Short, flat-tipped, viscous fingers: novel interfacial patterns in a Hele-Shaw channel with an elastic boundary

Abstract

Injecting a less viscous fluid into a more viscous fluid in a Hele-Shaw cell triggers two-dimensional viscous fingering patterns which are characterised by increasingly long fingers undergoing tip splitting and branching events. These complex structures are considered to be a paradigm for interfacial pattern formation in porous media flow and other related phenomena. Over the past five years, there has been a flurry of interest in manipulating these interfacial fingering patterns by altering the physical components of the Hele-Shaw apparatus. In this Focus on Fluids article, we summarise some of this work, concentrating on a very recent study in which the alterations include replacing one of the two bounding plates with an elastic membrane (Ducloué et al., J. Fluid Mech., vol. 826, 2017, R2). The resulting experimental set-up gives rise to a wide variety of novel interfacial patterns including periodic sideways fingers, dendritic-like patterns and short, flat-tipped viscous fingers that appear to resemble molar teeth. These latter fingers are similar to those observed in the printer's instability and when peeling off a layer of adhesive tape. This delightful work brings together a number of well-studied themes in interfacial fluid mechanics, including how viscous and surface tension forces compete to drive fingering patterns, how interfaces are affected by fluid-solid interactions and, finally, how novel strategies can be implemented to control interfacial instabilities.

Numerical investigation of controlling interfacial instabilities in non-standard Hele-Shaw configurations

“…Here, the morphology is significantly different from the standard case, and in fact this interfacial pattern resembles the short flat-tipped 'stubby' fingers observed both experimentally and numerically by Pihler-Puzović et al (2012, who considered a Hele-Shaw cell where the top plate is replaced by an elastic membrane. Other closely related studies, including links with the problems of opening an initially collapsed channel and peeling of a viscous strip, are outlined in , Lister et al (2013), Pihler-Puzović et al (2014, 2018, Ducloué et al (2017), and McCue (2018). These observations suggest there is a one-parameter family of solutions (α) joining the standard Hele-Shaw problem to one where the pattern formation is similar to that produced by a deformable boundary.…”

“…(2014, 2018), Ducloué et al. (2017), Juel, Pihler-Puzović & Heil (2018) and McCue (2018). These observations suggest there is a one-parameter family of solutions ( ) joining the standard Hele-Shaw problem to one where the pattern formation is similar to that produced by a deformable boundary.…”

Viscous fingering in a radial elastic-walled Hele-Shaw cell

“…Approximation of the geometry as a triangular wedge allowed the growth rate of the perturbations to be expressed analytically in terms of α i and the base-state capillary number Ca = µU/γ , where γ is the surface tension at the interface, U is its steady speed and µ is the dynamic viscosity of the viscous fluid. Ducloué et al (2017a,b) studied the two-phase displacement flow in a finite-width elastic-walled rectangular Hele-Shaw channel; see also McCue (2018). In a rigid channel this flow always results in the formation of a single Saffman-Taylor 166 D. Pihler-Puzović, G. G. Peng, J. R. Lister, M. Heil and A. Juel finger which propagates steadily and is linearly stable (Saffman & Taylor 1958;McLean & Saffman 1981;Combescot et al 1986).…”

Experimental and computational advances on the study of Viscous Fingering: An umbrella review

“…McCue studied a modified Hele-Shaw cell that had an elastic membrane instead of the classic top plate [48]. In this study, oil was displaced by air.…”