### Abstract

We consider the two-layer flow of immiscible, viscous, incompressible fluids in an inclined channel. We use long-wave theory to obtain a strongly nonlinear evolution equation which describes the motion of the interface. This equation includes the physical effects of viscosity stratification, density stratification, and shear. A weakly nonlinear analysis of this equation yields a Kuramoto-Sivashinsky equation, which possesses a quadratic nonlinearity. However, certain physical situations exist in two-layer flow for which modifications of the Kuramoto-Sivashinsky equation are physically pertinent. In particular, the presence of the second layer can mediate the wave-steepening instability found in single-phase falling films, requiring the inclusion of a cubic nonlinearity in the weakly nonlinear analysis. The introduction of the cubic nonlinearity destroys the symmetry-breaking bifurcations of the Kuramoto-Sivashinsky equation, and new isolated solution branches emerge as the strength of the cubic nonlinearity increases. Bistability between these new solutions and those associated with the Kuramoto-Sivashinsky equation is found, as well as the formation of a hysteresis loop from smaller-amplitude travelling waves to larger-amplitude travelling waves. The physical implications of these dynamics to the phenomenon of laminar flooding in a channel are discussed.

Original language | English (US) |
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Pages (from-to) | 55-83 |

Number of pages | 29 |

Journal | Journal of Fluid Mechanics |

Volume | 277 |

DOIs | |

State | Published - Jan 1 1994 |

### ASJC Scopus subject areas

- Condensed Matter Physics
- Mechanics of Materials
- Mechanical Engineering

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## Cite this

*Journal of Fluid Mechanics*,

*277*, 55-83. https://doi.org/10.1017/S0022112094002685