We study the dynamics of inertial particles in turbulence using datasets obtained from both direct numerical simulations and laboratory experiments of turbulent swirling flows. By analysing time series of particle velocity increments at different scales, we show that their evolution is consistent with a Markov process across the inertial range. This Markovian character enables a coarse-grained description of particle dynamics through a Fokker–Planck equation, from which we can extract drift and diffusion coefficients directly from the data. The inferred coefficients reveal scale-dependent relaxation and noise amplitudes, indicative of inertial filtering and intermittency effects. Beyond the kinematic description, we analyse the thermodynamic properties of particle trajectories by computing the trajectory-dependent entropy production. We show that the statistics of entropy fluctuations satisfy both the integral fluctuation theorem and, under certain conditions, the detailed fluctuation theorem. These results establish a quantitative bridge between stochastic thermodynamics and particle-laden flows, and open the door to modelling turbulent transport using effective stochastic theories constrained by data and physical consistency.

The accuracy obtained with computational fluid dynamics and process simulations of flotation critically depends on the quality and robustness of the underlying models for the non-resolved subprocesses. An important issue in flotation is the collision between particles and air bubbles. Many models have been developed, but their accuracy for applications in flotation is limited. In particular, the significant size difference between particles and bubbles and their intricate coupling to the turbulent flow field pose severe challenges. The present paper first reviews presently employed collision models, highlighting their advantages and disadvantages when applied to flotation. On this basis, the `integrated multisize collision model’ (IMSC) is proposed. After a detailed evaluation, it combines existing approaches from various sources and introduces new developments designed to address present shortcomings. The model is validated by own direct numerical simulation data as well as data from the literature. It is shown that, overall, the IMSC provides better predictions for the collision rate in typical flotation conditions than presently employed collision models and covers the entire parameter range of the flotation process very well. Using the available data, some of the underlying modelling assumptions are validated. Finally, a comprehensive overview of the model is provided for further use in Euler–Euler frameworks or process simulations also beyond flotation.

The stability analysis of multiphase capillary wavetrains on water of infinite depth is performed using two coupled fourth-order nonlinear evolution (NLE) equations. We have investigated analytically the influence of a second wavetrain travelling in a different direction to the first wavetrain. The propagation of multiphase modes is studied for the case when group velocity projections of two wavetrains overlap. Criteria are derived for capillary Stokes wave instabilities and for the existence of a multiphase solitary envelope solution. We have exhibited that the weakly nonlinear multiphase capillary wavetrains in deep water is unstable to oblique disturbances and presented that the dominant modulational instability is two-dimensional in deep water. It is found that the growth rate of modulational instability increases with the increase of the angle of interaction between two wavetrains. The existing fourth-order analysis provides significant deviations on the stability results when compared with the third-order analysis.

We derive a self-consistent hydrodynamic theory of coupled binary fluid–surfactant systems from the underlying microscopic physics using Rayleigh’s variational principle. At the microscopic level, surfactant molecules are modelled as dumbbells that exert forces and torques on the fluid and interface while undergoing Brownian motion. We obtain the overdamped stochastic dynamics of these particles from a Rayleighian dissipation functional, which we then coarse-grain to derive a set of continuum equations governing the surfactant concentration, orientation, fluid density and velocity. This approach introduces a polarisation field $\boldsymbol{p}(\boldsymbol{r},t)$, representing the average orientation of surfactants, which plays a central role in suppressing droplet coalescence. The remaining hydrodynamic equations are consistently obtained from a mesoscopic free energy functional. The resulting model accurately captures key surfactant phenomena, including surface tension reduction and droplet stabilisation, as confirmed by both perturbation theory and numerical simulations, and is thermodynamically consistent with both the Gibbs adsorption isotherm and Henry’s law for adsorbed surfactant concentration.

This study presents a novel extension of the Onsager variational principle to incorporate inertial and thermal effects in fluid dynamics, thereby establishing a unified variational framework for modelling non-isothermal two-phase flows with liquid–vapour phase transitions and wetting effects on solid substrates. From this framework, we naturally derive a thermodynamically consistent model for the fluid system, comprising two-phase Navier–Stokes equations, an equation for the total energy, and dynamic boundary conditions that account for thermal and wetting effects. The derivation is independent of the equation of state, and generalises the dynamic van der Waals theory. To address the computational complexity of the resulting dynamic system, we propose a lattice Boltzmann method based on double distribution functions, which enables accurate and robust simulations of coupled fluid and thermal transport. Numerical experiments – including droplet evaporation, bubble nucleation and departure, and Leidenfrost droplet impact – demonstrate good agreement with theoretical predictions and experimental data, indicating that the proposed numerical method can effectively capture complex thermohydrodynamic phenomena.

We investigate the three-dimensional melting dynamics of an initially spherical particle translating in a warmer liquid using sharp-interface simulations that fully resolve both solid and fluid phases with the Stefan condition. A wide parameter space is explored, spanning initial Reynolds number ($\textit{Re}_0$), Stefan number ($\textit{St}$) and Richardson number ($\textit{Ri}$). In the absence of buoyancy ($\textit{Ri}= 0$), the interface evolution is governed by canonical wake bifurcations. Four regimes are identified: an axisymmetric regime ($\textit{Re}_0\lt 212$) with a rounded front and planar rear; a steady planar-symmetric regime ($212\lt \textit{Re}_0\lt 273$) with an inclined rear plane; a periodic planar-symmetric regime ($273\lt \textit{Re}_0\lt 355$) where vortex shedding emerges in the wake; and a chaotic regime ($\textit{Re}_0\gt 355$) with fluctuating stagnation points and a more rounded rear. Despite these differences, all regimes exhibit a tendency towards melt-rate homogenisation over time. Besides, we introduce an aspect-ratio-based surface-area formulation that yields a predictive model, accurately capturing volume evolution across regimes. Hydrodynamic loads also reflect the coupling between shape and flow: drag follows rigid-sphere correlations only at moderate $\textit{Re}_0$; planar rears enhance drag at higher $\textit{Re}_0$; lift appears only in symmetry-broken regimes and reverses late in time; torque reorients the rear plane towards vertical, consistent with free-body experiments. When buoyancy is included, assisting configurations ($\textit{Ri}\gt 0$) suppress recirculation and maintain quasi-spherical shapes, whereas opposing or transverse buoyancy ($\textit{Ri}\lt 0$) destabilises wakes and promotes tilted planar rears. These results provide a unified framework for convection-driven melting across laminar, periodic and chaotic wakes, with implications for geophysical and industrial processes.

This article derives analytical expressions fully describing laminar flow through concentric pipe-within-pipe set-ups, focusing on scenarios where one tube is pressure driven, and the other serves as a lubricant. Both fluid zones are axially unbounded, therefore excluding recirculation, and are connected along longitudinal infinite slits situated on the inner pipe wall, representing fluid–fluid interfaces. Crucially, the viscous interaction along these interfaces is captured by means of a local slip length, for which explicit formulae are provided, allowing a straightforward evaluation. With that, these models provide a full description of the velocity field for slippery concentric pipes, taking into account the viscosity ratio of both fluids and the overall geometry, therefore extending beyond the common assumption of perfect slip applied to superhydrophobic surfaces. Thereby, they enable a precise analysis of the flow, offering important tools to decipher the intricate dynamics of the two coupled fluids within such set-ups. As a result, the insights acquired contribute to the design and optimisation of superhydrophobic and liquid-infused surfaces, with implications for numerous engineering applications such as microfluidic contactors or drag reduction. The analytical models are in excellent agreement with numerical simulations, thus confirming the selected approach. Therefore, our study further illustrates an effective methodology to derive additional analytical models through the presented mathematical techniques, which can serve as a useful template for modelling such surfaces.

Direct numerical simulations with two-way coupled Lagrangian tracking are carried out to study the bubble preferential concentration and the flow field modification. Simulations are conducted in an upward vertical turbulent channel driven by a constant pressure gradient, corresponding to a friction Reynolds number $Re_{\tau 0}=180$. Micro-sized bubbles with diameters ranging from 0.72 to 1.43 wall units are considered. Competition between lift force and wall-lift force in the wall-normal direction leads to significant near-wall bubble accumulation and directly results in distinct preferential concentration patterns across the channel. Below (above) the peak concentration height, the wall-lift (lift) force dominates, driving bubbles to accumulate in regions of high-speed sweep (low-speed ejection) events. In the vicinity of the wall, the wall-normal lift force exhibits a strong correlation with the local streamwise flow velocity, further reinforcing the preferential concentration of bubbles in high-speed regions. Additionally, bubbles show a strong preference for the low-enstrophy and high-dissipation nodal topologies. Furthermore, small bubbles primarily accumulate in the vicinity of the wall, reducing the work done on the flow and leading to a decrease in bulk velocity and turbulence statistics. In contrast, the turbulence statistics of large bubbles are nearly identical to those of the unladen flow. The impact of large bubbles on the flow field primarily manifests as an effective increase in the mean pressure gradient. These findings demonstrate that bubbles in the upward vertical channel flow exhibit strong preferential concentration behaviours, whereas their ability to modulate turbulence remains limited.

High-intensity focused ultrasound (HIFU) is a non-invasive alternative to traditional surgery for detection and treatment. When HIFU targets a specific area, ultrasonic cavitation occurs with mechanical stress, causing tissue damage, a process that is significantly influenced by the surroundings. This paper presents a numerical study on the cavitation initiation and evolution mechanisms under focused ultrasonic waves considering the influence of a solid surface. Firstly, the dynamic property of focused ultrasonic waves and the generation of diffraction waves is explained based on the Huygens–Fresnel principle, and the prefocused phenomenon is analysed. Notably, the scenario considering the existence of a solid wall is discussed, with the corresponding cavitation clouds in a ‘tree-like’ pattern that can be generally divided into three or four subregions. The different initiation mechanisms of the near-wall cavitation clouds under a different relative distance between the theoretical focal point and the solid wall are discussed in detail. Finally, by considering the effects of the incident waves, scattered waves and their reflected waves on the solid wall, a wave superposition model is established that can clearly explain the distribution characteristics of the near-wall cavitation clouds with different modes. The understanding of the ultrasonic cavitation mechanism may support precise control in future HIFU applications.

We investigate solute dispersion in a two-phase system comprising a Casson fluid flowing in a tube and its surrounding wall phase that allows interphase solute exchange to mimic solute transport in blood and tissue phases. A pulsatile pressure gradient is imposed, and Gill’s classical methodology is extended to two-phase flows to analyse solute transport. The key parameters are the diffusivity ratio between wall and fluid phases ($\lambda$), the partition coefficient ($\beta _p$), the Womersley number ($\alpha$), the yield stress ($\tau _y$), the wall thickness ($\delta _h$) and the initial dimensionless radius of the solute source ($a$). In the long-time limit, increasing $\lambda$, $\beta _p$ and $\delta _h$ reduces the phase-averaged convection ($K_1$) and dispersion ($K_2$) coefficients, owing to solute accumulation in the wall where convective and shear-induced transport are absent. Short-time behaviour is dictated by the rate of solute transfer to the wall. Larger $\alpha$ enhances both $K_1$ and $K_2$, while larger $\tau _y$ suppresses them. The presence of a wall phase permits $K_2$ to reach $O(10^{0})$, compared with $K_2 \sim O(10^{-3})$ without a wall, and can delay the onset of steady state to dimensionless time $t \sim O(10^{2})$. Strong solute exchange and increasing wall thickness diminish downstream solute penetration, while non-Newtonian effects promote interphase transfer. These results provide mechanistic insight into solute exchange across fluid–wall interfaces, relevant to solute transport in blood flow and engineered permeable systems.

Dynamics of spheroidal particle migration within the elasto-inertial square duct flow of Giesekus viscoelastic fluids were studied by using the direct forcing/fictitious domain method. The results show rich migration behaviours, a spheroidal particle gradually transitions from the corner (CO), channel centreline (CC), inertial rotational (IR), diagonal line and cross-section midline equilibrium positions with a decrease in the elastic number, depending on the initial particle position, initial particle orientation and fluid elasticity. From the effect of secondary flow, the IR equilibrium position is reported when the fluid inertia is relatively strong. Six (five) kinds of rotational behaviours are observed for the elasto-inertial migration of prolate (oblate) spheroids. Moreover, the critical elastic number is determined for the migration of spheroidal particles in Giesekus fluids. Near the critical elastic number, oblate and prolate spheroids can simultaneously maintain the CC, CO and IR equilibrium positions, and the initial orientation of particles affects their final rotational modes and equilibrium positions. Through comprehensive analysis, empirical formulas governing the ability of oblate and prolate spheroids to maintain the CC equilibrium position are proposed as $\textit{Wi} = 0.055\,\textit{Re}{-0.1}$ and Wi = 0.045 Re−0.35 when n = 0.5, 0.01 ≤ Wi ≤ 1. Due to the different directions of the pressure forces acting on the particles and the forces from the first normal stress difference and the second normal stress difference, the equilibrium position in Giesekus fluids is rapidly increased by increasing the secondary flow at higher elastic numbers, which is contrary to the phenomenon observed in the Oldroyd-B fluid.

We developed a two-phase lattice Boltzmann model by coupling the entropic multiple-relaxation-time (EMRT or KBC) collision operator enabling low fluid viscosity, with a source term (Wang et al. 2022, Phys. Rev. E vol. 105, no 4) to independently adjust surface tension. The coupling is implemented via the exact difference method (EDM), which allows full consideration of external-force effects on the entropic stabiliser in KBC, in contrast to the recent work of Wang et al. (2022 Phys. Rev. E vol. 105) and Xu et al. (2024 Comput. Math. Appl. vol. 159, 92–101). More importantly, we address a major drawback of the EDM by explicitly demonstrating how its high-order error terms influence the pressure tensor and surface tension. Using the developed model, we investigated droplet impact and splashing on a thin liquid film at a remarkably high Weber number of ${\textit{We}} = 5000$ and Reynolds number of ${\textit{Re}} = 5000$. Droplet impact and splashing on flat surfaces and mesh structures at very high ${\textit{Re}}$ (15 200) and ${\textit{We}}$ (1020) are also studied after validating four representative cases against experiments. For droplet impact on flat surfaces, hydrophobicity promotes the growth of peripheral instabilities, leading to fingering splashing. Corona splashing transitions to fingering splashing as the liquid–gas viscosity ratio increases. For droplet impact on mesh structures, large openings promote liquid penetration, whereas small openings enhance spreading. As the solid ratio increases, the maximum spreading ratio increases monotonically but nonlinearly, whereas the maximum penetrated liquid pillar length first rises and then drops. These simulations demonstrate the proposed model offers significant advantages for accurately capturing and elucidating complex droplet impact and splashing dynamics at high ${\textit{Re}}$ and ${\textit{We}}$.

The present study experimentally investigates the onset of ventilation of surface-piercing hydrofoils. Under steady-state conditions, the depth-based Froude number $\textit{Fr}$ and the angle of attack $\alpha$ define regions in which distinct flow regimes are either locally or globally stable. To map the boundary between these stability regions, the parameter space $(\alpha , \textit{Fr})$ was systematically surveyed by increasing $\alpha$ until the onset of ventilation while maintaining a constant $\textit{Fr}$. Two simplified model hydrofoils were examined: a semi-ogive with a blunt trailing edge and a modified NACA 0010-34. Tests were conducted in a towing tank under quasi-steady-state conditions for aspect ratios of $1.0$ and $1.5$, and for $\textit{Fr}$ ranging from $0.5$ to $2.5$. Ventilation occurred spontaneously for all test conditions as $\alpha$ increased. Three distinct trigger mechanisms were identified: nose, tail and base ventilation. Nose ventilation is prevalent at $\textit{Fr} \lt 1.0$ and $\textit{Fr} \lt 1.25$ for aspect ratios of $1.0$ and $1.5$, respectively, and is associated with an increase in the inception angle of attack. Tail ventilation becomes prevalent at higher $\textit{Fr}$, and the inception angle of attack exhibits a negative trend. Base ventilation was only observed for the semi-ogive profile, but it did not lead to the development of a stable ventilated cavity. Notably, the measurements indicate that the boundary between bistable and globally stable regions is not uniform and extends to significantly higher $\alpha$ than previously estimated. A revised stability map is proposed to reconcile previously published and current data, demonstrating how two alternative paths to a steady-state condition can lead to different flow regimes.

This work aims to complement the description of the atomisation process in a typical commercial pressure-swirl atomiser. Conventional characterisation focuses on the final spray, where established experimental techniques allow for measuring spherical droplets in a dilute regime. However, the early stages of atomisation involve distorted liquid structures with complex interface morphology that challenge both experimental and numerical approaches. While numerical simulations with interface-capturing methods have provided access to this region, they currently remain computationally prohibitive to follow the atomisation process until the formation of the final spherical droplets. To characterise the evolving interface morphology, we propose analysing the curvature distribution obtained from both simulations and two-photon laser-induced fluorescence (2P-LIF) imaging. This curvature-based methodology, recently developed to characterise numerical sprays (Palanti et al. Intl J. Multiphase Flow 147, 2022, 103879; Ferrando et al. Atomiz. Sprays 33, 2023, 1–28), is here extended to experimental data. Both approaches are compared with available phase Doppler anemometry (PDA) measurements performed further downstream on spherical droplets. The morphological evolution of the atomising spray is interpreted through curvature statistics, which provide a unified framework applicable to all atomisation stages. When applied to spherical droplets, the curvature distribution recovers the conventional drop size distribution, linking early interface deformation to the final spray structure. The birth of this final drop size distribution can thus be observed by comparing the three approaches – numerical simulation limited to the early stage of atomisation, curvature derived from 2P-LIF images limited to two-dimensional (2-D) contour analysis, and PDA measurements of the dilute spray. The results show that curvature properties evolve in a way that can be directly representative of the final spray even at early atomisation stages.

A lattice Boltzmann method is adopted to investigate the breakup of surfactant-free and surfactant-laden droplets in both regular and irregular T-junction microchannels. During droplet neck contraction, the neck thinning shifts from inertia dominated to interfacial tension dominated, causing spontaneous rapid neck collapse due to Rayleigh–Plateau instability. For the regular rectangular microchannels, we find that the prerequisite for the spontaneous breakup of a surfactant-free droplet is that the local capillary pressure in the triggering area exceeds the Laplace pressure difference between the inside and outside of the droplet neck. Results show that the critical neck thickness $\delta _\textit{cr}^{*}$ for the droplet spontaneous breakup increases with increasing height-to-width ratio $\chi$ of the microchannel in both surfactant-free and surfactant-laden systems. The presence of surfactants decreases $\delta _\textit{cr}^{*}$ at the identified $\chi$, while the surfactant effects are gradually enhanced as $\chi$ increases. Subsequently, a constriction section is incorporated into the upper microchannel wall to establish an irregular microchannel. As constriction depth (length) increases, $\delta _\textit{cr}^{*}$ linearly decreases (increases) in the surfactant-free system, while $\delta _\textit{cr}^{*}$ exponentially decreases (linearly increases) in the surfactant-laden system. Four empirical formulas are proposed to predict the values of $\delta _\textit{cr}^{*}$ under varying constriction depths and lengths in the two systems.

We present a study on the melting dynamics of neighbouring ice bodies by means of idealised simulations, focusing on collective effects, with the goal of obtaining fundamental insight into how collective interactions influence the melting of ice. Two neighbouring (vertically or horizontally aligned), square-shaped and equally sized ice objects (size of the order of centimetres) are immersed in quiescent fresh water at a temperature of ${20}\,^\circ \textrm {C}$. By performing two-dimensional direct numerical simulations, and using the phase-field method to model the phase change, the collective melting of these objects is studied. When the objects are horizontally aligned, no significant influence of the neighbouring object on the melting time is observed. On the other hand, when vertically aligned, although the melting of the upper object is mostly unaffected, the melting time and the morphology of the lower ice body strongly depends on the initial inter-object distance. We report that the melting of the bottom object can be enhanced by more than 10 %, or delayed more than 20 %, displaying a non-monotonic dependence on the initial object size. We show that this behaviour results from a non-trivial competition between layering of cold fluid, which lowers the heat transfer, and convective flows, which favour mixing and heat transfer. For this melting in mixed convection, we were able to collapse our data onto a single curve.

The energy of fluid turbulence is transported, on average, to smaller and larger scales in three-dimensional and two-dimensional flows, respectively. The motion along the flat free surface of a turbulent liquid shares similarities with both classes of flows, therefore the direction of the energy cascade along it is ambiguous. We show experimentally that the process is linked to the local divergence of the surface velocity field: expansive motions, associated with flow upwelling towards the surface, transfer energy to larger scales, while compressive motions, associated with fluid plunging into the bulk, do the opposite. The net inter-scale energy flux is therefore vanishingly small, in stark contrast with homogeneous turbulence in both two- and three-dimensional systems. Moreover, we find that rare and intense compressive/expansive events are chiefly responsible for the instantaneous inter-scale fluxes, which are much stronger than their counterparts at depth.

Pulsed gravity currents are generated by the sequential release of dense material into a lighter ambient. We investigate the dynamics of pulsed gravity currents using physical scale experiments, two-dimensional depth-averaged shallow water equation (SWE) based models and three-dimensional lattice Boltzmann method (LBM) simulations. Integrating these results we show for the first time that short duration pulsed releases generate intrusive layers, which accelerate front propagation relative to an instantaneously released current of the same total volume. Conversely, a long delay time between pulses produces a current that propagates slower than an equivalent instantaneous release. This finding is supported by physical experiments and depth-resolving LBM simulations. The depth-resolving simulations show that intrusions in pulsed flows experience less drag resistance than those generated by instantaneous releases. The depth-averaged model considered in the present study does not accurately capture the intrusive flow dynamics of pulsed currents. However, the limitations of the finite-depth SWE model may be mitigated by extensions to incorporate entrainment and density stratification. The results also motivate further research into the impact of buoyancy Reynolds number and channel slope on the propagation of pulsed currents.

Interfacial interactions between gas bubbles and the free surface are a hallmark of flows involving aqueous foams. In practice, bubble foams commonly arise from processes such as breaking waves at the ocean–atmosphere interface, plunging liquid jets and the effervescence of carbonated liquids. Once generated, bubbles within foam layers remain afloat at the free surface for finite durations before finally bursting into a fine spray of droplets. While the birth and bursting of bubble foams have received considerable attention, the understanding of floating bubbles is limited mainly to a single bubble. To build on this, in this article, we undertake numerical simulations of two or more floating bubbles in various canonical settings to examine their geometry and self-organising nature, with implications for real-world phenomena such as ocean spray production. Under lateral confinement, floating bubbles are prone to form vertically stacked layers. To this end, we analyse the geometry of coaxial pairs of floating bubbles and link geometrical differences between single and coaxial bubbles to various aspects of the ensuing bursting stage. Furthermore, we extend the existing theory of isolated floating bubbles to obtain unified analytical expressions for the shape parameters of single and coaxial bubbles of small sizes. Next, we investigate a pair of side-by-side floating bubbles, which serves as a minimal configuration to understand the formation of bubble rafts through self-organisation. We discover that Bond numbers in the range $10\leqslant \textit{Bo}\leqslant 50$ are more favourable for raft formation due to pronounced capillary attraction. The time required for two floating bubbles to assemble through capillary attraction grows exponentially with their initial separation. We also develop a linear model to capture the evolution of bubble spacing during capillary migration at low Bond numbers. Lastly, we extend the two-bubble configuration and showcase the emergent dynamics of a swarm of floating bubbles in mono- and bilayer configurations.

Nucleation phenomena associated with cloud cavitation about a three-dimensional (3-D) NACA$\,$16-029 hydrofoil are explored experimentally in a cavitation tunnel where susceptible free stream nuclei are absent. Microbubble nuclei are found to be intrinsically generated by cavity collapse and become sequestered in the low-momentum separated region ahead of the cavity leading edge. Nuclei dynamics upstream of a shedding sheet cavity was investigated using high-speed photography. Measurements were performed at zero incidence for cavitation numbers in the range of $0.55 \gt \sigma \gt 0.45$, and chord-based Reynolds numbers of $ \textit{Re} = 0.75\times 10^6$ and $ \textit{Re} = 1.5\times 10^6$. Nuclei are generated each shedding cycle due to cavity breakup from condensation shock-wave phenomena. These nuclei may undergo immediate activation or transport due to pressure gradients, local re-circulation and jetting. Some nuclei remain upstream of the cavity leading edge over multiple cycles. Several phenomena influence this behaviour, including cyclical variation of the boundary layer properties with each shedding cycle. A major conclusion of the work is that these nuclei are produced in a self-sustaining manner from near surface, small scale, interfacial or viscous phenomena rather than from surface or free stream nuclei. Additionally, these experiments reveal the low-momentum region upstream of the cavity to be above vapour pressure, despite the meta-stable tension developed in the boundary layer further upstream of the cavity.