We investigate the natural oscillations of sessile drops with a central trapped bubble on a plane using linear potential flow theory, considering both free and pinned contact lines. The system is governed by the contact angle $\alpha$ and the ratio $\tau$ of inner to outer contact line radii. For bubble-containing (BC) hemispherical drops with free contact lines (referred to as free BC semi-drops), the modes mirror half of those in concentric spherical BC drops due to plane symmetry. These modes are labelled ‘plus’ (with greater inner surface deformation) and ‘minus’ (with greater outer surface deformation). As $\tau \to 0$, minus modes converge to those of bubble-free drops. Results show that varying $\alpha$ from $90^\circ$ or pinning the contact line in free BC semi-drops alters the topology of spectral lines, turning original crossings of spectral lines between minus and plus modes into avoided crossings. This shift causes minus and plus modes to form spectral trends with avoided crossings, maintaining their original spectral shapes. In an avoided crossing, two coupled modes cannot be classified as plus or minus due to their comparable inner and outer surface deformations, resulting in mode beating when both are excited, as confirmed by our direct numerical simulations. This study on the impact of inner bubbles on the spectrum may help in predicting bubble size in opaque sessile drops.

The breakup of coaxial cone-jet interfaces to compound droplets in axisymmetric co-flow focusing (CFF) upon actuation is studied through numerical simulations. Due to the coupling effect of double interfaces, the response behaviours of coaxial cone-jet flow to actuation are more complex than those of a single-layered interface structure. Particularly, the coaxial jet presents totally different response modes between weak and strong interface coupling situations. In this work, the phase diagrams of response modes for coaxial jet breakup are depicted, considering the effect of perturbation frequency, amplitude and liquid flow rates. In particular, the breakup of a coaxial jet can be synchronized with actuation within a frequency range containing the natural breakup frequency, resulting in uniform compound droplets with a single core inside the shell, and the size of droplets can be adjusted by frequency. As the perturbation frequency exceeds the upper critical value, the external perturbation is unable to dominate the jet breakup, while below the lower critical frequency, the jet breaks up with multiple droplets generated in one period. The perturbation amplitude mainly affects the jet breakup length and also leads to the transition between different response modes. The coaxial cone upstream of the orifice can act as a buffer layer, regulating the perturbation amplitude of the coaxial jet downstream. The degree of buffering effect is affected by the perturbation frequency and amplitude. As the perturbation amplitude approaches unity, the decrease of perturbation frequency leads to the intermittent jet behaviour from the cone tip with a vibrating manner of the coaxial cone. Based on the linear instability analysis on the simplified single jet models for weak-coupled and strong-coupled jets, scaling analyses are carried out, which predict the jet breakup length and the natural frequency and critical frequency for the synchronized breakup. Finally, a strong pulse is added on the perturbation to produce compound droplets with a controllable number of cores. The present work provides valuable guidance for the practical application of on-demand compound droplet generation through active CFF.

In this work, the dynamics of two-dimensional rotating Janus drops in shear flow is studied numerically using a ternary-fluid diffuse interface method. The rotation of Janus drops is found to be closely related to their deformation. A new deformation parameter $D$ is proposed to assess the significance of the drop deformation. According to the maximum value of $D$ ($D_{max}$), the deformation of rotating Janus drops can be classified into linear deformation ($D_{max}\le 0.2$) and nonlinear deformation ($D_{max}> 0.2$). In particular, $D_{max}$ in the former depends linearly on the Reynolds and capillary numbers, which can be interpreted by a mass–spring model. Furthermore, the rotation period $t_R$ of a Janus drop is found to be more sensitive to the drop deformation than to the aspect ratio of the drop at equilibrium. By introducing a corrected shear rate and an aspect ratio of drop deformation, a rotation model for Janus drops is established based on Jeffery's theory for rigid particles, and it agrees well with our numerical results.

The dynamics of compound drops impacting on a flat substrate is numerically investigated using a ternary-fluid diffuse-interface method, with the aim of assessing the effect of a density difference between the inner and outer droplets (denoted by $\lambda$) on the evolution of the interfaces. With the help of numerical simulations, we find that, at the intermediate stage of drop impact, the inner droplet exhibits a self-similar deformation at $\lambda =1$ and relatively high Weber number, and experiences more or less a uniform acceleration for various $\lambda$. In particular, the acceleration magnitude at $\lambda \ne 1$ can be correlated with the acceleration at $\lambda =1$ and the Atwood number. When the inner droplet is denser than the outer one, a lamella occurs at the spreading front of the inner droplet. We present a scaling analysis of the thickness of the lamella, and the resultant theoretical prediction is in good agreement with numerical results. At the maximal spreading of the compound drop, a bulging structure is formed around the symmetry axis due to the presence of the inner droplet, thereby effectively reducing the liquid supply to the spreading front and leading to a decrease of maximal spreading ratio $\beta _{max}$ as compared with a pure drop. We proposed a corrected Weber number $We^*_\lambda$ by taking account of the combined effects of $\lambda$, volume fraction of the inner droplet, Weber number and morphology of the compound drop. Integrating $We^*_\lambda$ with the universal model of $\beta _{max}$ for impacting pure drops, we successfully build up a new model for predicting the maximal spreading ratio of impacting compound drops with various $\lambda$.

Natural oscillations of sessile drops with a free or pinned contact line in different gravity environments are studied based on a linear inviscid irrotational theory. The inviscid Navier–Stokes equations and boundary conditions are reduced to a functional eigenvalue problem by the normal-mode decomposition. We develop a boundary element method model to numerically solve the eigenvalue problem for predicting the natural frequencies. Emphasis is placed on the frequency shifts of modes due to gravity for a wide range of contact angles $\alpha$ and Bond numbers $Bo$. Three types of $\alpha$–$Bo$ diagrams reflecting how gravity shifts the frequency are identified. Specifically, the frequency of zonal modes shifts downwards (upwards) when $\alpha$ is smaller (larger) than a critical value, while the frequencies of most sectoral modes are shifted downwards regardless of $\alpha$. As a result, gravity can transform the lowest mode from a zonal mode to a sectoral mode. The spectral degeneracy of hemispherical drops inherited from the Rayleigh–Lamb spectrum is also broken by gravity. However, we discover that gravity has no effect on the mode associated with the horizontal motion of the centre of mass, whose frequency is always zero regardless of $\alpha$ and $Bo$. This implies that the ‘walking’ drop instability reported in previous literature does not exist.

The geometry of compound sessile drops at equilibrium on a flat substrate can exhibit a variety of complicated morphological configurations. In this paper, we first investigate the configuration boundaries of the compound sessile drops in a wide parameter space, where a specific configuration is not stable outside its boundaries. Then, we focus on the transitions among the axisymmetric configurations, i.e. encapsulation, lens and collars. The configuration transitions result from the variation of the wettability of the substrate and the volume ratio of the two component droplets. With the help of theoretical analysis and numerical simulations, we obtain previously unidentified criteria for the onset of configuration transition, identify the irreversible and reversible configuration transitions, reveal the dynamic behaviours of configuration transitions that are not accessible to theoretical analysis, and provide a further step towards the ultimate purpose of such work, which is the controllable reconfiguration of the compound sessile drops.

We investigate the dynamics of a self-rewetting drop placed on a substrate with a constant temperature gradient via three-dimensional numerical simulations using a conservative level-set approach to track the interface of the drop. The surface tension of a so-called self-rewetting fluid exhibits a parabolic dependence on temperature with a well-defined minimum. Two distinct drop behaviours, namely deformation and elongation, are observed when it is placed at the location of the minimum surface tension. The drop spreads slightly and reaches a pseudo-steady state in the deformation regime, while it continuously spreads until breakup in the elongation regime. Theoretical models based on the forces exerted on the drop have been developed to predict the critical condition at which the drop undergoes the transition between the two regimes, and the predictions are in good agreement with the numerical results. We also investigate the effect of the initial position of the drop with respect to the location of the minimum surface tension on the spreading and migration dynamics. It is found that, at early times, the migration of the drop obeys an exponential function with time, but it diverges at the later stage due to an increase in the drop deformation.

We numerically investigate the mechanism resulting in fate change of a hydrophobic sphere impacting onto a confined pool, that is, at the same impact speed, it does not submerge in a wide pool but does in a narrow pool. We find that the reflection of the impact-induced gravity-capillary waves from the pool boundary is responsible for this phenomenon. In particular, the return of the wave to the symmetry axis may coincide with the rising of the impacting sphere to the water surface, which corresponds to the critical conditions of the fate change. Moreover, for the spheres at the onset of submersion in a wide pool, our analysis suggests that this scenario also accounts for an interesting observation in the numerical simulations. That is, the effective pool size $S_{c}$, beyond which the submersion of impacting spheres is no longer affected by the pool size $S$, is mainly dependent on the sphere diameter, no matter whether the surface waves are the capillary or gravity waves. For $S<S_{c}$, two important pool sizes ($S_{w,1}$ and $S_{w,2}$, and $S_{w,2}\geqslant S_{w,1}$) are identified at the same impact speed, and the sphere submersion takes place at $S_{w,1}\leqslant S\leqslant S_{w,2}$. Based on the flow features identified in simulations, a scaling law is proposed to correlate the Weber number and Bond number with $S_{w}$. The theoretical prediction is shown to agree well with the numerical results.

We report direct numerical simulation results that clearly elucidate the mechanism that leads to curvature dependence of drag enhancement (DE) in viscoelastic turbulent Taylor–Couette flow. Change in the angular momentum transport and its inherent link to transitions in vortical flow structures have been explored to depict the influence of the curvature of the flow geometry on DE. Specifically, it has been demonstrated that a transition in vortical structures with increasing radius ratio leads to weakening and elimination of the small-scale Görtler vortices and development and better organization (occupying the entire gap) of large-scale Taylor vortices as also evinced by the patterns of angular momentum current. The commensurate change in DE and its underlying mechanism are examined by contributions of convective flux and polymeric stress to the angular momentum current. The present finding paves the way for capturing highly localized elastic turbulence structures in direct numerical simulation by increasing geometry curvature in traditional turbulent curvilinear flows.

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.

We numerically investigate the mechanism leading to the entrapment of spheres at the gas–liquid interface after impact. Upon impact onto a liquid pool, a hydrophobic sphere is seen to follow one of the three regimes identified in the experiment (Lee & Kim, Langmuir, vol. 24, 2008, pp. 142–145): sinking, bouncing or being entrapped at the interface. It is important to understand the role of wettability in this process of flow–structure interaction with dynamic wetting, and in particular, to what extent the wettability can determine whether the sphere is entrapped at the interface. For this purpose, a diffuse-interface immersed boundary method is adopted in the numerical simulations. We expand the parameter space considered previously, provide the phase diagrams and identify the key phenomena in the impact dynamics. Then, we propose the scaling models to interpret the critical conditions for the occurrence of sphere entrapment, accounting for the wettability of the sphere. The models are shown to provide a good correlation among the impact inertia of the drop, the surface tension, the wettability and the density ratio of the sphere to the liquid.

In this paper, we investigate the ratchet mechanism of drops climbing a vibrated oblique plate based on three-dimensional direct numerical simulations, which for the first time reproduce the existing experiment (Brunet et al., Phys. Rev. Lett., vol. 99, 2007, 144501). With the help of numerical simulations, we identify an interesting and important wetting behaviour of the climbing drop; that is, the breaking of symmetry due to the inclination of the plate with respect to the acceleration leads to a hysteresis of the wetted area in one period of harmonic vibration. In particular, the average wetted area in the downhill stage is larger than that in the uphill stage, which is found to be responsible for the uphill net motion of the drop. A new hydrodynamic model is proposed to interpret the ratchet mechanism, taking account of the effects of the acceleration and contact angle hysteresis. The predictions of the theoretical analysis are in good agreement with the numerical results.

In this article a comparison study of the numerical methods for compressible two-phase flows is presented. Although many numerical methods have been developed in recent years to deal with the jump conditions at the fluid-fluid interfaces in compressible multiphase flows, there is a lack of a detailed comparison of these methods. With this regard, the transport five equation model, the modified ghost fluid method and the cut-cell method are investigated here as the typical methods in this field. A variety of numerical experiments are conducted to examine their performance in simulating inviscid compressible two-phase flows. Numerical experiments include Richtmyer-Meshkov instability, interaction between a shock and a rectangle SF6 bubble, Rayleigh collapse of a cylindrical gas bubble in water and shock-induced bubble collapse, involving fluids with small or large density difference. Based on the numerical results, the performance of the method is assessed by the convergence order of the method with respect to interface position, mass conservation, interface resolution and computational efficiency.

In this paper, drop impact onto a sphere is numerically investigated at moderate Reynolds and Weber numbers. It is naturally expected that the aspect ratio of the sphere to the drop, $\unicode[STIX]{x1D706}_{r}$, would make a big difference to drop spreading and retraction on the sphere, compared with drop impact onto a flat substrate. To quantitatively assess the effect of $\unicode[STIX]{x1D706}_{r}$, a diffuse-interface immersed-boundary method is adopted after being validated against experiments. With the help of numerical simulations, we identify the key regimes in the spreading and retraction, analyse the results by scaling laws, and quantitatively evaluate the effect of $\unicode[STIX]{x1D706}_{r}$ on the impact dynamics. We find that the thickness of the liquid film spreading on the sphere can be well approximated by $h_{L,\infty }(1+3/4\unicode[STIX]{x1D706}_{r}^{-3/2})$, where $h_{L,\infty }$ represents the film thickness of drop impact on a flat substrate. At the early stage of spreading, the temporal variation of the wetted area is independent of $\unicode[STIX]{x1D706}_{r}$ when the time is rescaled by the thickness of the liquid film. Drops are observed to retract on the sphere at a roughly constant speed, and the predictions of theoretical analysis are in good agreement with numerical results.

A 100-J-level Nd:glass laser system in nanosecond-scale pulse width has been constructed to perform as a standard source of high-fluence-laser science experiments. The laser system, operating with typical pulse durations of 3–5 ns and beam diameter 60 mm, employs a sequence of successive rod amplifiers to achieve 100-J-level energy at 1053 nm at 3 ns. The frequency conversion can provide energy of 50-J level at 351 nm. In addition to the high stability of the energy output, the most valuable of the laser system is the high spatiotemporal beam quality of the output, which contains the uniform square pulse waveform, the uniform flat-top spatial fluence distribution and the uniform flat-top wavefront.

We investigate the entrainment of liquid films on a partially wetting plate vertically withdrawn from a reservoir of viscous liquid using a combination of diffuse-interface numerical simulation and lubrication analysis. So far available theoretical investigations were commonly conducted by focusing on separate parameter regions, and a complete description of the flow regimes with increasing plate speed is still missing. By solving the full Stokes equations, we present a complete scenario of film transition in the presence of moving contact line. With increasing plate speed, we identify numerically four successive flow regimes in terms of the interfacial morphologies: (1) a stationary meniscus, (2) a speed-independent thick film connected to the liquid bath through a stationary dimple, (3) coexistence of a thick film and the classical Landau–Levich–Derjaguin (LLD) film connected by a propagating capillary shock and (4) a film with a monotonically varying thickness. The characteristics of the film profiles in different regions of the interfaces are analysed with lubrication theory as applicable, and satisfactory agreements with the numerical results are obtained. In particular, we confirm that the onset of film deposition occurs at a vanishing apparent contact angle, consistent with the predictions of lubrication theory. Numerical results suggest that the critical capillary number for the onset of film deposition is smaller than that for the onset of LLD film despite the fact that it is higher than the experimentally observed one, showing that the thick film can be realized in the two-dimensional model. We also demonstrated that the LLD film is triggered by the bifurcation of the stationary dimple, which is found to admit multiple branches of stable and unstable solutions.

We investigate here what happens beyond the onset of motion of a droplet on a wall by the action of an imposed shear flow, accounting for inertial effects and contact-angle hysteresis. A diffuse-interface method is used for this purpose, which alleviates the shear stress singularity at a moving contact line, resulting in an effective slip length. Various flow regimes are investigated, including steadily moving drops, and partial or entire droplet entrainment. In the regime of quasi-steadily moving drops, the drop speed is found to be linear in the imposed shear rate, but to exhibit an apparent discontinuity at the onset of motion. The results also include the relation between a local maximum angle between the interface and the wall and the instantaneous value of the contact-line speed. The critical conditions for the onset of entrainment are determined for pinned as well as for moving drops. The corresponding critical capillary numbers are found to be in a rather narrow range, even for quite substantial values of a Reynolds number. The approach to breakup is then investigated in detail, including the growth of a ligament on a drop, and the reduction of the radius of a pinching neck. A model based on an energy argument is proposed to explain the results for the rate of elongation of ligaments. The paper concludes with an investigation of detachment of a hydrophobic droplet from the solid wall.

We investigate the critical conditions for the onset of motion of a three-dimensional droplet on a wall in shear flows at moderate Reynolds number. A diffuse-interface method is used for this purpose, which also circumvents the stress singularity at the moving contact line, and the method allows for a density and viscosity contrast between the fluids. Contact-angle hysteresis is represented by the prescription of a receding contact angle θR and an advancing contact angle value θA. Critical conditions are determined by tracking the motion and deformation of a droplet (initially a spherical cap with a uniform contact angle θ0). At sufficiently low values of a Weber number, We (based on the applied shear rate and the drop volume), the drop deforms and translates for some time, but subsequently reaches a stationary position and attains a steady-state shape. At sufficiently large values of We no such steady state is found. We present results for the critical value of We as a function of Reynolds number Re for cases with the initial value of the contact angle θ0=θR as well as for θ0=θA. A scaling argument based on a force balance on the drop is shown to represent the results very accurately. Results are also presented for the static shape, transient motion and flow structure at criticality. It is shown that at low Re our results agree (with some qualifications) with those of Dimitrakopoulos & Higdon (1998, J. Fluid Mech. vol. 377, p. 189). Overall, the results indicate that the critical value of We is affected significantly by inertial effects at moderate Reynolds numbers, whereas the steady shape of droplets still shows some resemblance to that obtained previously for creeping flow conditions. The paper concludes with an investigation into the complex structure of a steady wake behind the droplet and the occurrence of a stagnation point at the upstream side of the droplet.

Axisymmetric droplet spreading is investigated numerically at relatively large rates of spreading, such that inertial effects become important. Results from two numerical methods that use different means to alleviate the stress singularity at moving contact lines (a diffuse interface, and a slip-length-based level-set method) are shown to agree well. An initial inertial regime is observed to yield to a regime associated with Tanner's law at later times. The spreading rate oscillates during the changeover between these regimes. This becomes more significant for a fixed (effective) slip length when decreasing the value of an Ohnesorge number. The initial, inertia-dominated regime is characterized by a rapidly extending region affected by the spreading, giving the appearance of a capillary wave travelling from the contact line. The oscillatory behaviour is associated with the rapid collapse that follows the point at which this region extends to the entire droplet. Results are presented for the apparent contact angle as a function of dimensionless spreading rate for various values of Ohnesorge number, slip length and initial conditions. The results indicate that there is no such universal relation when inertial effects are important.