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We have developed a numerical package to simulate particle motions in fluid interfaces. The particles are moved in a direct simulation respecting the fundamental equations of motion of fluids and solid particles without the use of models. The fluid–particle motion is resolved by the method of distributed Lagrange multipliers and the interface is moved by the method of level sets. The present work fills a gap since there are no other theoretical methods available to describe the nonlinear fluid dynamics of capillary attraction.Two different cases of constrained motions of floating particles are studied here. In the first case, we study motions of floating spheres under the constraint that the contact angle is fixed by the Young–Dupr´e law; the contact line must move when the contact angle is fixed. In the second case, we study motions of disks (short cylinders) with flat ends in which the contact line is pinned at the sharp edge of the disk; the contact angle must change when the disks move and this angle can change within the limits specified by the Gibbs extension to the Young–Dupré law. The fact that sharp edged particles cling to interfaces independent of particle wettability is not fully appreciated and needs study.The numerical scheme presented here is at present the only one which can move floating particles in direct simulation. We simulate the evolution of single heavier-than-liquid spheres and disks to their equilibrium depth and the evolution to clusters of two and fours spheres and two disks under lateral forces, collectively called capillary attraction. New experiments by Wang, Bai & Joseph on the equilibrium depth of floating disks pinned at the edge are presented and compared with analysis and simulations.

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In microfluidic devices the fluid can be manipulated either as continuous streams or droplets. The latter is particularly attractive as individual droplets can not only move but also split and fuse, thus offering great flexibility for applications such as laboratory-on-a-chip. We consider the transport of liquid drops immersed in a surrounding liquid by means of the dielectrophoretic force generated by electrodes mounted at the bottom of a microdevice. The direct numerical simulation (DNS) approach is used to study the motion of droplets subjected to both hydrodynamic and electrostatic forces. Our technique is based on a finite element scheme using the fundamental equations of motion for both the droplets and surrounding fluid. The interface is tracked by the level set method and the electrostatic forces are computed using the Maxwell stress tensor. The DNS results show that the droplets move, and deform, under the action of nonuniform electric stresses on their surfaces. The deformation increases as the drop moves closer to the electrodes. The extent to which the isolated drops deform depends on the electric Weber number. When the electric Weber number is small, the drops remain spherical; otherwise, the drops stretch. Two droplets, however, that are sufficiently close to each other, can deform and coalesce, even if the electric Weber number is small. This phenomenon does not rely on the magnitude of the electric stresses generated by the bulk electric field, but instead is due to the attractive electrostatic drop-drop interaction overcoming the surface tension force. Experimental results are also presented and found to be in agreement with the DNS results.

A numerical scheme based on the distributed Lagrange multiplier method (DLM) is used to study the motion of nano-sized particles of dielectric suspensions subjected to uniform and nonuniform electric fields. Particles are subjected to both electrostatic and hydrodynamic forces, as well as Brownian motion. The results of the simulations presented in this paper show that uniform electric fields the evolution of the particle structures depends on the ratio of electrostatic particle-particle interactions and Brownian forces. When this ratio is of the order of 100 or greater, particles form stable chains and columns, whereas when it is of the order of 10 or smaller the particle distribution is random. For the nonuniform electric field cases considered in this paper, the relative magnitude of Brownian forces is in the range such that it does not influence the eventual collection of particles by dielectrophoresis and the particular locations where the particles are collected. However, Brownian motion is observed to influence the transient particle trajectories. The deviation of the particle trajectories compared to those determined by the electrostatic and hydrodynamic forces alone is characterized by the ratio of Brownian and dielectrophoretic forces.

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A finite element code based on the level-set method is used to perform direct numerical simulations (DNS) of the transient and steady-state motion of bubbles rising in a viscoelastic liquid modelled by the Oldroyd-B constitutive equation. The role of the governing dimensionless parameters, the capillary number (Ca), the Deborah number (De) and the polymer concentration parameter c, in both the rising speed and the deformation of the bubbles is studied. Simulations show that there exists a critical bubble volume at which there is a sharp increase in the terminal velocity with increasing bubble volume, similar to the behaviour observed in experiments, and that the shape of both the bubble and its wake structure changes fundamentally at that critical volume value. The bubbles with volumes smaller than the critical volume are prolate shaped while those with volumes larger than the critical volume have cusp-like trailing ends. In the latter situation, we show that there is a net force in the upward direction because the surface tension no longer integrates to zero. In addition, the structure of the wake of a bubble with a volume smaller than the critical volume is similar to that of a bubble rising in a Newtonian fluid, whereas the wake structure of a bubble with a volume larger than the critical value is strikingly different. Specifically, in addition to the vortex ring located at the equator of the bubble similar to the one present for a Newtonian fluid, a vortex ring is also present in the wake of a larger bubble, with a circulation of opposite sign, thus corresponding to the formation of a negative wake. This not only coincides with the appearance of a cusp-like trailing end of the rising bubble but also propels the bubble, the direction of the fluid velocity behind the bubble being in the opposite direction to that of the bubble. These DNS results are in agreement with experiments.

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