On Viscous Fluid Flow Down an Inclined Plane and the Development of Roll Waves

Abstract

We develop a tractable, and mathematically rigorous, asymptotic theory for the development of instability in viscous fluid flow down an inclined plane, at supercritical Reynolds numbers. Our theory involves consideration of an ill posed problem, and provides a new example of the way such problems come up in applied mathematics. We show how one may use additional information, inherent in the problem one really wishes to solve, in order to regularize this improperly posed problem. We discuss several numerical experiments and compare the results with experimental observations of “roll waves” and “slug flows” in inclined channels, at similar slopes and Reynolds numbers.