The Motion of Immiscible Drops and the Stability of Annular Flow

Abstract

The creeping motion of a neutrally buoyant drop in Poiseuille flow is studied numerically using the boundary integral technique. The effects of the viscosity ratio, interfacial tension and drop size on steady shapes and velocities of the deformed drop are considered. Particular attention is given to cases involving large deformation which occurs when the interfacial tension becomes small. The critical value of the capillary number, for a given viscosity ratio, above which a steady shape for the drop does not exist is determined. The stability of annular flow of two fluids of different viscosities through a circular tube is studied. The instability considered in the present study occurs at the interface between two fluids. Linear stability analysis is carried out for axisymmetric disturbances when the mechanisms of instability due to a viscosity difference between two fluids and interfacial tension are simultaneously present. The growth factor of instability is nonlinear in the viscosity ratio and the interfacial tension because the governing equations and boundary conditions are linearized with respect to a disturbance amplitude function, but not linearized with respect to the viscosity ratio and the interfacial tension. The effects of the viscosity ratio, interfacial tension, radius ratio and Reynolds number on the stability of the interface as well as the modes of maximum instability are studied. Numerical study of the axisymmetric approach of a deformable drop toward a deformable interface under the action of a constant buoyancy force is carried out using the boundary integral technique. Unlike the previous film drainage theories, governing equations are applied in all fluids including the drop, lower bulk and upper bulk fluids. Therefore, physical properties of the drop fluid and the upper bulk fluid, which are neglected in the film drainage theories, are included in the present study. The influence of the viscosity ratio and interfacial tension is considered. Three distinct mechanisms of film drainage are identified: rapid drainage where the film between the drop and the interface is thinnest at the centerline and the film thickness increases with radial distance, uniform drainage where the region of uniform film thickness appears and persists during further approach of the drop toward the interface, and dimpled drainage where the film is thinnest at a rim radius rather than at the centerline.