Lattice Boltzmann Simulation on Droplet Flow through 3D Metal Foam

The hydrodynamics of droplets passing through metal foam is investigated using the lattice Boltzmann method (LBM). The accurate 3D porous structure for the simulation is generated by X-ray micro-computed tomography. The simulated results are in good agreement with the experimental ones using high-speed video. The simulated results show that for droplets passing metal foam, there is a critical capillary number, Cac (around 0.061), above which the droplet continues to deform until it breaks up. The simulated results show that the capillary number, droplet size, pores diameter, and thickness of metal foam have the significant effect of droplets deforming and breaking up when the droplets pass through the metal foam. To avoid the calescence of two droplets at the inlet zone of the metal foam, the distance between droplets should be larger than three times the diameter of the droplet.

Drops with insoluble surfactant squeezing through interparticle constrictions

The interfacial behaviour of surfactant-laden drops squeezing through tight constrictions in a uniform far-field flow is modelled with respect to capillary number, drop-to-medium viscosity ratio and surfactant contamination. The surfactant is treated as insoluble and non-diffusive, and drop surface tension is related to surfactant concentration by a linear equation of state. The constriction is formed by three solid spheres held rigidly in space. A characteristic aspect of this confined and contaminated multiphase system is the rapid development of steep surfactant-concentration gradients during the onset of drop squeezing. The interplay between two physical effects of surfactant, namely the greater interface deformability due to decreased surface tension and interface immobilization due to Marangoni stresses, results in particularly rich drop-squeezing dynamics. A three-dimensional boundary-integral algorithm is used to describe drop hydrodynamics, and accurate treatment of close squeezing and trapped states is enabled by advanced singularity subtraction techniques. Surfactant transport and hydrodynamics are coupled via the surface convection equation (or convection–diffusion equation, if artificial diffusion is included), the interfacial stress balance and a solid-particle contribution based on the Hebeker representation. For extreme conditions, such as drop-to-medium viscosity ratios significantly less than unity, it is found that upwind-biased methods are the only stable approaches for modelling surfactant transport. Two distinct schemes, upwind finite volume and flow-biased least squares, are found to provide results in close agreement, indicating negligible numerical diffusion. Surfactant transport is enhanced by low drop-to-medium viscosity ratios, at which extremely sharp concentration gradients form during various stages of the squeezing process. The presence of surfactant, even at low degrees of contamination, significantly decreases the critical capillary number for droplet trapping, due to the accumulation of surfactant at the downwind pole of the drop and its subsequent elongation. Increasing the degree of contamination significantly affects surface mobility and further decreases the critical capillary number as well as drop squeezing times, up to a threshold above which the addition of surfactant negligibly affects squeezing dynamics.

Electrocoalescence of a pair of conducting drops in an insulating oil

The effect of an electric field on the coalescence of two water drops suspended in an insulating oil is investigated. We report four new results. (i) The cone angle for the non-coalescence of drops can be significantly smaller (as small as $19^{\circ }$) than the value of $30.8^{\circ }$ reported by Bird et al. (Phys. Rev. Lett., vol. 103 (16), 2009, 164502). (ii) A surprising observation of the dependence of the mode of coalescence/non-coalescence on the type of insulating oil is seen. A cone–cone mode for silicone oil is observed as against cone–dimple mode for castor oil. (iii) The critical capillary number for non-coalescence decreases with increase in the conductivity of the droplet phase. (iv) Systematic experiments prove that the apparent bridge during non-coalescence is indeed transitory and not permanent, as reported elsewhere. Theoretical calculations using analytical theory and the boundary integral method explain the formation of the cone–dimple mode as well as the transitory bridge length. The numerical calculation and thereby the physical mechanism to explain the occurrence of very small non-coalescence angles as well as the dependence of the phenomenon on the conductivity of the insulating oil and the water droplets remain unexplained.

Forced dewetting in a capillary tube

Liquid films can be entrained when the dewetting velocity attains a threshold, and this dynamical wetting transition has been well studied in the situation of plane substrates. We investigate the forced dewetting in a capillary tube using diffuse-interface simulations and lubrication analysis, focusing on the onset of wetting transition and subsequent interface evolution. Results show that the meniscus remains stable when the displacing rate is below a threshold, beyond which film entrainment occurs and eventually leads to the formation of Taylor bubbles separated by liquid slugs, as has also been observed in the recent experiments of Zhao et al. (Phys. Rev. Lett., vol. 120, 2018, 084501). We derive an analytical solution of the critical capillary number, and demonstrate that the wetting transition is accompanied by a vanishing apparent contact angle and an abrupt drop of the contact-line velocity. Both the bubble and slug lengths are found to depend on the capillary number and the wettability of the wall. A theoretical formula for the bubble length is also proposed and compares favourably with numerical and experimental results.

Dynamic drying transition via free-surface cusps

We study air entrainment by a solid plate plunging into a viscous liquid, theoretically and numerically. At dimensionless speeds $Ca=U\unicode[STIX]{x1D702}/\unicode[STIX]{x1D6FE}$ of order unity, a near-cusp forms due to the presence of a moving contact line. The radius of curvature of the cusp’s tip scales with the slip length multiplied by an exponential of $-Ca$. The pressure from the air flow drawn inside the cusp leads to a bifurcation, at which air is entrained, i.e. there is ‘wetting failure’. We develop an analytical theory of the threshold to air entrainment, which predicts the critical capillary number to depend logarithmically on the viscosity ratio, with corrections coming from the slip in the gas phase.

The role of surface charge convection in the electrohydrodynamics and breakup of prolate drops

The deformation of a weakly conducting, ‘leaky dielectric’, drop in a density matched, immiscible weakly conducting medium under a uniform direct current (DC) electric field is quantified computationally. We exclusively consider prolate drops, for which the drop elongates in the direction of the applied field. Furthermore, for the majority of this study, we assume the drop and medium to have equal viscosities. Using axisymmetric boundary integral computations, we delineate drop deformation and breakup regimes in the $Ca_{E}-Re_{E}$ parameter space, where $Ca_{E}$ is the electric capillary number (ratio of the electric to capillary stresses); and $Re_{E}$ is the electric Reynolds number (ratio of charge relaxation to flow time scales), which characterizes the strength of surface charge convection along the interface. For so-called ‘prolate A’ drops, where the surface charge is convected towards the ‘poles’ of the drop, we demonstrate that increasing $Re_{E}$ reduces the critical capillary number for breakup. Moreover, surface charge convection is the cause of an abrupt transition in the breakup mode of a drop from end pinching, where the drop elongates and develops bulbs at its ends that eventually detach, to a breakup mode characterized by the formation of conical ends. On the contrary, the deformation of ‘prolate B’ drops, where the surface charge is convected away from the poles, is essentially unaffected by the magnitude of $Re_{E}$.

Splitting of a two-dimensional liquid plug at an airway bifurcation

Certain medical treatments involve the introduction of exogenous liquids in the lungs. These liquids can form plugs within the airways. The plugs propagate throughout the branching network in the lungs being forced by airflow. They leave a deposited film on the airway walls and split at bifurcations. Understanding the resulting distribution of liquid throughout the lungs is important for the effective administration of the prescribed treatments. In this paper, we investigate numerically the splitting of a liquid plug by a two-dimensional pulmonary bifurcation under the influence of a transverse gravitational field. The splitting is characterized by the splitting ratio, which is the ratio of volume of the liquid plug in the daughter channels and depends on the capillary number and the orientation of the bifurcation plane with respect to a three-dimensional gravitational field. It is observed that gravity induces asymmetry in the splitting, causing the splitting ratio to be reduced. This effect is mitigated as the capillary number is increased. It is also observed that there exists a critical capillary number where the plug will not split and will instead propagate entirely into the gravitationally favoured daughter channel. We also compute the wall stresses on the bifurcation walls and observe the locations where stresses and their gradients are the highest in magnitude.

Numerical simulations of immiscible displacement in the cavities via lattice Boltzmann method

In this paper, the immiscible displacements in the different cavities are studied by the pseudo-potential lattice Boltzmann (LB) model. We first validate the model with a two-dimensional (2D) layered flow, and find that the numerical results agree well with the corresponding analytical solutions. Then, we perform some numerical simulations to study the immiscible displacements in the cavities, and focus on the effects of the surface wettability, capillary number and density ratio on the displacement efficiency. The numerical results show that the displacement efficiency increases with the increase of the capillary number at first and then presents a decrease with the capillary number when it is large enough. The increase of the contact angle θ1 or decrease of the density ratio increases the displacement efficiency but decreases the critical capillary number. Finally, it is also found that both the size and geometry of cavity have a significant influence on the displacement efficiency.

Control of Slag-Dragging Effects at the Metal–Slag Interface through Electromagnetic Brake in a Slab Mold

Turbulent flow when steel is delivered through a nozzle in a slab mold induces dragging forces at the metal–slag interface that entrain slag droplets into the metal bulk. These dragging effects are discontinuous and correspond to the velocity fluctuations of turbulence at that interface which themselves, are dependent on nozzle immersion, nozzle design, mold width and casting speed. Slag viscosity and density, metal viscosity and slag layer thickness are employed to estimate that critical velocity which is embodied in a critical capillary number for some established mold operating conditions. This approach permits the link between all operating variables including flux chemistry and nozzle design with the interface instability. A relationship between the capillary number and the magnetic field strength used to brake the liquid steel is established which is used to assure the interface stability for any operating condition and flux chemistry.