### Abstract

Combined effects of shear-flow and thermocapillary instabilities in a two-layer Couette flow are asymptotically examined in the thin-layer limit. The basic features of the system instability are revealed by first analyzing the two-dimensional stability problem. A scaling analysis is devised to identify dominant mechanisms in various parameter regimes. With an appropriate scaling, the leading order linear stability is reduced to a one-dimensional evolution equation containing a nonlocal contribution from viscosity stratification. Viscosity stratification destabilizes (stabilizes) the system with a more (less) viscous film, but the effect can be compromised by thermocapillary stabilization (destabilization) as the film is cooled (heated). Thermocapillary effects dominate over viscosity stratification effects for short-wave perturbations albeit the latter could be stronger than the former for long waves. The competition between these two effects gives rise to the critical Reynolds number for the onset of stability/instability. A nontrivial interplay is found within a window in the weak interfacial-tension regime. It demonstrates a possibility of the existence of two neutral states in the wavenumber space. The three-dimensional problem is also examined. For the first time, a two-dimensional film evolution equation with the inclusion of a nonlocal term is systematically derived for the corresponding stability. It can be shown analytically that three-dimensional perturbations can be more unstable than two-dimensional ones due to thermocapillarity in line with the nonexistence of Squires' theorem. The three-dimensional problem has the critical Reynolds number larger than the two-dimensional problem, but an instability in the latter does not necessarily suggest an instability in the former. An extension of each problem to the weakly nonlinear regime is also discussed in the context of the Kuramoto-Sivashinsky equation.

Original language | English |
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Article number | 064109 |

Journal | Physics of Fluids |

Volume | 18 |

Issue number | 6 |

DOIs | |

Publication status | Published - 2006 Jun |

### All Science Journal Classification (ASJC) codes

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

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## Cite this

*Physics of Fluids*,

*18*(6), [064109]. https://doi.org/10.1063/1.2213279