The stability of liquid films coating the walls of a parallel-plate channeland sheared by a pressure-driven gas flow along the channel centre is studied.The films are susceptible to a long-wavelength instability, whose dynamicbehaviour is found - for sufficiently low Reynolds numbers and thick gas layers- to be described by two coupled non-linear partial differential equations. Tothe best of our knowledge, such coupled fully non-linear equations for the filmthicknesses have not been derived previously. A linear stability analysisconducted under the condition that the material properties and the initialundisturbed liquid film thicknesses are equal can be utilized to determinewhether the interfaces are predominantly destabilized by the variations of theshear stress or by the pressure gradient acting upon them. The analysis of theweakly non-linear equations performed for this case shows that instabilitiescorresponding to a vanishing Reynolds number are absent from the system.Moreover, for this configuration, the patterns emerging along the twointerfaces are found to be identical in the long-time limit, implying that thefilms are fully synchronized. A different setup, where the liquid films haveidentical material properties but their undisturbed thicknesses differ, isstudied numerically. The results show that even for this configuration theinterfacial waves remain phase-synchronized and closely correlated for a broadrange of parameters. These findings are particularly relevant for multiphaseflow in narrow ducts, for example in the respiratory system or in microfluidicchannels.
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