Viscous flows in two dimensions

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Abstract
This review is an expository treatment of the displacement of one fluid by another in a two-dimensional geometry (a Hele-Shaw cell). The Saffman-Taylor equations modeling this system are discussed. They are simulated by random-walk techniques and studied by methods from complex analysis. The stability of the generated patterns (fingers) is studied by a WKB approximation and by complex analytic techniques. The primary conclusions reached are that (a) the fingers are linearly stable even at the highest velocities, (b) they are nonlinearly unstable against noise or an external perturbation, the critical amplitude for the noise being an exponential function of a power of the velocity for high velocities, (c) such exponentials seem to dominate high-velocity behavior, as can be seen from a WKB analysis, and (d) the results of the Saffman-Taylor equations disagree with experiments, apparently because they leave out film-flow phenomena.

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