Pattern selection in an anisotropic Hele-Shaw cell

Phase field approach to spatial perturbations in normal Saffman-Taylor fingers

We make a numerical study of the effect that spatial perturbations have in normal Saffman-Taylor fingers driven at constant pressure gradients. We use a phase field model that allows for spatial variations in the Hele-Shaw cell. We find that, regardless of the specific way in which spatial perturbations are introduced, a lateral instability develops on the sides of the propagating Saffman-Taylor finger. Moreover, the instability exists regardless of the intensity of spatial perturbations in the cell as long as the perturbations are felt by the finger tip. If, as the finger propagates, the spatial perturbations felt by the tip change, the instability is nonperiodic. If, as the finger propagates, the spatial perturbations felt by the tip are persistent, the instability developed is periodic. In the later case, the instability is symmetrical or asymmetrical depending on the intensity of the perturbation.

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.

Effect of anisotropy on viscous fingering patterns of polymer solutions in linear Hele-Shaw cells

Viscous fingering patterns of hydroxypropyl methyl cellulose (HPMC) solutions were investigated by forcing air in linear geometry isotropic and anisotropic Hele-Shaw cells as functions of HPMC concentration and injection pressure of air. The resulting patterns depended on the HPMC concentration and the cell anisotropy. The characteristic quantities of pattern growth, such as the finger velocity and the finger width, were evaluated. The finger velocity in the anisotropic linear cell was faster than that in the isotropic one, and the finger velocity and the finger width were well correlated with the pattern morphological changes. (c) 1999 American Institute of Physics.