Unsteady growth and relaxation of viscous fingers

Viscous fingering occurs when a less viscous fluid is driven into a more viscous fluid in a porous medium or a thin layer geometry known as a Hele-Shaw cell. The problem is studied because of its applications to filtration and oil extraction, because of its relationship to other moving interface problems, such as solidification and flame fronts, and because the mathematical idealization of the problem has a number of unusual properties. One set of our experiments was performed in rectangular Hele-Shaw channels where fingers travel down the long axis of the cell. These experiments revealed several phenomena that were not observed in previous experiments. At low flow rates, growing fingers exhibited width fluctuations that intermittently narrowed the finger as they evolved. The magnitude of these fluctuations was proportional to Ca−0.64, where Ca is the capillary number, which is proportional to the finger velocity. This relation held for all aspect ratios studied (60 < (channel width)/(channel thickness) < 500) up to the onset of tip instabilities in the fingers. At higher flow rates, finger pinch-off and reconnection events were observed. These events appear to be caused by an interaction between the actively growing finger and suppressed fingers at the back of the channel. Both the fluctuation and pinch-off phenomena were robust but not explained by current theory. Our other experiments were performed in a Hele-Shaw cell with radial symmetry. These fingers generate highly ramified patterns as the air enters the cell from a central point. We have examined the relaxation of the fine structure of these patterns due to surface tension by removing the forcing after a pattern was grown. This relaxation, unlike most coarsening processes, was not dynamically scale invariant. Rather, it exhibits two distinct dynamic length scales that grow as different powers of time: l1(t) ∼ t0.22, l2(t) ∼ t0.31. These lengths correspond respectively to the scale below which the pattern is smooth (non-ramified) and the distance between different fingers in the pattern. The measured exponents were in agreement with the results of recent numerical studies of diffusion-controlled coarsening of fractal clusters [Lipshtat et al. Phys. Rev. E 65, 050501 (2002)]. A consequence of the existence of two length scales is that the patterns at late times depended on the structural form at the onset of coarsening, providing information on the age of the fractal. We also present preliminary work on the fractal dimension of viscous fingering patterns.