Viscous fingering in an anisotropic Hele-Shaw cell

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Abstract
The effect of anisotropy, induced by a periodic modulation of the plate separation in circular Hele-Shaw geometry, on the viscous fingering patterns is studied. Experimentally the modulation of the separation is induced by etching a regular pattern of channels on one of the confining plates. We show that for etching patterns of sufficiently high symmetry, the bulk properties of the flow are isotropic on the length scale of investigations of the interfacial pattern. Hence anisotropic effects are due to the boundary conditions at the interface only. Such boundary conditions are modeled by considering the effect of an anisotropic, effective surface tension, and the macroscopic-interface equations are solved numerically to study the morphological phase diagram. We discuss our choice of the constant-flux operating mode, in which case the phase diagram is swept by adjusting one anisotropy parameter. Some qualitative arguments are offered for the nature of the phase diagram, which includes a transition region from radial growth characterized by a diffusion-limited-aggregation-like fractal dimension at low anisotropy to a dendritic region with evident sidebranching.