Non-uniqueness and stability of the configuration of flow of immiscible fluids with different viscosities

Abstract

High-viscosity liquids hate to work. Low-viscosity liquids are the victims of the laziness of high-viscosity liquids because they are easy to push around.The arrangement of components in steady flow of immiscible liquids is typically non-unique. The problem of selection of arrangements is defined here and is studied by variational methods under the hypothesis that the realized arrangements are the ones that maximize the speed on exterior boundaries for prescribed boundary tractions, or the ones that minimize the tractions for prescribed speeds. The arrangements which minimize tractions also minimize the dissipation by putting low-viscosity liquid in regions of high shear. The variational problem is used as a guide to intuition in the design and interpretation of experiments when results of analysis of stability are unavailable. In fact we always observe some kind of shielding of high-viscosity liquid. This can occur by sheet coating in which low-viscosity liquid encapsulates high-viscosity liquid, or through the formation of rigidly rotating masses of high-viscosity liquid which we call rollers. In other cases we get emulsions of low-viscosity liquid in a high-viscosity foam. The emulsions arise from a fingering instability. The low-viscosity liquid fingers into the high-viscosity liquid and then low-viscosity bubbles are pinched off the fingers. The emulsions seem to have a very low effective viscosity and they shield the high-viscosity liquid from shearing. In the problem of Taylor instability with two fluids, low-viscosity Taylor cells are separated by stable high-viscosity rollers.