Interfacial instability of liquid films coating the walls of a parallel-plate channel and sheared by a gas flow

Abstract: The stability of liquid films coating the walls of a parallel-plate channel and sheared by a pressure-driven gas flow along the channel centre is studied. The films are susceptible to a long-wavelength instability, whose dynamic behaviour is found - for sufficiently low Reynolds numbers and thick gas layers - to be described by two coupled non-linear partial differential equations. To the best of our knowledge, such coupled fully non-linear equations for the film thicknesses have not been derived previously. A linear stability analysis conducted under the condition that the material properties and the initial undisturbed liquid film thicknesses are equal can be 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 non-linear 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 a broad range of parameters. These findings are particularly relevant for multiphase flow in narrow ducts, for example in the respiratory system or in microfluidic channels.