Interfacial dynamics in pressure-driven two-layer laminar channel flow with high viscosity ratios

The large-scale dynamics of an interface separating two immiscible fluids in a channel is studied in the case of large viscosity contrasts. A long-wave analysis in conjunction with the Karman-Polhausen method to approximate the velocity profile in the less viscous fluid is used to derive a single equation for the interface. This equation accounts for the presence of interfacial stress, capillarity, and viscous retardation as well as inertia in the less viscous fluid layer where the flow is considered to be quasistatic; the equation is shown to reduce to a Benney-type equation and the Kuramoto-Sivashinskiy equation in the relevant limits. The solutions of this equation are parametrized by an initial thickness ratio h(0) and a dimensionless parameter S, which measures the relative significance of inertial to capillary forces. A parametric continuation technique is employed, which reveals that nonuniqueness of periodic solutions is possible in certain regions of (h(0),S) space. Transient numerical simulations are also reported, whose results demonstrate good agreement with the bifurcation structure obtained from the parametric continuation results.