Saffman-Taylor fingers with anisotropic surface tension

Abstract

We have qualitatively explained the experiments of McCloud and Maher for the viscous fingering problem in which an anisotropy in the surface tension parameter was imposed by engraving a grid in one of the plates of the Hele-Shaw cell. We approached the problem in an analytical form by extending solvability theory to incorporate the effect of anisotropy. For the case in which the surface tension has a maximum at the finger tip, our theory provides two possible solutions: one corresponding to the solution of the isotropic case and a new solution which, below a threshold of the surface tension parameter, predicts a wider finger than the isotropic solution. Intuitivelt, we expect the “old” solution, namely the one that does not differ from the isotropic case, to be the selected solution for large values of the surface tension parameter and we expect the new solution to be selected for small values of the surface tension parameter. This was confirmed by dynamical simulations of the interface. The simulations predict that for the case in which the surface tension has a maximum at the finger tip, anisotropy is irrelevant for large values of the surface tension parameter. Furthermore, below a threshold in this surface tension parameter, the selected finger width is systematically wider than the corresponding isotropic case.