Simulation of Hele-Shaw fingering with finite-capillary-number effects included

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Abstract
Results of a numerical simulation of unstable two-phase displacement in a Hele-Shaw cell are presented and compared with experimental data. Previous flow simulations, while reproducing many of the qualitative features of viscous fingering, consistently underpredicted the asymptotic value of finger width observed in a linear channel. We introduce here a modified boundary condition, as indicated by an asymptotic analysis of Bretherton, so that the pressure jump at the fluid interface is now a function of the instantaneous value of the particle velocity. This modification makes the imbedded problem nonlinear; it can, however, be treated successfully using Newton iteration. The results of the model depend parametrically on both the cell depth-to-width ratio and the capillary number of the displacement. Most of the discrepancy between theory and experiment is removed except at high speeds, where the numerically predicted finger width is somewhat higher than is observed. The basic qualitative features of the fingering phenomenon remain unchanged by the addition of the new effect.