The stability and coupling of liquid films coating the walls of a parallel-plate channel and sheared by a pressure-driven gas flow along the channel center plane is studied. The films are susceptible to a long-wavelength instability. For sufficiently low Reynolds numbers and thick gas layers, the dynamic behavior is found to be described by two coupled nonlinear partial differential equations. A linear stability analysis is conducted under the condition that the material properties and the initial undisturbed liquid-film thicknesses are equal. The linear analysis is utilized to determine whether the interfaces are predominantly destabilized by the variations of the shear stress or by the pressure gradient acting upon them. The analysis of the weakly nonlinear equations performed for this case shows that instabilities corresponding to a vanishing Reynolds number are absent from the system. Moreover, for this configuration, the patterns emerging along the two interfaces are found to be identical in the long-time limit, implying that the films are fully synchronized. A different setup, where the liquid films have identical material properties but their undisturbed thicknesses differ, is studied numerically. The results show that, even for this configuration, the interfacial waves remain phase-synchronized and closely correlated for an extended period of time. These findings are particularly relevant for gaseous flow through narrow ducts with liquid-coated walls.
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This study was supported by the Deutsche Forschungsgemeinschaft (DFG) under Grant number DI 1689/1-1, which is gratefully acknowledged.