Viscous fingering: A singularity in Laplacian growth models

Saffman-Taylor instability of viscoelastic fluids: from viscous fingering to elastic fractures

We study the linear stability of an air front pushing on a viscoelastic upper convected Mawxell fluid inside a Hele-Shaw cell. Both theory and experiments involving several viscoelastic fluids prove that a unique dimensionless time parameter lambda[over] controls all elastic effects. For small values of lambda[over], Newtonian behavior dominates, while for higher values of lambda[over] viscoelastic effects appear. We show that the linear growth rate of a small initial perturbation diverges for a critical value lambda[over]=lambda(c)[over] approximately 10. Experiments prove that this divergence is associated to a fracturelike pattern instability of the interface. We conclude that the observed fractures come from the Saffman-Taylor instability and that they directly emerge from the linear regime of it.

Saffman-Taylor instability for generalized Newtonian fluids

We study theoretically the linear Saffman-Taylor instability for non-Newtonian fluids in a Hele-Shaw cell. After introducing the notion of generalized Newtonian fluid we calculate the associated Darcy's law. We derive the relation governing the growth rate of normal modes for a large class of non-Newtonian flows. For shear-thinning fluids at high shear rate our theory provides Darcy's laws free of the nonphysical divergences appearing in the classical approaches. We characterize fluids which develop instabilities faster than Newtonian fluids under the same hydrodynamical conditions. Another primary result that this paper provides is that for some shear-thickening fluids, all normal modes are stable.

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.

Viscous fingering in a yield stress fluid

We study the Saffman-Taylor or viscous fingering instability in yield stress fluids. The theory for yield stress fluids shows that the dispersion equation of the instability is similar to that for Newtonian fluids; however, the capillary number governing the instability now contains the yield stress. Experiments using gels and foams reveal very branched fingers in the gel. The results are in excellent agreement with theory for the gel, with, in addition, a crossover from yield stress dominated to viscous behavior. The results for foams are very different due to the existence of wall slip.