Nonuniqueness in Singular Viscous Fingering

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Abstract
With use of scaled variables, the equation of viscous fingering in radial geometry for zero surface tension is written in a form which admits singular solutions. It is then shown in an example that there exist singular solutions which are tree graphs. The dynamics of these solutions is not uniquely determined by the differential equation. The nonuniqueness corresponds to kinking, branching, and tip splitting at the growing ends, which need not happen, but which may happen. The qualitative resemblance of these ramified, underdetermined solutions to the experimental viscous fingering patterns in radial geometry is pointed out.