Dendritic growth in the viscous fingering instability

The displacement of a more viscous fluid by a less viscous one in a quasi-two dimensional geometry leads to the formation of complex fingering patterns. In isotropic system the pattern forms by dense branching growth characterized by repeated tip-splitting of the evolving finger. When anisotropy is introduced into the system, the growth morphology changes dramatically to a highly ordered dendritic growth characterized by stable needle-like protrusions decorated with regular side-branches. We investigate dendritic structure in anisotropic environments by engraving six-fold symmetric lattice on the Hele-Shaw cell. The morphology transition from dense-branching growth to dendritic growth is controlled by two parameters: the degree of anisotropy and the viscosity ratio between the less-viscous inner fluid and the more-viscous outer one. Remarkably, the imposed six-fold symmetry only leads to six-fold symmetric dendrites at low viscosity ratio. At higher viscosity ratio, the pattern instead adopts a twelve-fold symmetry. In addition to the six main branches evolving in the imposed direction, an additional six sub-branches emerge, at a 30° angle to the preferred growth direction. We discuss how this transition is related to an intrinsic length scale that depends on the viscosity ratio.