Particle-laden horizontal turbulent pipe flow is studied experimentally in the two-way coupling regime with a focus on delineating the effects of particle-to-fluid density ratio $\rho _{p}/\rho _{f}=1$ and 1.05 on the fluid and particle statistics. Particle volume fraction $\phi _{v}$ up to $1\,\%$ and viscous Stokes numbers ranging from $St^+ \approx 1.2$ to $St^+ \approx 3.8$ are investigated at friction Reynolds number $Re_\tau \approx 195$ using time-resolved two-dimensional particle image and tracking velocimetry. Substantial differences are observed between the statistics of neutrally buoyant (i.e. $\rho _{p}/\rho _{f}=1$) and denser (i.e. $\rho _{p}/\rho _{f}=1.05$) settling particles (with settling velocities 0.12–0.32 times the friction velocity), which, at most instances, show opposing trends compared to unladen pipe flow statistics. Neutrally buoyant particles show a slightly increased overall drag and suppressed turbulent stresses, but elevated particle–fluid interaction drag and results in elongated turbulent structures compared to the unladen flow, whereas $\rho _{p}/\rho _{f}=1.05$ particles exhibit a slight overall drag reduction even with increased radial turbulent stresses, and shorter streamwise structures compared to the unladen flow. These differences are enhanced with increasing $St^+$ and $\phi _v$, and can be attributed to the small but non-negligible settling velocity of denser particles, which also leads to differing statistics in the upper and lower pipe halves.

We perform simulations of a two-fluid–structure interaction problem involving liquid–gas flow past a fully submerged stationary circular cylinder. Interactions between the liquid–gas interface with finite surface tension and flow disturbances arising from the cylinder induce a variety of interfacial phenomena and wake structures. We map different interface regimes in a parameter space defined by the Bond number $Bo \in [100, 5000]$ and the submergence depth $h/D \in [1, 2.5]$ of the cylinder while keeping the Reynolds (Re) and Weber (We) numbers fixed at 150 and 1000, respectively. The emerging interface features are classified into three distinct regimes: interfacial waves generated by Strouhal vortices, the entrainment of multi-scale gas bubbles and the reduced deformation state. In the interfacial wave regime, we demonstrate that the frequency of transverse interface fluctuations at a specific streamwise location is identical to the vortex shedding frequency. Additionally, the wavelength of interfacial waves is determined by the size of vortex pairs consisting of alternating Strouhal vortices. In the gas entrainment regime at $ Bo = 1000$, our bubble-size distributions reveal that the entrained bubbles have sizes ranging from one to two orders of magnitude smaller than the cylinder. These multi-scale bubbles are formed primarily through plunging and surfing breakers at $h/D = 2.5$. In contrast, at $h/D = 1$, smaller bubbles initially emerge from the breakup of a gas finger. Over time, some of these bubbles grow in size through coalescence cascades. The influence of $ Re \in [50, 150]$ and $ We \in [700, 1100]$ on gas entrainment is quantified in terms of mean bubble size and count. Lastly, we demonstrate how the deformability of the liquid–gas interface drives the hydrodynamic lift force acting on the cylinder. The net downward lift materializes only in the gas entrainment and reduced deformation regimes due to the broken symmetry of the front stagnation point. While our study focuses on two-dimensional simulations, we also provide insights into the three-dimensional gas entrainment mechanism for one of the extreme cases at $h/D = 1$.

When an oblate droplet translates through a viscous fluid under linear shear, it experiences a lateral lift force whose direction and magnitude are influenced by the Reynolds number, the droplet’s viscosity and its aspect ratio. Using a recently developed sharp interface method, we perform three-dimensional direct numerical simulations to explore the evolution of lift forces on oblate droplets across a broad range of these parameters. Our findings reveal that in the low-but-finite Reynolds number regime, the Saffman mechanism consistently governs the lift force. The lift increases with the droplet’s viscosity, aligning with the analytical solution derived by Legendre & Magnaudet (Phys. Fluids, vol. 9, 1997, p. 3572), and also rises with the droplet’s aspect ratio. We propose a semi-analytical correlation to predict this lift force. In the moderate- to high-Reynolds-number regime, distinct behaviours emerge: the $L\hbox{-}$ and $S\hbox{-}$mechanisms, arising from the vorticity contained in the upstream shear flow and the vorticity produced at the droplet surface, dominate for weakly and highly viscous droplets, respectively. Both mechanisms generate counter-rotating streamwise vortices of opposite signs, leading to observed lift reversals with increasing droplet viscosity. Detailed force decomposition based on vorticity moments indicates that in the $L\hbox{-}$mechanism-dominated regime for weakly to moderately viscous droplets, the streamwise vorticity-induced lift approximates the total lift. Conversely, in the $S\hbox{-}$mechanism-dominated regime, for moderately to highly viscous droplets, the streamwise vorticity-induced lift constitutes only a portion of the total lift, with the asymmetric advection of azimuthal vorticity at the droplet interface contributing additional positive lift to counterbalance the $S\hbox{-}$mechanism’s effects. These insights bridge the understanding between inviscid bubbles and rigid particles, enhancing our comprehension of the lift force experienced by droplets in different flow regimes.

We investigate the fluid–solid interaction of suspensions of Kolmogorov-size spherical particles moving in homogeneous isotropic turbulence at a microscale Reynolds number of $Re_\lambda \approx 140$. Two volume fractions are considered, $10^{-5}$ and $10^{-3}$, and the solid-to-fluid density ratio is set to $5$ and $100$. We present a comparison between interface-resolved (PR-DNS) and one-way-coupled point-particle (PP-DNS) direct numerical simulations. We find that the modulated energy spectrum shows the classical $-5/3$ Kolmogorov scaling in the inertial range of scales and a $-4$ scaling at smaller scales, with the latter resulting from a balance between the energy injected by the particles and the viscous dissipation, in an otherwise smooth flow. An analysis of the small-scale flow topology shows that the particles mainly favour events with axial strain and vortex compression. The dynamics of the particles and their collective motion studied for PR-DNS are used to assess the validity of the PP-DNS. We find that the PP-DNS predicts fairly well both the Lagrangian and Eulerian statistics of the particle motion for the low-density case, while some discrepancies are observed for the high-density case. Also, the PP-DNS is found to underpredict the level of clustering of the suspension compared with the PR-DNS, with a larger difference for the high-density case.

This study investigates the spatial distribution of inertial particles in turbulent Taylor–Couette flow. Direct numerical simulations are performed using a one-way coupled Eulerian–Lagrangian approach, with a fixed inner-wall Reynolds number of 2500 for the carrier flow, while the particle Stokes number ($St$) varies from 0.034 to 1 for the dispersed phase. We first examine the issue of preferential concentration of particles near the outer-wall region. Employing two-dimensional Voronoï analysis, we observe a pronounced particle clustering with increasing $St$, particularly evident in regions of low fluid velocity. Additionally, we investigate the concentration balance equation, inspired by the work of Johnson et al. (J. Fluid Mech., vol. 883, 2020, A27), to examine the particle radial distribution. We discern the predominant sources of influence, namely biased sampling, turbophoresis and centrifugal effects. Across all cases, centrifugal force emerges as the primary driver, causing particle migration toward the outer wall. Biased sampling predominantly affects smaller inertial particles, driving them toward the inner wall due to sampling within Taylor rolls with inward radial velocity. Conversely, turbophoresis primarily impacts larger inertial particles, inducing migration towards both walls where turbulent intensity is weaker compared with the bulk. With the revealed physics, our work provides a basis for predicting and controlling particle movement and distribution in industrial applications.

Understanding the mechanism of hydrodynamic cloud cavitation is crucial to reducing noise, vibration and wear. Recent studies have clarified the physics of two distinct formation mechanisms of cloud cavitation. Ganesh et al. (J. Fluid Mech., vol. 802, 2016, pp. 37–78) identified the propagation of bubbly shockwaves as a cloud detachment mechanism. Pelz et al. (J. Fluid Mech., vol. 817, 2017, pp. 439–454) explained the influence of Reynolds number and cavitation number on asymptotic growth of the cavity sheet and its periodic shedding caused by re-entrant flow. In this paper the two mechanisms are set in relation to each other. For this, we show firstly that the transition from re-entrant flow to shockwave-driven cloud cavitation is given by a kinematic condition, namely the asymptotic sheet length equates to the chord length, $\hat {a}=L$. For $\hat {a}>L$ shockwave-driven cloud cavitation dominates. For $\hat {a}< L$ re-entrant flow-driven cloud cavitation dominates. As the cavitation number decreases, the closure region of the cavity sheet reaches the trailing edge of the hydrofoil, identifying the trailing edge as a trigger for condensation shockwaves, particularly as re-entrant flow-driven cavitation diminishes. Since the sheet length is an implicit function of the cavitation number, the kinematic condition $\hat {a}/L=1$ results in a critical cavitation number ${\sigma _\mathrm {II,III}}$ that is calculated analytically and validated by experiments. Secondly, we derive the relationship between the Strouhal number and the asymptotic sheet length for re-entrant flow-driven cloud cavitation. The model presented here is thoroughly validated by experiments.

The added mass force resulting from the acceleration of a body in a fluid is of fundamental and practical interest in dispersed multiphase flows. Euler–Lagrange (EL) and Euler–Euler (EE) simulations require closure terms for the added mass force in order to accurately couple the conserved variables between phases. Presently, a more thorough understanding of the added mass force in a multi-particle system is developed based on potential flow resulting in a resistance matrix formulation analogous to Stokesian dynamics. This formulation is then used to generate a dataset of added mass resistance matrices for large systems of randomly generated particles. This methodology is used to create a volume fraction corrected binary model for predicting the added mass force in large systems as well as generate statistics of the added mass force in such systems. This work provides clarification to the theory of the added mass force for particle clouds, and modelling options that may be implemented in existing EL and EE codes.

When one fluid is injected into a confined geometry such as a porous medium filled with another immiscible fluid, even at an extremely low injection speed, rapid filling of several pore spaces accompanied by retraction of multiple fluid–fluid interfaces can be observed. Such processes with fast liquid redistribution within the solid structure, called Haines jumps, are ubiquitous in many multiphase flow systems, which can impact fluid trapping, energy dissipation and hysteretic saturation in various engineering applications. Inspired by this mechanism, here, we propose a dual-channel structure to realise controlled Haines jumps during fluid displacement processes. Via theoretical analysis and numerical simulations, we show that the dynamics of fluid interfaces during Haines jumps can be quantitatively correlated with the driving capillary pressure and dissipating viscous stress, which enables simultaneous determination of the fluid viscosity and interfacial tension in the dual-channel multiphase system.

The presence of dispersed-phase droplets can result in a notable increase in a system's drag. However, our understanding of the mechanism underlying this phenomenon remains limited. In this study, we use three-dimensional direct numerical simulations with a modified multi-marker volume-of-fluid method to investigate liquid–liquid two-phase turbulence in a Taylor–Couette geometry. The dispersed phase has the same density and viscosity as the continuous phase. The Reynolds number $Re\equiv r_i\omega _i d/\nu$ is fixed at 5200, the volume fraction of the dispersed phase is up to $40\,\%$, and the Weber number $We\equiv \rho u^2_\tau d/\sigma$ is approximately 8. It is found that the increase in the system's drag originates from the contribution of interfacial tension. Specifically, droplets experience significant deformation and stretching in the streamwise direction due to shear near the inner cylinder. Consequently, the rear end of the droplets lags behind the fore head. This causes opposing interfacial tension effects on the fore head and rear end of the droplets. For the fore head of the droplets, the effect of interfacial tension appears to act against the flow direction. For the rear end, the effect appears to act in the flow direction. The increase in the system's drag is attributed primarily to the effect of interfacial tension on the fore head of the droplets which leads to the hindering effect of the droplets on the surrounding continuous phase. This hindering effect disrupts the formation of high-speed streaks, favouring the formation of low-speed ones, which are generally associated with higher viscous stress and drag of the system. This study provides new insights into the mechanism of drag enhancement reported in our previous experiments.

In this paper, aerobreakup in the stagnation region of high-Mach-number flow over a bluff body is studied with experiments and computations. Water drops of diameter 0.51–2.30 mm were acoustically levitated at sea level along the flight path of a rectangular $100\ {\rm mm} \times 150\ {\rm mm}$ rail-gun launched projectile. This enabled the study of aerobreakup at high Mach (3.03–5.12), post-shock Mach (1.5–1.9), Weber $(5 \times 10^4\unicode{x2013}4 \times 10^5)$ and Reynolds $(6 \times 10^4\unicode{x2013}3 \times 10^5)$ numbers. High-speed backlit shadowgraphy is used to record the flow structure. Computations are made for two cases, and it was found that the drop behaviour is not dominated by viscous or surface-tension effects and can be adequately captured by treating the gas as calorically perfect with the ratio of specific heats set to 1.3 to account for thermochemical effects. To assess drop surface stability at early breakup times, results from Newton's inclination method are used to determine the flow along the drop surface and input to a linear-stability analysis; from this, it was found that viscosity and surface tension can be neglected. Moreover, the acceleration term dominates the shear term at the stagnation point, a point accentuated as a drop flattens; this relation inverts closer to the drop equator. Linear-stability analysis was insufficient for modelling late-stage aerobreakup because the predicted wavelengths were too small and the expected aerobreakup times were non-physically short. To address this discrepancy, a nonlinear instability model with constant-rate growth is used that treats the accelerated drop surface as analogous to bubbles rising through a liquid; agreement with computations is good.

In this study, direct numerical simulation of the particle dispersion and turbulence modulation in a sonic transverse jet injected into a supersonic cross-flow with a Mach number of 2 was carried out with the Eulerian–Lagrangian point-particle method. One single-phase case and two particle-laden cases with different particle diameters were simulated. The jet and particle trajectories, the dispersion characteristics of particles, and the modulation effect of particles on the flow were investigated systematically. It was found that large particles primarily accumulate around shear layer structures situated on the windward side of the jet trajectory. In contrast, small particles exhibit radial transport, accessing both upstream and downstream recirculation zones. Moreover, small particles disperse extensively within the boundary layer and large-scale shear layers, evidently influenced by the streamwise vortices. The particles increase the mean wall-normal velocity near the wall in the wake region of the transverse jet, while reducing the mean streamwise and wall-normal velocities in outer regions. Particles significantly alter the flow velocity adjacent to shock fronts. In particular, the turbulent fluctuations near the windward barrel shock and bow shock are reduced, while those around the leeward barrel shock are increased. An upward displacement of the bow shock in the wall-normal direction is also observed due to particles. In the regions away from the shocks, small particles tend to amplify the Reynolds stress, while large particles attenuate the turbulent kinetic energy.

We study numerically the flow around a spherical droplet set fixed in a linear shear flow with moderate shear rates ($Sr\leq 0.5$, $Sr$ being the ratio between the velocity difference across the drop and the relative velocity) over a wide range of external Reynolds numbers ($0.1<{{Re}}\leq 250$, ${{Re}}$ based on the slip velocity and the viscosity of the external fluid) and drop-to-fluid viscosity ratios ($0.01\leq \mu ^\ast \leq 100$). The flow structure, the vorticity field and their intrinsic connection with the lift force are analysed. Specifically, the results on lift force are compared with the low-${{Re}}$ solution derived for droplets of arbitrary $\mu ^\ast$, as well as prior data at finite ${{Re}}$ available in both the clean-bubble limit ($\mu ^\ast \to 0$) and the solid-sphere limit ($\mu ^\ast \to \infty$). Notably, at ${{Re}}=O(100)$, the lift force exhibits a non-monotonic transition from $\mu ^\ast \to 0$ to $\mu ^\ast \to \infty$, peaking at $\mu ^\ast \approx 1$. This behaviour is related to an internal three-dimensional flow bifurcation also occurring under uniform-flow conditions, which makes the flow to evolve from axisymmetric to biplanar symmetric. This flow bifurcation occurs at low-but-finite $\mu ^\ast$ when the internal Reynolds number (${{Re}}^i$, based on the viscosity of the internal fluid) exceeds approximately 300. In the presence of shear, the corresponding imperfect bifurcation enhances the extensional rate of the flow in the wake. Consequently, the streamwise vortices generated behind the droplet can be more intense compared with those behind a clean bubble. Given the close relation between the lift and these vortices, a droplet with ${{Re}}=O(100)$ and $\mu ^\ast \approx 1$ typically experiences a greater lift force than that in the inviscid limit.

Depth-limited overturning wave shape affects water turbulence and sediment suspension. Experiments have shown that wind affects shoaling and overturning wave shape, with uncertain mechanism. Here, we study wind effects (given by the wind Reynolds number) on solitary wave shoaling and overturning with the two-phase direct numerical simulations model Basilisk run in two dimensions on steep bathymetry for fixed wave Reynolds number and Bond number. For all wind, the propagating solitary wave sheds a two-dimensional turbulent air wake and has nearly uniform speed with minimal wave energy changes over the rapidly varying bathymetry. Wave-face slope is influenced by wind, and shoaling wave shape changes are consistent with previous studies. As overturning jet impacts, wind-dependent differences in overturn shape are quantified. The non-dimensional breakpoint location and overturn area have similar wind dependence as previous experience, whereas the overturn aspect ratio has opposite wind dependence. During shoaling, the surface viscous stresses are negligible relative to pressure. Surface tension effects are also small but grow rapidly near overturning. In a wave frame of reference, surface pressure is low in the lee and contributes 2–5 % to the velocity potential rate of change in the surface dynamic boundary condition, which, integrated over time, changes the wave shape. Reasons why the overturn aspect ratio is different than in experiment and why a stronger simulated wind is required are explored. The dramatic wind effects on overturning jet area, and thus to the available overturn potential energy, make concrete the implications of wind-induced changes to wave shape.

We investigate the coupling effects of the two-phase interface, viscosity ratio and density ratio of the dispersed phase to the continuous phase on the flow statistics in two-phase Taylor–Couette turbulence at a system Reynolds number of $6\times 10^3$ and a system Weber number of 10 using interface-resolved three-dimensional direct numerical simulations with the volume-of-fluid method. Our study focuses on four different scenarios: neutral droplets, low-viscosity droplets, light droplets and low-viscosity light droplets. We find that neutral droplets and low-viscosity droplets primarily contribute to drag enhancement through the two-phase interface, whereas light droplets reduce the system's drag by explicitly reducing Reynolds stress due to the density dependence of Reynolds stress. In addition, low-viscosity light droplets contribute to greater drag reduction by further reducing momentum transport near the inner cylinder and implicitly reducing Reynolds stress. While interfacial tension enhances turbulent kinetic energy (TKE) transport, drag enhancement is not strongly correlated with TKE transport for both neutral droplets and low-viscosity droplets. Light droplets primarily reduce the production term by diminishing Reynolds stress, whereas the density contrast between the phases boosts TKE transport near the inner wall. Therefore, the reduction in the dissipation rate is predominantly attributed to decreased turbulence production, causing drag reduction. For low-viscosity light droplets, the production term diminishes further, primarily due to their greater reduction in Reynolds stress, while reduced viscosity weakens the density difference's contribution to TKE transport near the inner cylinder, resulting in a more pronounced reduction in the dissipation rate and consequently stronger drag reduction. Our findings provide new insights into the physics of turbulence modulation by the dispersed phase in two-phase turbulence systems.

Linear instability analysis of a viscous swirling liquid jet surrounded by ambient gas is carried out by considering the significant influence of axial shear effect. The jet azimuthal flow is assumed as a Rankine vortex, and the non-uniform velocity distribution in the jet axial direction is approximated by parabolic and error functions. The enhancement of jet rotation is found to promote the predominant mode with larger azimuthal wavenumbers, and the mode transition is decided by the competition between centrifugal force and axial shear stress. Subsequently, the influence of the axial shear effect is examined through changing the degree of shear stress and the thickness of the gas velocity boundary layer. It is found that an increase of jet average velocity or surface velocity in the axial direction leads to the predominant mode transition to smaller azimuthal wavenumbers, due to the combined effects of shear stress and gas pressure perturbation. A larger velocity difference between ambient gas and liquid jet also promotes the predominant modes with smaller azimuthal wavenumbers, and the physical mechanism is attributed to gas pressure perturbation. Phase diagrams of different azimuthal modes are given and compared with the study of Kubitschek & Weidman (J. Fluid Mech., vol. 572, 2007, pp. 261–286), where a static swirling column without axial shear stress was considered. The strengthened axial shear effect is found to delay the transition of predominant modes with the increase of angular velocity. Experimental studies considering the swirling jets with different axial velocities are further carried out, which validate the theoretical findings. Different instability mechanisms and their transition rules are also identified through energy budget analysis. This study is expected to give scientific guidance on understanding the instability mechanisms of the swirling jets that widely exist in natural phenomena and engineering applications.

Droplet impingement on a heated substrate is the fundamental process underlying various technologies, ranging from spray cooling to inkjet printing. Understanding the coupled effects of fluid dynamics and heat transfer patterns during droplet jumping, boiling and evaporation, which determine the outcomes of the impingement process, is essential. Here, we developed two-colour planar laser-induced fluorescence and micro-particle image velocimetry technologies to measure quantitatively the velocity and temperature distributions inside the droplet during an impingement process with high temporal and spatial resolution. With our novel measuring system, the hot spots at the solid–liquid interface are discovered for the first time. The influence of contact boiling on the droplet internal mixing, which impedes droplet recoiling and reduces the rebounding velocity, is discussed. A significant enhancement in heat absorption for partially rebounding droplets is discovered, where the impingement heat transfer rate is doubled compared to other vapour-layer-covered droplets. The scaling correlations of viscous dissipation rate and contact time of rebounding droplets, as well as the time variation of droplet temperature rise, are proposed. More detailed patterns inside droplets can be captured by these experimental methods, which will help to reveal more intrinsic mechanisms lying in thermally induced flow, complex fluids and droplet-impacting-based technologies.

This study conducts a numerical investigation into the three-dimensional film boiling of liquid under the influence of external magnetic fields. The numerical method incorporates a sharp phase-change model based on the volume-of-fluid approach to track the liquid–vapour interface. Additionally, a consistent and conservative scheme is employed to calculate the induced current densities and electromagnetic forces. We investigate the magnetohydrodynamic effects on film boiling, particularly examining the pattern transition of the vapour bubble and the evolution of heat transfer characteristics, exposed to either a vertical or horizontal magnetic field. In single-mode scenarios, film boiling under a vertical magnetic field displays an isotropic flow structure, forming a columnar vapour jet at higher magnetic field intensities. In contrast, horizontal magnetic fields result in anisotropic flow, creating a two-dimensional vapour sheet as the magnetic strength increases. In multi-mode scenarios, the patterns observed in single-mode film boiling persist, with the interaction of vapour bubbles introducing additional complexity to the magnetohydrodynamic flow. More importantly, our comprehensive analysis reveals how and why distinct boiling effects are generated by various orientations of magnetic fields, which induce directional electromagnetic forces to suppress flow vortices within the cross-sectional plane.

The present study aims to examine the temporal linear stability analysis of isothermal plane Couette flow over a porous layer using the two-domain approach. The flow in the porous layer is described by the unsteady Darcy–Brinkman equations, whereas it is characterised by the Navier–Stokes equations in the fluid layer. In contrast to the Darcy model, it is observed that the isothermal plane Couette flow becomes unstable for such a superposed system on the inclusion of the Brinkman term. From the stability analysis, the two-dimensional mode is found to be least stable, and two modes of instability, namely porous mode and mixed mode are obtained under the consideration of the Darcy–Brinkman model along with advection term (DBA model). For Darcy number $(\delta )=0.01$, depending on the value of the stress-jump coefficient, mixed mode controls the instability of the system at small values of depth ratio $(\hat {d})$, and it disappears for relatively high values of $\hat {d}$, where the porous mode dominates. In addition, it has been observed that when $\hat {d}=0.1$, the critical mode of instability is found to be mixed for $\delta >0.02$ and porous for $\delta \le 0.02$. The stress-jump coefficient destabilises the flow in terms of energy production through perturbed stresses at the interface. As observed in the case of isothermal plane Poiseuille flow studied by Chang, Chen & Straughan (J. Fluid Mech., vol. 564, 2006, pp. 287–303), here also depth ratio (Darcy number) stabilises (destabilises) the flow. However, this characteristic does not remain valid when the advection term is eliminated from the considered momentum equation. For a certain range of $\hat {d} (\delta )$, the destabilising (stabilising) characteristic of the respective parameters are encountered when the fluid mode of instability prevails.

We study the stability and dewetting dynamics of a thin free-surface film composed of two miscible liquids placed on a solid substrate. Our study focuses on the development of a self-consistent model such that the mixture concentration influences both free-surface and wetting energies. By assuming a simple relation between these energies and the bulk and surface concentrations, we analyse their effect on the concentration distribution and dewetting down to the equilibrium film thickness determined by the fluid–solid interaction potential. The model, developed within the gradient dynamics formulation, includes the dependence of the free-surface energy on surface concentration leading to the Marangoni effect, while a composition-dependent Hamaker constant describes the wetting energy resulting from the fluid–solid interaction. We analyse the restrictions that must be fulfilled to ensure an equilibrium state for a flat film of a binary fluid. Then, we proceed by studying its linear stability. First, we consider the Marangoni effect while assuming that wetting energy depends only on the fluid thickness. Then, we include a dependence of wetting energy on concentration and study its effects. We find that the linear stability results compare very well with those of numerical simulations of the full nonlinear problem applied to the particular case of a binary melted metal alloy, even close to breakup times. Therefore, in practice, most of the evolution can be studied by using the linear theory, simplifying the problem considerably.

This study investigates the interaction between a freely rising, deformable bubble and a freely settling particle of the same size due to gravity. Initially, an in-line configuration is considered while varying the Bond, Galilei and Archimedes numbers. The study shows that as the bubble and particle approach each other, a liquid film forms between them that undergoes drainage. The formation of the liquid film leads to dissipation of kinetic energy, and for sufficiently large bubble velocities, particle flotation takes place. Increasing the Bond number causes the bubble to deform more severely, which may allow the particle to pass through the bubble as it ruptures. This work also considers an offset configuration, which shows that the bubble slides away from the particle, affecting its settling trajectory.