Abbreviated Title: Viscous Ngering: Bound on Growth Rate of Mixing Zone Viscous Ngering: an Optimal Bound on the Growth Rate of the Mixing Zone

We consider the ow of two immiscible uids of diierent mobility in a porous medium. If the more mobile uid displaces the other, a macroscopically sharp interface is unstable. By growing a network of ngers 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 eeect of a large but nite mobility ratio is strong enough to limit the growth rate of the mixing zone. This is done by rigorously deriving an a priori estimate for the Saaman{Taylor model. In this geometry of an innnite channel, the estimate essentially states that the mobility ratio itself (in the nondimensionalized stetting with unit velocity imposed at innnity) is the optimal bound on the velocity by which the penetrating phase progresses in direction of the channel. Since the introduction of DLA, various stochastic algorithms simulating this two{ phase ow have been developed. The generated clusters, which correspond to the distribution of the highly mobile displacing phase, are fractal in the limiting case of = 1 and \compact" for = 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 nite 2 (1; 1). Our result may be interpreted as another connrmation of this conjecture. Introduction. We are interested in the ow of two immiscible uids of dif