Fractal dimension and unscreened angles measured for radial viscous fingering

Olivier Praud, Harry L. Swinney
2005 Physical Review E  
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 ≤ ∆P ≤ 1.75 atm. We find that an asymptotic structure is reached at large values of the ratio of the pattern radius to the instability length scale, r/λc = πb σ µV , where b is the gap between the plates, σ is the surface tension, µ the viscosity, and V the tip
more » ... d V the tip velocity. The ratio r/λc reaches 240, which is an order of magnitude larger than in past experiments. The fractal dimension D0 of the pattern for large r/λc is 1.70 ± 0.02. Further, the generalized dimensions Dq of the pattern are independent of q, Dq 1.70 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 DLA (Diffusion Limited Aggregation) 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.
doi:10.1103/physreve.72.011406 pmid:16089960 fatcat:fvj3ivecovbexfsiksjkqevqam