Theory of the Saffman-Taylor ‘‘finger’’ pattern. II

We generalize the stability results of the preceding paper to arbitrary λ. Coupled with a new approach to the selection of ‘‘finger’’ width, this stability computation allows us to predict the dependence of allowed widths on capillary number. At the same time, we show that of all possible finger shapes at fixed capillary number, only one [denoted ${\lambda }^{\mathrm{*}}$(γ)] is stable. Finally, we show how finite noise can shift ${\lambda }^{\mathrm{*}}$(γ), causing observed widths and stability to diverge from the noiseless calculation at large velocity. We conclude with a discussion of the paradigm of microscopic solvability for diffusive controlled pattern formation, into which the above analysis neatly falls.