Shapes and paths of an air bubble rising inside a liquid are investigated experimentally. About three hundred experiments are conducted in order to generate a phase plot in the Galilei and Eötvös numbers plane, which separates distinct regimes in terms of bubble behaviour. A wide range of the Galilei and Eötvös numbers are obtained by using aqueous glycerol solutions of different concentrations as the surrounding fluid, and by varying the bubble size. The dynamics is investigated in terms of shapes, topological changes and trajectories of the bubbles. Direct numerical simulations are conducted to study the bubble dynamics, which show excellent agreement with the experiments. To the best of our knowledge, this is the first time an experimentally obtained phase plot showing the distinct behaviour of an air bubble rising in a quiescent medium is reported for such a large range of Galilei and Eötvös numbers.

A non-uniform surface slip can cause a symmetry breaking in the geometry of an otherwise homogeneous spherical particle to give rise to an anisotropic hydrodynamic resistance to the particle. Here, we develop a more general theoretical framework capable of decoding the surface-pattern-dependent hydrodynamic features for single heterogeneous spheres having arbitrary non-uniform slip length distributions in small variations, especially for those of weakly stick–slip or slip–slip Janus spheres in either the two-faced or striped type. Utilizing the Lorentz reciprocal theorem in conjunction with surface spherical harmonic expansion, we derive a new coupled set of Faxen formulas for the hydrodynamic force and torque on a non-uniform slip sphere by expressing impacts of slip anisotropy in terms of surface dipole and quadrupole without solving detailed flow fields. Our results reveal not only how various additional forces/torques arise from surface dipole and quadrupole, but also that it is the anti-symmetric dipole responsible for distinctive force-rotation/torque-translation coupling. These features are very distinct from those of no-slip or uniform-slip particles, possibly spurring new means to characterize or sort Janus particles in microfluidic experiments. In addition, the coupled Faxen relations with surface moment contributions reported here may infer potential changes in the collective nature of hydrodynamic interactions between non-uniform slip spheres. Furthermore, the present framework can also be readily applied to heterogeneous self-propelled squirmers whose swimming velocities are sensitive to slip anisotropy.

The dynamics of an initially nonspherical liquid droplet falling in air under the action of gravity is investigated via three-dimensional numerical simulations of the Navier-Stokes and continuity equations in the inertial regime. The surface tension is considered to be high enough so that a droplet does not undergo breakup. Vertically symmetric oscillations which decay with time are observed for low inertia. The amplitude of these oscillations increases for high Gallilei numbers and the shape asymmetry in the vertical direction becomes prominent. The reason for this asymmetry has been attributed to the higher aerodynamic inertia. Moreover, even for large inertia, no path deviations or oscillations are observed.

A pair of bubbles starting from rest and rising side-by-side in a liquid have been shown earlier to display spherical and ellipsoidal shapes. In contrast to earlier computational studies on the two-dimensional dynamics of a pair of bubbles, we study the fully three-dimensional motion of the bubbles in the inertial regime. We reveal the destabilizing nature of the interaction between the wakes of the bubbles, which causes them to rise in an oscillatory path. Such three dimensionality sets in earlier in time than for a single bubble and also at a lower inertia. The interaction leads to a mirror symmetry in the trajectories of the two bubbles, which persists for some time even in the high inertia regime where each path is chaotic. The e↵ect of the inertia and initial separation on the mirror symmetry of the path, the vortex shedding pattern and the attraction/repulsion between the bubbles are examined. The bubble rise has been interestingly observed to be symmetrical about the plane perpendicular to the separation vector for all separation distances considered in the present study.

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