On the tip-splitting instability of viscous fingers

Abstract

The instabilities of Saffman–Taylor viscous fingers are revisited experimentally in the standard linear channel as well as in wedges of angle θ₀. The local destabilization of a finger occurs by a splitting of its tip and results in the formation of two branches separated by a fjord. It is shown that, in a first approximation, the central line of a fjord follows a curve normal to the successive profiles of stable fingers. These normal curves are computed analytically for the Saffman–Taylor finger in a linear cell and numerically for the wedges. The length of a fjord is critically sensitive to the position of the initial destabilization of the finger. The nearer to the tip it occurs, the longer the fjord will be. Assuming a uniform spatial distribution of the disturbances in the central part of the finger front it is possible to predict the size distribution of the lateral branches. In the linear channel the probability of branches larger than the channel width is negligible. For wedges of increasing angle the probability of large secondary branches increases. Finally, for wedges with θ₀ larger than approximately 90° infinite fjords separating two long-lived structures are observed. Our experimental results also suggest a generalization of the definition of virtual cells. With this new definition it is possible to show that the increasing complexity of the patterns corresponds to a hierarchy of virtual cells of various sizes.