Roll waves and plugs in two-layer flows

In this paper I consider a system of equations describing two stratified fluids flowing in closed, slightly inclined ducts. In the framework of the shallow water approximation with turbulent friction acting on the wall and at the interface, I investigate a special class of periodic travelling waves containing stable moving jumps, namely, roll waves and periodic slug flows. The modulation equations of roll waves and slugs are derived and a nonlinear stability criterion is obtained. As for plug flow, only a gas-liquid system is considered. The stability criterion is expressed in terms of an integro-differential relation. For self-similar cross-section, this criterion is simplified to a relation for a function of one variable only.