Selection of Saffman-Taylor fingers in the sector geometry

Interfacial dynamics and pinch-off singularities for axially symmetric Darcy flow

We study a model for the evolution of an axially symmetric bubble of inviscid fluid in a homogeneous porous medium otherwise saturated with a viscous fluid. The model is a moving boundary problem that is a higher-dimensional analog of Hele-Shaw flow. Here we are concerned with the development of pinch-off singularities characterized by a blowup of the interface curvature and the bubble subsequently breaking up into two; these singularities do not occur in the corresponding two-dimensional Hele-Shaw problem. By applying a numerical scheme based on the level set method, we show that solutions to our problem can undergo pinch-off in various geometries. A similarity analysis suggests that the minimum radius behaves as a power law in time with exponent α=1/3 just before and after pinch-off has occurred, regardless of the initial conditions; our numerical results support this prediction. Further, we apply our numerical scheme to simulate the time-dependent development and translation of axially symmetric Saffman-Taylor fingers and Taylor-Saffman bubbles in a cylindrical tube, highlighting key similarities and differences with the well-studied two-dimensional cases.

Controlling Viscous Fingering Using Time-Dependent Strategies

Control and stabilization of viscous fingering of immiscible fluids impacts a wide variety of pressure-driven multiphase flows. We report theoretical and experimental results on a time-dependent control strategy by manipulating the gap thickness b(t) in a lifting Hele-Shaw cell in the power-law form b(t)=b(1)t(1/7). Experimental results show good quantitative agreement with the predictions of linear stability analysis. By choosing the value of a single time-independent control parameter, we can either totally suppress the viscous fingering instability or maintain a series of nonsplitting viscous fingers during the fluid displacement process. In addition to the gap thickness of a Hele-Shaw cell, time-dependent control strategies can, in principle, also be placed on the injection rate, viscosity of the displaced fluid, and interfacial tension between the two fluids.

Control of viscous fingering patterns in a radial Hele-Shaw cell

We study numerically and experimentally the dynamics and control of viscous fingering patterns in a circular Hele-Shaw cell. The nonlocality and nonlinearity of the system, especially interactions among developing fingers, make the emergent pattern difficult to predict and control. By controlling the injection rate of the less viscous fluid, we can precisely suppress the evolving interfacial instabilities. There exist denumerable attractive, self-similarly evolving symmetric, universal shapes. Experiments confirm the feasibility of the control strategy, which is summarized in a morphology diagram.

Selection rules for the tip-splitting instability

The local destabilization of a Saffman-Taylor viscous finger occurs by a splitting of its tip and results in the formation of two branches separated by a fjord. The accumulation of such instabilities leads to complex patterns. In this paper we present a detailed analysis of a dynamical model that accounts for the selection of both the width and the orientation of the fjords growing in a wedge of angle theta(0). It is shown that the selection rules have a dynamical origin and are related to the existence of attracting sets that disappear in the absence of surface tension. We also infer the existence of a critical angle theta(c)=60 degrees such that if theta(0)<theta(c), the symmetric tip-splitting becomes unstable.

Mean-field diffusion-limited aggregation: a "density" model for viscous fingering phenomena

We explore a universal "density" formalism to describe nonequilibrium growth processes, specifically, the immiscible viscous fingering in Hele-Shaw cells (usually referred to as the Saffman-Taylor problem). For that we develop an alternative approach to the viscous fingering phenomena, whose basic concepts have been recently published in a Rapid Communication [Phys. Rev. E 63, 045305(R) (2001)]. This approach uses the diffusion-limited aggregation (DLA) paradigm as a core: we introduce a mean-field DLA generalization in stochastic and deterministic formulations. The stochastic model, a quasicontinuum DLA, simulates Monte Carlo patterns, which demonstrate a striking resemblance to natural Hele-Shaw fingers and, for steady-state growth regimes, follow precisely the Saffman-Taylor analytical solutions in channel and sector configurations. The relevant deterministic theory, a complete set of differential equations for a time development of density fields, is derived from that stochastic model. As a principal conclusion, we prove an asymptotic equivalency of both the stochastic and deterministic mean-field DLA formulations to the classic Saffman-Taylor hydrodynamics in terms of an interface evolution.