Nonuniqueness and stability of the configuration of flow of immiscible fluids with different viscosities

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