On symmetric long-wave patterns in two-layer flows

The doubly periodic pattern formation of nonlinear partial differential equations that describe the small-amplitude, long wave, interfacial dynamics of a two-layer fluid flow in a channel for small volumetric-flow rates is investigated. This system consists of a Kuramoto-Sivashinsky type equation governing the evolution of the interfacial deflection coupled to a Poisson equation for the disturbance pressure field. The system has reflectional and translational symmetries in the direction of the flow and transverse to the flow. A local analysis near the mutual bifurcation of a two-dimensional wave and oblique waves yields criteria for different bifurcation scenarios near criticality. A numerical study away from the initial bifurcation reveals that for linearly coupled systems, the patterns appear to be steady for moderate values of the bifurcation parameter. When the equations are coupled nonlinearly, instabilities to the local solutions lead to a low-dimensional chaotic attractor through a period-doubling scenario. Oscillatory solutions, such as an exchange of energy from the two-dimensional wave to oblique modes are also seen, along with a travelling-square pattern that is quasi-periodic.