Analytic theory for the linear stability of the Saffman–Taylor finger

Publisher Website
Google Scholar
Abstract
An analytic theory is presented for the linear stability of the Saffman–Taylor finger in a Hele–Shaw cell. Eigenvalues of the stability operator are determined in the limit of zero surface tension and it is found that all modes for the McLean–Saffman branch of solutions [J. Fluid Mech. 1 0 2, 455 (1980)] are neutrally stable, whereas other branches first calculated by Romero (Ph.D. thesis, California Institute of Technology, 1982) and Vanden‐Broeck [Phys. Fluids 2 6, 2033 (1983)] are unstable to arbitrary infinitesimal disturbances. It is also shown that the Saffman–Taylor discrete set of eigenvalues is a special case of a continuous unstable spectrum for zero surface tension. The introduction of any amount of surface tension perturbs the corresponding eigenmodes such that the finger boundary is a nonanalytic curve in general. Only transcendentally small terms in surface tension are responsible for the nonanalyticity of the finger boundary as in the case of Saffman–Taylor steady finger solutions of arbitrary finger width.

This publication has 10 references indexed in Scilit: