A differential equation for the Saffman-Taylor finger

TL;DR

This paper introduces a nonlinear differential equation model for the Saffman-Taylor finger in Hele-Shaw flow, accurately capturing observed finger widths and instability characteristics.

Contribution

It develops a novel stream function approach leading to a differential equation that models finger formation and instability in Hele-Shaw flow.

Findings

Derived finger widths match observed range 1 > λ > 1/√5

Equation captures the correct dispersion relation for instability

Numerical solutions of stationary states are provided in a companion paper

Abstract

We develop a stream function approach for the horizontal Hele-Shaw, Saffman-Taylor finger. The model yields a nonlinear time-dependent differential equation. The finger widths derived from the equation are $1>\lambda>\frac{1}{\sqrt{5}}$, in units of half the width of the Hele-Shaw cell, in accordance with observation. The equation contains the correct dispersion relation for the creation of the finger instability. In an accompanying paper the stationary solutions of the equation are found numerically.