AN ASYMPTOTIC THEORY FOR THE LINEAR-STABILITY OF A CORE ANNULAR-FLOW IN THE THIN ANNULAR LIMIT

We study the linear stability of a vertical, perfectly concentric, core-annular flow in the limit in which the gap is much thinner than the core radius. The analysis includes the effects of viscosity and density stratification, interfacial tension, gravity and pressure-driven forcing. In the limit of small annular thicknesses, several terms of the expression for the growth rate are found in order to identify and characterize the competing effects of the various physical mechanisms present. For the sets of parameters describing physical situations they allow immediate determination of which mechanisms dominate the stability. Comparisons between the asymptotic formula and available full numerical computations show excellent agreement for non-dimensional ratio of undisturbed annular thickness to core radius as large as 0.2. The expansion leads to new linear stability results (an expression for the growth rate in powers of the capillary number to the 1/3 power) for wetting layers in low-capillary-number liquid-liquid displacements. The expression includes both capillarity and viscosity stratification and agrees well with the experimental results of Aul & Olbrich (1990). Finally, we derive Kuramoto-Sivashinsky-type integro-differential equation for the later nonlinear stages of the interfacial dynamics, and discuss their solutions.