Inertial settling of a sphere through an interface. Part 2. Sphere and tail dynamics

Abstract

Selected situations in which a rigid sphere settles through a two-layer system obtained by superimposing two immiscible Newtonian fluids are studied using a combination of experiments and direct numerical simulations. By varying the viscosity of the two fluids and the sphere size and inertia, the flow conditions cover situations driven by capillary and viscous effects, in which case the sphere detaches slowly from the interface and may even rise for a period of time, as well as highly inertial cases where its motion is barely affected by the interface and essentially reacts to the change in the fluid viscosity and density. The evolutions of the sphere velocity, effective drag force and entrained volume of upper fluid are analysed. In most cases considered here, this entrained volume first takes the form of an axisymmetric tail which elongates as time proceeds until it pinches off at some point. We examine the post-pinch-off dynamics of this tail under various conditions. When the viscosity of the lower fluid is comparable or larger than that of the upper one, an end-pinching process initiated near the initial pinch-off position develops and propagates along the tail, gradually transforming it into a series of primary and satellite drops; the size of the former is correctly predicted by the linear stability theory. In contrast, when the lower fluid is much less viscous than the upper one, the tail recedes without pinching off again. During a certain stage of the process, the tip velocity keeps a constant value which is significantly underpredicted by the classical Taylor–Culick model. An improved theoretical prediction, shown to agree well with observations, is obtained by incorporating buoyancy effects resulting from the density difference between the two fluids. Spheres with large enough inertia settling in a low-viscosity lower fluid are found to exhibit specific tail dynamics prefiguring wake fragmentation. Indeed, an interfacial instability quickly develops near the top of the sphere, resulting in the formation of thin axisymmetric corollas surrounding the central part of the tail and propagating upwards. A simplified inviscid model considering the role of the boundary layer around the tail and including surface tension effects is found to predict correctly the characteristics of the observed instability which turns out to be governed by the Kelvin–Helmholtz mechanism.