## Abstract

In this paper, we examine the effect of viscosity stratification on wave propagation in three-layer flow down an inclined plane at vanishingly small Reynolds number and at finite wavelengths, for cases of negligible liquid - liquid interfacial tensions. We have found that the long-wavelength interface mode inertialess instability of Weinstein and Kurz [Phys. Fluids A 3, 2680 (1991)] persists into the finite wavelength domain in the form of nearly complex conjugate wave speed pairs; in certain limits, the interface modes are precisely complex conjugates. As in the case of Weinstein and Kurz, the physical configuration necessary to achieve inertialess instability is a low viscosity and thin internal layer with respect to the other layers in the film. The largest growth rate of the inertialess instability is found at finite wavelengths on the order of the total thickness of the film, and is orders of magnitude larger than the maximum growth rates identified by Lowenhurz and Lawrence [Phys. Fluids A 1, 1686 (1989)] for two-layer flows. We have also found an additional configuration exhibiting extremely large growth rates, also characterized by nearly complex conjugate behavior, that is not accessible via a long or short wavelength asymptotic limit; these three-layer structures have thin, high viscosity internal layers. The characteristic wavelengths associated with the largest growth rates are on the order of ten times smaller than those for the low viscosity internal layer cases. The influence of the deformable free surface on the growth rates of these interface modes is studied and found to be significant.

Original language | English (US) |
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Pages (from-to) | 3270-3282 |

Number of pages | 13 |

Journal | Physics of Fluids |

Volume | 11 |

Issue number | 11 |

DOIs | |

State | Published - Nov 1999 |

## ASJC Scopus subject areas

- Computational Mechanics
- Condensed Matter Physics
- Mechanics of Materials
- Mechanical Engineering
- Fluid Flow and Transfer Processes