Dendritic growth of viscous fingers under linear anisotropy

As a hydrodynamic analog of the dendritic crystal growth we have performed viscous fingering experiments under the influence of linear anisotropy through a single straight groove. One finger always tends to grow faster on the groove, accompanied by side branches. It was found that (1) the finger-tip profile is well approximated by a parabola, (2) experimental data of the dimensionless radius-of-curvature of the tip ρ/b (b is the plate separation of the Hele-Shaw cell, the only fundamental unit of length of the system)_collapse into a single curve when plotted to the capillary number Ca (proportional to tip speed v), regardless of various values of b and μ (fluid viscosity), (3) ρ/b is proportional to Ca−sol:12 for small values of Ca, (4) a crossover takes place around some critical value of Ca, beyond which ρ/b becomes almost constant, (5) side-branch spacing λ is approximately proportional to ρ. These behaviors of viscous finger dendrites are directly comparable with those of crystal dendrites. In addition, the crossing angle ϑ made by envelope lines for side-branch tips near a main-finger tip was found to increase with the increase of Ca. It was also observed that if the groove is terminated halfway, the finger tip cannot maintain the parabolic shape and the finger grows isotropically with the tip-splitting mode immediately after passing by the groove end.