstract

We investigate the nonlinear dynamics of a moving interface in a Hele-Shaw cell subject to an in-plane applied electric field. We develop a spectrally accurate boundary integral method where a coupled integral equation system is formulated. Although the stiffness due to the high order spatial derivatives can be removed, the long-time simulation is still expensive since the evolving velocity of the interface drops dramatically as the interface expands. We remove this physically imposed stiffness by employing a rescaling scheme, which accelerates the slow dynamics and reduces the computational cost. Our nonlinear results reveal that positive currents restrain finger ramification and promote overall stabilization of patterns. On the other hand, negative currents make the interface more unstable and lead to the formation of thin tail structures connecting the fingers and a small inner region. When no flux is injected, and a negative current is utilized, the interface tends to approach the origin and break up into several drops. We investigate the temporal evolution of the smallest distance between the interface and the origin and find that it obeys an algebraic law $\displaystyle (t_*-t)^b$, where $t_*$ is the estimated pinch-off time.