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The impact of interfacial deformations on the hydrodynamic force and torque exerted on a spherical particle with surface slip moving along a gas-liquid interface is investigated. Following a two-parameter asymptotic modeling approach, we perturb the interface from its planar state and apply the Lorentz reciprocal theorem to the zeroth- and first-order approximations to analytically calculate the drag and torque on the particle. This allows us to explicitly account for the effect of physical parameters like the three-phase contact angle, the Bond number, and the slip coefficient on the particle motion. In addition, we study the interactions between two translating and rotating particles at a large separation. The interaction forces and torques exerted by the flow-induced deformations are calculated via the linear superposition approximation, where the interaction forces are identified as dipolar in terms of the azimuthal angle. 

The encapsulation of active particles, such as bacteria or active colloids, inside a droplet gives rise to a non-trivial shape dynamics and droplet displacement. To understand this behaviour, we derive an asymptotic solution for the fluid flow about a deformable droplet containing an active particle, modelled as a Stokes-flow singularity, in the case of small shape distortions. We develop a general solution for any Stokes singularity and apply it to compute the flows and resulting droplet velocity due to common singularity representations of active particles, such as Stokeslets, rotlets and stresslets. The results show that offsetting of the active particle from the centre of the drop breaks symmetry and excites a large number of generally non-axisymmetric shape modes as well as particle and droplet motion. In the case of a swimming stresslet singularity, a run-and-tumble locomotion results in superdiffusive droplet displacement. The effect of interfacial properties is also investigated. Surfactants adsorbed at the droplet interface counteract the internal flow and arrest the droplet motion for all Stokes singularities except the Stokeslet. Our results highlight strategies to steer the flows of active particles and create autonomously navigating containers. 

Nonlinear electrokinetic phenomena, where electrically driven fluid flows depend nonlinearly on the applied voltage, are commonly encountered in aqueous suspensions of colloidal particles. A prime example is the induced-charge electro-osmosis, driven by an electric field acting on diffuse charge induced near a polarizable surface. Nonlinear electrohydrodynamic flows also occur in non-polar fluids, driven by the electric field acting on space charge induced by conductivity gradients. Here, we analyse the flows about a charge-neutral spherical solid particle in an applied uniform electric field that arise from conductivity dependence on local field intensity. The flow pattern varies with particle conductivity: while the flow about a conducting particle has a quadrupolar pattern similar to induced-charge electro-osmosis, albeit with opposite direction, the flow about an insulating particle has a more complex structure. We find that this flow induces a force on a particle near an electrode that varies non-trivially with particle conductivity: while it is repulsive for perfectly insulating particles and particles more conductive than the suspending medium, there exists a range of particle conductivities where the force is attractive. The force decays as the inverse square of the distance to the electrode and thus can dominate the dielectrophoretic attraction due to the image dipole, which falls off with the fourth power with the distance. This electrohydrodynamic lift opens new possibilities for colloidal manipulation and driven assembly by electric fields. 

A charge-free particle in a uniform electric field experiences no net force in an unbounded domain. A boundary, however, breaks the symmetry and the particle can be attracted or repelled to it, depending on the applied field direction [Z. Wang et al., Phys. Rev. E, 2022, 106, 034607]. Here, we investigate the effect of a second boundary because of its common occurrence in practical applications. We consider a spherical particle suspended between two parallel walls and subjected to a uniform electric field, applied in a direction either normal or tangential to the surfaces. All media are modeled as leaky dielectrics, thus allowing for the accumulation of free charge at interfaces, while bulk media remain charge-free. The Laplace equation for the electric potential is solved using a multipole expansion and the boundaries are accounted for by a set of images. The results show that in the case of a normal electric field, which corresponds to a particle between two electrodes, the force is always attractive to the nearer boundary and, in general, weaker that the case of only one wall. Intriguingly, for a given particle-wall separation we find that the force may vary nonmonotonically with confinement and its magnitude may exceed the one-wall value. In the case of tangential electric field, which corresponds to a particle between insulating boundaries, the force follows the same trends but it is always repulsive.