Determination of finger shape using the dynamic capillary pressure

Abstract

Descriptions of fingered flow typically are based on linear instability theory, which becomes inappropriate once a finger has formed. In contrast, we obtain a self-similar solution to a moving finger with a curved interface by using the dynamic capillary pressure, where the capillary pressure depends on the velocity of the moving interface. From this solution we find analytical expressions for the shape and width of gravitationally driven fingers when they are fully developed. The solution has an unphysical limit of zero width at the limit of zero velocity owing to the simple dependence of capillary pressure on velocity. We discuss modifications to this dependence and compare the results to previous observations of fingering by Glass et al. [1989].