Viscous Fingering: An Optimal Bound on the Growth Rate of the Mixing Zone

We consider the flow of two immiscible fluids of different mobility in a porous medium. If the more mobile fluid displaces the other, a macroscopically sharp interface is unstable. By growing a network of fingers on a mesoscopic scale, the two phases mix on a macroscopic scale. We are interested in the evolution of this mixing zone. We show that the effect of a large but finite mobility ratio λ is strong enough to limit the growth rate of the mixing zone. This is done by rigorously deriving an a priori estimate for the Saffman–Taylor model. In this geometry of an infinite channel, the estimate essentially states that the mobility ratio λ itself (in the nondimensionalized setting with unit velocity imposed at infinity) is the optimal bound on the velocity by which the penetrating phase progresses in direction of the channel. Since the introduction of diffusion-limited aggregation, various stochastic algorithms simulating this two-phase flow have been developed. The generated clusters, which correspond to the distribution of the highly mobile displacing phase, are fractal in the limiting case of λ = ∞ and “compact” for λ = 1. With support of numerical experiments and renormalization-group arguments, it had been conjectured that they eventually cross over from fractal to compact for all finite λ ∈ (1, ∞). Our result may be interpreted as another confirmation of this conjecture.