Wave regimes in two-layer microchannel flow

We consider the dynamics of an interface separating two immiscible fluids in a vertically oriented channel. We use boundary-layer theory together with the Kármán–Pohlhausen approximation to derive coupled nonlinear evolution equations for the interface shape and the flow rate in one of the layers. These equations reduce to those that govern the evolution of falling films at moderate flow rates. We use bifurcation theory to determine the ‘families’ of travelling waves for given flow rate and physical properties of the fluids; these waves are characterised by their speed and frequency. The solutions are obtained via continuation from those calculated for the falling film case.