Fractal dimension and unscreened angles measured for radial viscous fingering

We have examined fractal patterns formed by the injection of air into oil in a thin (0.1270.3em0exmm) layer contained between two cylindrical glass plates of 2880.3em0exmm diameter (a Hele-Shaw cell), for pressure differences in the range 0.25ΔP1.750.3em0exatm. We find that an asymptotic structure is reached at large values of the ratio $r∕b$, where r is the pattern radius and b the gap between the plates. Both the driving force and the size of the pattern, which reaches $r∕b=900$, are far larger than in past experiments. The fractal dimension D₀ of the pattern for large $r∕b$ is 1.70±0.02. Further, the generalized dimensions Dq of the pattern are independent of q, Dq1.70 for the range examined, -11<q<17; thus the pattern is self-similar within the experimental uncertainty. The results for Dq agree well with recent calculations for diffusion-limited aggregation (DLA) clusters. We have also measured the probability distribution of unscreened angles. At late times, the distribution approaches a universal (i.e., forcing and size-independent) asymptotic form that has mean 145^∘{} and standard deviation 36^∘{}. These results indicate that the distribution function for the unscreened angle is an invariant property of the growth process.