Spreading or contraction of viscous drops between plates: single, multiple or annular drops

The behaviour of a viscous drop squeezed between two horizontal planes (a squeezed Hele-Shaw cell) is treated by both theory and experiment. When the squeezing force $F$ is constant and surface tension is neglected, the theory predicts ultimate growth of the radius $a\sim t^{1/8}$ with time $t$. This theory is first reviewed and found to be in excellent agreement with experiment. Surface tension at the drop boundary reduces the interior pressure, and this effect is included in the analysis, although it is negligibly small in the squeezing experiments. An initially elliptic drop tends to become circular as $t$ increases. More generally, the circular evolution is found to be stable under small perturbations. If, on the other hand, the force is reversed ($F<0$), so that the plates are drawn apart (the ‘contraction’, or ‘lifting plate’, problem), the boundary of the drop is subject to a fingering instability on a scale determined by surface tension. The effect of a trapped air bubble at the centre of the drop is then considered. The annular evolution of the drop under constant squeezing is still found to follow a ‘one-eighth’ power law, but this is unstable, the instability originating at the boundary of the air bubble, i.e. the inner boundary of the annulus. The air bubble is realised experimentally in two ways: first by simply starting with the drop in the form of an annulus, as nearly circular as possible; and second by forcing four initially separate drops to expand and merge, a process that involves the resolution of ‘contact singularities’ by surface tension. If the plates are drawn apart, the evolution is still subject to the fingering instability driven from the outer boundary of the annulus. This instability is realised experimentally by levering the plates apart at one corner: fingering develops at the outer boundary and spreads rapidly to the interior as the levering is slowly increased. At a later stage, before ultimate rupture of the film and complete separation of the plates, fingering spreads also from the boundary of any interior trapped air bubble, and small cavitation bubbles appear in the very low-pressure region, far from the point of leverage. This exotic behaviour is discussed in the light of the foregoing theoretical analysis.

Controlled locomotion of a droplet propelled by an encapsulated squirmer

We work out the propulsion of a viscous drop which is driven by two mechanisms: the active velocity of an encapsulated squirmer and an externally applied force acting on the squirmer. Of particular interest is the existence of a stable comoving state of drop and squirmer, allowing for controlled manipulation of the viscous drop by external forcing. The velocities of droplet and squirmer, as well as the conditions for a stable comoving state are worked out analytically for the axisymmetric configuration with a general displacement of the squirmer from the center of the droplet Graphic abstract

Towards an analytical description of active microswimmers in clean and in surfactant-covered drops

Geometric confinements are frequently encountered in the biological world and strongly affect the stability, topology, and transport properties of active suspensions in viscous flow. Based on a far-field analytical model, the low-Reynolds-number locomotion of a self-propelled microswimmer moving inside a clean viscous drop or a drop covered with a homogeneously distributed surfactant, is theoretically examined. The interfacial viscous stresses induced by the surfactant are described by the well-established Boussinesq-Scriven constitutive rheological model. Moreover, the active agent is represented by a force dipole and the resulting fluid-mediated hydrodynamic couplings between the swimmer and the confining drop are investigated. We find that the presence of the surfactant significantly alters the dynamics of the encapsulated swimmer by enhancing its reorientation. Exact solutions for the velocity images for the Stokeslet and dipolar flow singularities inside the drop are introduced and expressed in terms of infinite series of harmonic components. Our results offer useful insights into guiding principles for the control of confined active matter systems and support the objective of utilizing synthetic microswimmers to drive drops for targeted drug delivery applications. Graphical abstract

Capillary ripples in thin viscous films

Capillary ripples in thin viscous films are important features of coating and lubrication flows. Here, we present experiments based on digital holographic microscopy, measuring with nanoscale resolution the morphology of capillary ripples ahead of a viscous drop spreading on a prewetted surface. Our experiments reveal that upon increasing the spreading velocity, the amplitude of the ripples first increases and subsequently decreases. Above a critical spreading velocity, the ripples even disappear completely and this transition is accompanied by a divergence of the ripple wavelength. These observations are explained quantitatively using linear wave analysis, beyond the usual lubrication approximation, illustrating that new phenomena arise when the capillary number becomes of the order of unity.

Gliding on a layer of air: impact of a large-viscosity drop on a liquid film

In this paper we contrast the early impact stage of a highly viscous drop onto a liquid versus a solid substrate. Water drops impacting at low velocities can rebound from a solid surface without contact. This dynamic is mediated through lubrication of a thin air layer between the liquid and solid. Drops can also rebound from a liquid surface, but only for low Weber numbers. Impacts at higher velocities in both cases lead to circular contacts which entrap an air disc under the centre of the drop. Increasing the drop viscosity produces extended air films for impacts on a smooth solid surface even for much larger velocities. These air films eventually break through random wetting contacts with the solid. Herein we use high-speed interferometry to study the extent and thickness profile of the air film for a large-viscosity drop impacting onto a viscous film of the same liquid. We demonstrate a unified scaling of the centreline height of the air film for impacts on both solid and liquid, when using the effective impact velocity. On the other hand, we show that the large-viscosity liquid film promotes air films of larger extent. Furthermore, the rupture behaviour becomes fundamentally different, with the air film between the two compliant surfaces being more stable, lacking the random wetting patches seen on the solid. We map the parameter range where these air films occur and explore the transition from gliding to ring contact at the edge of the drop dimple. After the air film ruptures, the initial contraction occurs very rapidly and for viscosities greater than 100 cSt the retraction velocity of the air film is ${\sim}0.3~\text{m}~\text{s}^{-1}$, independent of the liquid viscosity and impact velocity, in sharp contrast with theoretical predictions.