Harmonic moments of radial viscous fingering patterns

We measure the displacement of oil by air in a Hele-Shaw cell constructed of circular glass plates 288 mm in diameter, separated by a gap of 0.38 mm [1]. Oil is removed from the cell perimeter and is replaced by air that enters through a hole in the bottom plate. High resolution digital imaging is used to track the growth of the air bubble, which develops viscous fingers. We determine the pattern's harmonic moments, which are integrals of integer powers of z = x + iy over the oil domain. The results for the harmonic moments are in accord with Richardson's theory [2], which predicts that harmonic moments should be time invariant in the absence of surface tension. Extending the theory to include surface tension, we obtain from measurements of the time evolution of the harmonic moments a value for the surface tension that is within 10 percent of the accepted value. \newline [1] L. Ristroph, M. Thrasher, M. B. Mineev-Weinstein, and H. L. Swinney, Phys. Rev. E 74, 015201 (2006). \newline [2] S. Richardson, J. Fluid Mechanics 56, 609 (1972).