Buoyancy-driven bubbly flows naturally have spatially dependent density fields, which allow for multiple definitions of the scale-dependent (or filtered) energy. A priori, it is not obvious which of these provide the most physically apt scale-by-scale budget. In the present study, we compare two such definitions, based on (i) filtered momentum and filtered velocity (Pandey et al., J. Fluid Mech., 2020, vol. 884, p. R6), and (ii) Favre-filtered energy (Aluie, Phys. D: Nonlinear Phenom., 2013, vol. 247, pp. 54–65; Pandey et al., Phys. Rev. Lett., 2023, vol. 131, p. 114002). We also derive a Kármán–Howarth–Monin relation using the momentum–velocity correlation function and contrast it with the scale-by-scale energy budget obtained in (i). We find that, for the volume fraction and Atwood number explored, irrespective of the definition, energy transfers due to the advective nonlinearity and surface tension are identical. However, discrepancies arise for the buoyancy and pressure contributions. We show that the Favre-filtered definition is the more appropriate choice, within which buoyancy injects energy, pressure transfers energy to large scales and both advective nonlinearity and surface tension transfer energy downscales where it is dissipated by viscosity.

This study presents an in-depth analysis of the energy dissipation and momentum balance during a laminar planar hydraulic jump in a viscous free surface flow, with shallow flow theory used to estimate the relevant jump parameters. The inclusion of momentum and kinetic energy correction factors incorporates the influence of the fluid nature. The fluid is described by the generalised Herschel–Bulkley model with Papanastasiou regularisation, which reduces to the Bingham plastic, power-law and Newtonian models under relevant limiting conditions. The analysis, extensively validated against experimental and simulated data, is explored to understand the physics of free surface flow during jump formation. Energy dissipation increases with an increase in the flow behaviour index n, flow consistency index k and yield stress τo since each of them increases the apparent viscosity. Interestingly, it is higher in the supercritical (upstream) compared with the subcritical (downstream) zone. For constant discharge rate and film thickness, the specific energy depends on the velocity profile and is thus a function of n and τo but not k, and the mechanism of influence of n and τo are also different. For a generalised approach, energy dissipation and jump parameters are discussed as a function of relevant non-dimensional numbers obtained from SFT. Energy dissipation during a hydraulic jump in non-Newtonian liquids is a hitherto unexplored aspect. In fact, energy dissipation during a planar jump in a viscous Newtonian liquid is also rare, although hydraulic jumps are primarily used as energy dissipators in free surface flows.

The rheology of bubble suspensions is critical for the prediction and control of bubbly flows in various industrial processes. It is well known that bubble suspensions exhibit shear-thinning behaviour due to bubble shape deformation under pure shear, but the shear rheological response to dilatation under time-varying pressure remains unexplored. Here, we propose a constitutive equation for dilute bubble suspensions that accounts for both shear and dilatational effects, demonstrating that bubble compressibility can significantly influence the shear viscosity under time-varying pressures. Under constant-rate pressure variations, compression leads to a progressive reduction in viscosity, whereas decompression induces an increase. This peculiar compression-thinning behaviour arises microscopically from the fact that a shrinking bubble surface effectively weakens the flow resistance of the surrounding liquid. Under oscillatory pressure, the amplitude of the dilatation-induced viscosity grows with frequency, and the phase shift of the shear stress response transitions from $-\pi /2$ to $-\pi$. Notably, viscosity troughs emerge during oscillation cycles, leading to transient flow resistance lower than that in pure shear, indicating a potential route for drag reduction when combined with shear-thinning. These findings highlight bubble compressibility as a controllable factor for tuning bubbly flow rheology, with potential for enhanced flow efficiency in practical applications.

The present study investigates the mass transfer of a dilute species from a dispersed bubbly phase to a carrier liquid phase using interface-resolved simulations and proposes a phenomenological model for its transient dynamics. To this end, we individually vary several input parameters, i.e. the species diffusivity (the Schmidt number, ${\textit{Sc}}$), the void fraction $(\alpha )$ and the bubble size (which affects the Galilei number ${\textit{Ga}}$ and Bond number ${\textit{Bo}}$) – while maintaining conditions representative of air bubbles in water. For the parametric range – ${\textit{Sc}}$ = $1,5$ and 10; $\alpha$ = 0.5 %, 1.9 % and 3.6 %; and three initial bubble diameters $d_0$ = $0.63\, \text{mm}$ $(\text{i.e.}\ Ga$ = $1.75$ and ${\textit{Bo}}$ = $0.0125)$, $1.2\, \text{mm}$ (${\textit{Ga}}$ = $4.6$ and ${\textit{Bo}}$ = 0.045) and $1.58\, \text{mm}$ (${\textit{Ga}}$ = $7$ and ${\textit{Bo}}$ = $0.07921$) – we show that increasing diffusivity and decreasing bubble size decreases scalar advection as compared with diffusion and hence the time needed for the species concentration to saturate in the carrier phase, which can be well represented by a Péclet number $Pe = {\sqrt {gr_0} r_0}/{D_c}$ based on the bubble rising speed, its radius and the species diffusivity. We also document the increase of the mass transfer rate with the void fraction when the carrier-phase scalar concentration is low, driven by a larger interfacial area and enhanced velocity fluctuations. When the species-rich bubble wakes start to interact, however, the transfer rate decreases, which occurs earlier at higher values of $\alpha$. The proposed phenomenological model agrees closely with simulations, capturing a self-similar temporal evolution of the mean carrier-phase concentration. By rescaling the non-dimensional time $\tau$ as ${Sc^{{2}/{3}}Ga}/{\alpha ^{0.45}}$, the numerical results collapse onto a master curve.

We present a comprehensive experimental and theoretical investigation of the evaporation dynamics of freely levitated water droplets in an upward airstream under varying temperature and relative humidity conditions, using a custom-designed wind tunnel that replicates natural rainfall scenarios. A high-speed imaging system captures the temporal evolution of morphology, shape oscillations and size reduction of the droplet undergoing evaporation. Our observations reveal that larger droplets exhibit persistent shape oscillations due to the interplay between inertia and surface tension in the presence of convective airflow, which significantly alters the evaporation rate compared with that of a stationary spherical droplet in quiescent air. To quantify the effects of air convection, complex morphology and shape oscillations of the levitated droplet at different temperatures and humidity, we develop a modified evaporation model that extends the classical $d^2$ law. This model incorporates (i) a generalised Sherwood number that accounts for the variation in Reynolds number, Schmidt number, temperature and relative humidity; and (ii) a shape factor that captures the time-averaged surface area of oscillating droplets. The model is validated against experimental findings across a wide range of droplet sizes and environmental conditions, showing excellent agreement in predicting the temporal evolution of droplet diameter and total evaporation time. Furthermore, we construct a regime map showing the variation in the lifetime of the droplet in the temperature–humidity space. The present study establishes a framework that integrates convective transport and morphological deformation, offering new insights into the microphysics of raindrop evaporation.

We investigate a nonlinear interaction between viscous fingering (VF) and phase separation (PS) in a binary fluid system within a radial Hele-Shaw cell. Through nonlinear simulations, we analyse displacement under favourable viscosity contrast, in which a more viscous fluid displaces a less viscous one, and PS may induce VF. The system undergoes PS when the concentration lies within the spinodal region, where the second derivative of free energy is negative. In the absence of viscosity contrast, the flow exhibits distinct morphologies, rings under the strongest PS conditions and droplets otherwise. A distinct composition of separated droplets results from uphill and downhill diffusion imbalances. The prominence of PS is characterised by the interfacial tension within the spinodal region, while the extent of pattern rupture is quantified by the interfacial length in the fully separated region. We identify interfacial tension as a reliable and experimentally accessible indicator of PS. It offers a practical alternative to the second derivative of free energy, which is challenging to quantify directly. We find that higher miscibility stabilises the overall pattern, as evidenced by reductions in both interfacial tension and interfacial length. In contrast, viscosity contrast plays a complex role: while a favourable viscosity contrast generally stabilises the flow by reducing interfacial length, there are specific flow conditions under which the interfacial length increases despite a weaker PS condition. Our results reveal instability patterns consistent with experimental observations, reinforcing the reliability of our findings.

Surfactants at the air–sea interface are known to alter surface wave dynamics by modifying surface tension and Marangoni stresses. In this study, we perform two-dimensional direct numerical simulations of gravity-capillary waves with insoluble surfactants using a coupled phase field and volume-of-fluid method. We consider a nonlinear equation of state for surface tension and resolve Marangoni stresses induced by surfactant concentration gradients. We explore a broad parameter space characterised by initial wave steepness $ak$, Bond number $\textit{Bo}$ (comparing gravity and surface tension), Reynolds number $\textit{Re}$ (comparing inertia and viscosity), and the importance of surfactant concentration and strength of the gradient, characterised by a surfactant parameter $\beta$. We analyse the impact of surfactants on wave patterns, surface roughness, wave breaking, energy dissipation and surface vorticity. Our results reveal a non-monotonic dependence of wave shape, roughness, vorticity and energy dissipation on $\beta$, which is found to be governed by Marangoni effects that peak at intermediate surfactant concentrations. Wave regime transition at high $\textit{Bo}$ is governed by an effective $\textit{Bo}$, which accounts for the reduction in surface tension induced by surfactants. We further introduce a rescaled parameter $\textit{Bo}\,\textit{Re}^{-1/2}\,(ak)^{-1}$ based on force balance, which collapses the transition boundaries across different $\textit{Re}$. These findings provide a systematic understanding of surfactant-modulated wave dynamics for both laboratory and geophysical applications.

Despite its significance in biology and materials science, the dynamics of multicomponent vesicles under shear flow remains poorly understood because of its nonlinear and strongly coupled nature, especially regarding the role of membrane heterogeneity in driving non-equilibrium behaviour. Here we present a thermodynamically consistent phase-field model, which is validated against experiments, for the quantitative investigation of the dynamics. While prior research has primarily focused on viscosity or bending rigidity contrasts, we demonstrate that surface tension heterogeneity can also trigger swinging and tumbling in vesicles under shear. Additionally, our systematic phase diagram reveals three previously unreported dynamical regimes arising from the interplay between bending rigidity heterogeneity and shear flow. Overall, our model provides a robust framework for understanding multicomponent vesicle dynamics, with findings offering new physical insights and design principles for tuneable vesicle-based carriers.

We investigate the interfacial fluid dynamics and heat transfer at nanoscales using an improved diffuse interface approach for liquid–vapour interfaces in non-equilibrium. Conventional Navier–Stokes–Korteweg (NSK) formulations often fail to accurately capture transport phenomena across extremely thin interfaces due to underestimation of interface resistances. In this work, we improve the NSK model by adding a production term in the momentum equation based on higher-order corrections. To enhance interface resistances, viscosity and thermal conductivity are made dependent on the density gradient, increasing resistance only within the interface region. The gradient-based coefficients are determined by fitting to solutions of the Enskog–Vlasov equation for Couette flow of Struchtrup & Frezzotti (2022 J. Fluid Mech., vol. 940, p. A40). Applying these fitted equations to pure heat conduction and planar evaporation problems shows that the model accurately captures interfacial transport, making it a useful tool for studying nanoscale evaporation, thermal management and the droplet dynamics on solid surfaces.

Dense granular flows exhibit pronounced non-local behaviours, particularly in creeping regions and shear-localised zones, which challenges classical local inertial rheologies. In this work, we develop a continuum framework for dense granular flows by extending the $\mu (I)$ rheology through the inclusion of granular temperature as an explicit state variable, thereby establishing a direct link between grain-scale velocity fluctuations and macroscopic stresses, and enabling the representation of non-local effects. The model is implemented within a finite-volume computational framework, and systematically validated against three canonical configurations spanning steady and transient regimes: heap flows, split-bottom Couette flows, and granular column collapse. Across these benchmarks, the formulation captures key non-local features observed experimentally and numerically, including sustained creeping below yield, shear-band broadening and migration, and the transient evolution of free surfaces and runout dynamics. Overall, the granular-temperature-extended $\mu (I)$ rheology provides a unified continuum description that reconciles local and non-local behaviour in dense granular flows, retains the predictive capability of inertial rheology in rapid regimes, and extends its applicability to creeping and shear-localised flows. The proposed framework offers a physically interpretable and scalable basis for modelling granular processes in both geophysical and industrial contexts.

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.