Interfacial instabilities and fingering formation in Hele-Shaw flow

The interfacial instability of Hele-Shaw flow has been a crucial issue for the understanding of the pattern formation of viscous fingers in a Hele-Shaw cell. By using a unified asymptotic approach, we derive two different types of instability mechanisms for ‘slightly’ time-dependent finger solutions; namely, (i) the global-trapped-wave (GTW) instability; and (ii) the zero-frequency (null-f) instability. On the basis of these instability mechanisms, the selection of viscous finger formation is clarified; the apparent contradiction between the previous linearstability analysis by Tanveer (1987, Phys. Fluid30, 1589) and others and the numerical simulations by DeGregoria & Schwartz (1986, J. Fluid Mech.164, 383)and the experimental evidence is reconciled.