Effective description of the interaction between anisotropic viscous fingers

Abstract

We study patterns formed by viscous fingering in a rectangular network of microfluidic channels. Due to the strong anisotropy of such a system, the emerging patterns have a form of thin needle-like fingers, which interact with each other, competing for an available flow. We develop an upscaled description of this system in which only the fingers are tracked and the effective interactions between them are introduced, mediated through the evolving pressure field. Due to the quasi-2d geometry of the system, this is conveniently accomplished using conformal mapping techniques. A complex two-phase flow problem is thus reduced to a much simpler task of tracking evolving shapes in a 2d complex plane. This description, although simplified, turns out to capture all the key features of the system's dynamics and allows for the effective prediction of the resulting growth patterns.

Competition Is the Underlying Mechanism Controlling Viscous Fingering and Wormhole Growth

“…We validate the competition model against the viscous fingers experimental data of Pecelerowicz et al () and the wormhole numerical data from P. Szymczak (personal communication, September 2018). We start the simulation using the measured initial lengths and separations.…”

Thin-finger growth and droplet pinch-off in miscible and immiscible displacements in a periodic network of microfluidic channels

“…One of such models has been proposed in Ref. 17. It focuses on the thin-finger (large D and S) regime and approximates the fingers by thin lines growing at their tips only and interacting through the Laplacian pressure field.…”

Laplacian networks: Growth, local symmetry, and shape optimization

“…Although they follow an optimal trajectory, they end up decreasing their flux as they grow. Mathematically, this indicates that the right-hand term in equation (51) changes sign during growth. The trajectory starting from the optimal straight bifurcation belongs to this family: optimal growth drives it away from the opti-mal static configuration it started at, to bring it to the fixed point.…”