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 (0.127 mm) layer contained between two cylindrical glass plates of 288 mm diameter (a Hele-Shaw cell), for pressure differences in the range 0.25 <=Delta P <= 1.75 atm. 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 D-0 of the pattern for large r/b is 1.70 +/- 0.02. Further, the generalized dimensions D-q of the pattern are independent of q, D-q similar or equal to 1.70 for the range examined, -11 < q < 17; thus the pattern is self-similar within the experimental uncertainty. The results for D-q 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 degrees and standard deviation 36 degrees. These results indicate that the distribution function for the unscreened angle is an invariant property of the growth process.