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Inertial settling of a sphere through an interface. Part 1. From sphere flotation to wake fragmentation
- Jean-Lou Pierson, Jacques Magnaudet
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- Journal:
- Journal of Fluid Mechanics / Volume 835 / 25 January 2018
- Published online by Cambridge University Press:
- 28 November 2017, pp. 762-807
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- Article
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Experiments are performed to better understand the characteristics of the flow induced by the gravity-driven settling of a rigid sphere through a two-layer arrangement of immiscible Newtonian fluids, mostly in inertia-controlled regimes. High-speed video imaging is employed to follow the sphere motion and the deformation of the interface separating the two fluids. The viscosity ratio between the lower and upper fluids is varied by four orders of magnitude, making it possible to observe highly contrasting interface patterns. Depending on the properties of the sphere and the fluids, the sphere may either float steadily at the interface or cross it by pulling a column of the upper fluid into the lower one. This column, which may be axisymmetric or three-dimensional depending on the relative magnitude of inertia effects in the upper fluid, generally pinches off at some position located either close to the initial interface or, more frequently, close to the sphere. Its lower part then recedes towards the sphere, forming a drop which remains attached to its top half. However, when inertia effects in the lower fluid are large enough and the upper fluid is not ‘too’ viscous, the tail quickly undergoes a complete fragmentation, giving birth to a large quantity of filaments and droplets. These various interface configurations are qualitatively analysed using the five independent dimensionless parameters characterizing the system, and regime maps based on the most relevant of them are provided. The influence of several of these parameters on four specific features observed in the course of the experiments, namely the pinch-off position, the floating/sinking transition, the volume of the attached drops and the average size of the droplets formed during the fragmentation process, is examined in detail. A simple model providing qualitative or quantitative predictions is established in each case, and its validity and limitations are assessed against experimental observations.
Inertial settling of a sphere through an interface. Part 2. Sphere and tail dynamics
- Jean-Lou Pierson, Jacques Magnaudet
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- Journal:
- Journal of Fluid Mechanics / Volume 835 / 25 January 2018
- Published online by Cambridge University Press:
- 28 November 2017, pp. 808-851
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- Article
- Export citation
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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.