Long wave and lubrication theories for core-annular flow

Abstract

Different nonlinear amplitude equations for long waves in core-annular flow are compared. Each equation has its own limits of validity that can be critically assessed by comparing the linearization of approximate and exact theories. Long wave theory gets the dispersion relation for the longest waves correctly but cannot accommodate cases like capillary instability, in which the most dangerous wave is not surpassingly long. Small gap lubrication based theories accommodate shorter waves of the size of the core when various extra conditions are satisfied, but various stabilizing mechanisms associated with inertia may not be well represented. One theory in which lubrication theory is used in the water film but not in the core captures the shear stabilization of inertia when the gap is small enough. The criterion for small enough is not uniform in the viscosity ratio and surpassingly small films are required for validity when the oil viscosity is large. The results of lubrication theory are not robust with respect to changes to larger gaps outside the regime of asymptotic validity; for example, the stabilizing effects of the inertia of the core and annulus may reverse for larger, but still small thicknesses.