Capillary waves and dendritic instability on radial Hele–Shaw displacements

Abstract

The displacement of a fluid by another less viscous one in isotropic quasi-two-dimensional Hele–Shaw cells typically leads to viscous fingering patterns characterized by repeated tip-splitting and side-branching. When anisotropy is present, the viscous fingering patterns are replaced by dendritic instability. In isotropic cells, Couder et al. [“Dendritic growth in the Saffman-Taylor experiment,” Europhys. Lett. 2, 437 (1986)] induced the growth of a dendritic finger by placing a small isolated bubble in contact with the tip of a finger. Moreover, in rectangular displacements, Kopf-Sill and Homsy [“Narrow fingers in a Hele–Shaw cell,” Phys. Fluids 30, 2607–2609 (1987)] observed for specific ranges of modified capillary numbers, Ca, that narrow fingers develop into dendritic patterns after carefully cleaning the Hele–Shaw cell with a soap solution. In the current work, by injecting air to radially displace a lubricant oil with strong adsorption properties, we observe for the first time the growth of dendritic instability in displacements confined in isotropic Hele–Shaw cells in the absence of air bubbles driving the fingertip. These structures are driven by the formation of capillary waves inside the finger adjacent to the Hele–Shaw surfaces. Furthermore, we compare displacements of different materials at similar high Ca values and obtain significantly different morphological results, indicating the pattern formation may not scale with this parameter in this range. Discrepancies are explained based on both wetting and inertia effects.