The geometry of interfacial waves in vertical core-annular flow

A deterministic approach for the interface shape and position in vertical core annular flow of viscous oil and water is proposed, Through the inclusion of interfacial tension and neglecting inertial terms in the annulus, equations are obtained which govern the shape of the liquid-liquid interface. First, it is shown from the equations that the interface must be circular in the pipe cross section, as expected. Furthermore, the condition that the pressure is continuous on each side of the interface leads to the conclusion that the interface profile generally forms longitudinal waves, which are also symmetric along the vertical direction. Solution of the corresponding Laplace-Young equation reveals interface shapes entirely compatible with the observed "bamboo waves" for upward flow,whereas a pattern of pearl necklace waves is obtained for downward flow. A simple correlation for predicting the wavelength is suggested, which is in good agreement with presently available data.