Viscous fingering as a paradigm of interfacial pattern formation: Recent results and new challenges

Abstract

We review recent results on dynamical aspects of viscous fingering. The Saffman-Taylor instability is studied beyond linear stability analysis by means of a weakly nonlinear analysis and the exact determination of the subcritical branch. A series of contributions pursuing the idea of a dynamical solvability scenario associated to surface tension in analogy with the traditional selection theory is put in perspective and discussed in the light of the asymptotic theory of Tanveer and co-workers. The inherently dynamical singular effects of surface tension are clarified. The dynamical role of viscosity contrast is explored numerically. We find that the basin of attraction of the Saffman-Taylor finger depends on viscosity contrast, and that the sensitivity to this parameter is maximal in the usual limit of high viscosity contrast. The competing attractors are identified as closed bubble solutions. We briefly report on recent results and work in progress concerning rotating Hele-Shaw flows, topological singularities and wetting effects, and also discuss future directions in the context of viscous fingering. © 2004 American Institute of Physics. ͓DOI: 10.1063/1.1784931͔The study of viscous fingering in a Hele-Shaw cell is a longstanding problem which has become an archetype of interfacial pattern formation, but continues to bring up new surprises which challenge our understanding of nonlocal, nonlinear pattern dynamics of interfaces. The problem refers to the dynamics of the interface between two immiscible viscous fluids confined in a quasi-twodimensional geometry, the Hele-Shaw cell, leading to pattern formation through a morphological instability. In this article we briefly review some recent developments on the dynamics of fingering patterns. We discuss the effects of surface tension as a singular perturbation showing that the problem with and without surface tension are essentially different. Within a dynamical systems approach, we describe how the introduction of surface tension dramatically modifies the global "topological… structure of the phase space flow of the system. We also address, in more detail, the effect of varying the parameter viscosity contrast. We show that the dynamics of fingered structures is highly sensitive to this parameter, and that the long time asymptotics is dominated by the competition between the usual Saffman-Taylor single-finger stationary solution and other attractors defined by closed bubbles. In this context and also taking into account recent results on rotating Hele-Shaw flows, we discuss future perspectives in the field concerning the existence of topological singularities in the form of interface pinchoff, wetting effects and applications to other problems such interface roughening in fluid invasion of porous media.

Multiple finger propagation modes in Hele-Shaw channels of variable depth

“…The steady finger shapes found by McLean & Saffman are linearly stable for modest B and are thought to be linearly stable whenever the surface tension is non-zero (Kessler & Levine 1986;Tanveer 1987;Casademunt 2004). However, for noisy systems with small B, the interface is unstable to tip-splitting behaviour both experimentally (Park & Homsy 1985;Maxworthy 1987;Tabeling, Zocchi & Libchaber 1987) and numerically (Casademunt 2004).…”

“…The instability of the double-fingered state with respect to asymmetric perturbations is consistent with the absence of states with two long Multiple finger propagation modes in Hele-Shaw channels of variable depth 141 fingers from experimental data for the rigid channel. We note that tip splitting and competition between fingers is a classic feature of experimental observations of fingering in Hele-Shaw cells at low surface tension (Park & Homsy 1985;Maxworthy 1987;Tabeling et al 1987) and in numerical investigations of evolution towards a single finger (Casademunt 2004). For these low-surface-tension cases, the tip-splitting behaviour is essentially triggered by experimental perturbations or numerical noise.…”

“…However, for noisy systems with small B, the interface is unstable to tip-splitting behaviour both experimentally (Park & Homsy 1985;Maxworthy 1987;Tabeling, Zocchi & Libchaber 1987) and numerically (Casademunt 2004). Tip splitting is followed by competition between fingers; once one finger begins to propagate ahead of the others, it screens the others and prevents them from growing.…”

Problems of in vitro SPF Measurements Brought about by Viscous Fingering Generated during Sunscreen Applications

“…Viscous fingering is one example of spontaneous pattern formation in far-from-equilibrium systems. Fingering patterns are seen in moving interfaces as the result of the growth of morphological fluctuations [12,13,14,15]. Stripe patterns are thus spontaneously generated when a thin film of viscous fluid is produced by passing it through a small gap [16,17,18,19,20] or when two pieces of adhesive tapes that are stuck together are peeled [21,22,23].…”

A lattice Boltzmann method for immiscible multiphase flow simulations using the level set method

“…Since the governing equations for the gap-averaged velocity are similar to Darcy's law, the Hele-Shaw cell also serves as a 2D model for flow through porous media, where the flow is governed by Darcy's law. The investigation of interface formation in immiscible flows with and without surface tension effects has revealed that this can be used as a generic model problem for analytical studies as well as for benchmarking numerical simulations of free surfaces [7,14,48,25,30] In our simulations we considered the flow in the narrow gap between the plates and investigated the finger by assuming that the tip can be fitted by an exponential shape as described in [30]. We used a channel [0, 32] × [−1, 1] with aspect ratio 16:1 using 16N grid points in x-direction, and N in y-direction.…”