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An Isaac Newton Institute Workshop

Developments in Experimental Pattern Formation

"Fractal growth of viscous fingers"

12th August 2005

Authors: Harry L Swinney (U. Texas at Austin), Olivier Praud (U. Texas at Austin)


It has long been noted that similar highly ramified structures are formed by Diffusion Limited Aggregation (DLA) clusters, viscous fingering patterns, dielectric breakdown, flame fronts, and the growth of bacterial colonies, but it remains unknown whether these patterns are in the same or different universality classes. We address that question in a comparison of DLA with radial viscous fingering patterns in a Hele-Shaw cell. In our experiments a thin oil layer is contained between closely spaced (0.127 mm separation) circular plates (diameter 288 mm). When the oil is withdrawn from the edges and replaced by air supplied through a hole in the center of a plate, the air bubble evolves into an asymptotic multi-fingered structure with a fractal dimension 1.70+/-0.02 [1]. Similarly, the dimension of Diffusion Limited Aggregation (DLA) clusters is 1.713+/-0.003 [2]. The asymptotic behavior of the viscous fingering patterns is achieved for structures larger than about 400 times the layer thickness, a regime not studied in previous experiments. Further, preliminary results indicate that the harmonic measure for the viscous fingering structure has the same multifractal [f(alpha)] spectrum as DLA aggregates. Thus DLA and viscous fingering patterns are in the same universality class [3]. References: [1] O. Praud & H.L. Swinney, Phys. Rev. E, in press (2005). [2] B. Davidovitch, A. Levermann, & I. Procaccia, Phys. Rev. E 62, R5919 (2000). [3] Work in progress with I. Procaccia, J. Mathieson, L. Ristroph, M. Thrasher, & M.G. Moore.

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