Theory of the Saffman-Taylor finger pattern. I

In a series of two papers, we present a comprehensive approach to the pattern formed via viscous fingering in a Hele Shaw cell, the Saffman-Taylor finger pattern. We explain how the finger width is selected and why this selection cannot be computed via asymptotic analysis in the (large) capillary number. Also, we derive the spectrum of small oscillations around the steady-state shape and show that the selected shape is stable. In this paper, we set up the mathematical and numerical formalism, and demonstrate stability for the =(1/2) solution. We also consider the effects of noise on the stability analysis.