Delayed onset and the transition to late time growth in viscous fingering

TL;DR

This study investigates the transition from initial instability to late-time growth in viscous fingering across different geometries and fluid types, revealing discrepancies between theory and experiments and proposing a predictive framework for pattern development.

Contribution

It provides new experimental measurements of onset lengths in viscous fingering and highlights limitations of existing theories, especially in radial geometries and miscible fluids.

Findings

Onset length matches linear stability theory for immiscible fluids in rectilinear geometry.

Radial geometries show onset lengths much larger than theoretical predictions.

The combined onset length and growth rate can predict late-time pattern formation.

Abstract

Viscous fingering patterns form in confined geometries at the interface between two fluids as the lower-viscosity fluid displaces the one with higher viscosity. Previous studies have examined the most unstable wavelength of the patterns that form using both linear-stability analysis and the dynamics of finger growth in the nonlinear regime. Interesting differences in dynamics have been seen between rectilinear and radial geometries as well as between fluid pairs that are immiscible (with interfacial tension) or miscible (with negligible interfacial tension). This paper reports measurements of how all of these systems transition from the linearly unstable regime to their late time, nonlinear dynamics. In all four cases there is a region of stable or slow growth characterized by an onset length scale before fingers enter the late-time regime. For immiscible fluids in a rectilinear geometry this onset length is consistent with linear-stability analyses. All other cases are not adequately described by existing theory. In radial geometries, the onset length predicted from theory is an order of magnitude smaller than what is experimentally observed and has the incorrect scaling with dimensionless numbers. For miscible fluids in a rectilinear geometry the onset length is related to the development of steady-state structures within the confining dimension and cannot be explained by quasi-two dimensional theories. By combining the onset length with the finger growth rate measured after onset, the global patterns that form well into the late-time dynamics can be predicted.