Phase change materials (PCMs) hold considerable promise for thermal energy storage applications. However, designing a PCM system to meet a specific performance presents a formidable challenge, given the intricate influence of multiple factors on the performance. To address this challenge, we hereby develop a theoretical framework that elucidates the melting process of PCMs. By integrating stability analysis with theoretical modelling, we derive a transition criterion to demarcate different melting regimes, and subsequently formulate the melting curve that uniquely characterises the performance of an exemplary PCM system. This theoretical melting curve captures the key trends observed in experimental and numerical data across a broad parameter space, establishing a convenient and quantitative relationship between design parameters and system performance. Furthermore, we demonstrate the versatility of the theoretical framework across diverse configurations. Overall, our findings deepen the understanding of thermo-hydrodynamics in melting PCMs, thereby facilitating the evaluation, design and enhancement of PCM systems.

In this work, we study the effect of flow curvature, or angular momentum, on the propagation and trapping characteristics of near-inertial waves (NIWs) in a curved front. The curved front is idealised as a baroclinic vortex in cyclogeostrophic balance. Motivated by ocean observations, we employ a Gaussian base flow, which by construction possesses a shield of oppositely signed vorticity surrounding its core, and we consider both cyclonic and anticyclonic representations of this flow. Following two main assumptions, i.e. that (i) the horizontal wavelength of the NIW is smaller than the length scale of the background flow (the WKBJ approximation), and (ii) the vertical wavelength of the NIW is smaller than the radial distance of interest, we derive the NIW dispersion relation and discuss the group velocity and direction of energy propagation. We show that the curvature can (i) increase the critical depth and horizontal extent of the trapping region, (ii) reduce NIW activity at the centre of the anticyclonic vortex core and enhance it in the cyclonic shield surrounding the core for high curvatures, (iii) lead to NIW trapping in the anticyclonic shield surrounding the cyclonic core, and (iv) increase the available band of NIW frequencies that are trapped. The solutions from the ray-tracing method are supported by numerical solutions of the governing equations linearised about the cyclogeostrophic base state. Finally, these methods are applied to an idealised model of oceanic mesoscale Arctic eddies showing an increase in the critical depth of trapping. Our results – while applied to polar eddies – equally apply at lower latitudes in both oceans and atmospheres, highlighting the potential importance of flow curvature in controlling the propagation of NIW energy.

This paper presents a theoretical investigation of vortex modes in acoustofluidic cylindrical resonators with rigid boundaries and viscous fluids. By solving the Helmholtz equation for linear pressure, incorporating boundary conditions that account for no-slip surfaces and vortex and non-vortex excitation at the base, we analyse both single- and dual-eigenfunction modes near system resonance. The results demonstrate that single-vortex modes generate spin angular momentum exclusively along the axial direction, while dual modes introduce a transverse spin component due to the nonlinear interaction between axial and transverse ultrasonic waves, even in the absence of vortex excitation. We find that nonlinear acoustic fields, including energy density, radiation force potential and spin, scale with the square of the shear wave number, defined as the ratio of the cavity radius to the thickness of the viscous boundary layer. Theoretical predictions align closely with finite element simulations based on a model for an acoustofluidic cavity with adiabatic and rigid walls. These findings hold particular significance for acoustofluidic systems, offering potential applications in the precise control of cells and microparticles.

Direct numerical simulations have been conducted to explore the coupling effect of the thermoelectric effect and vertical convection (VC) in a square cavity composed of liquid lithium and stainless steel under different Hartmann numbers at $Ra=10^5$. By leveraging thermoelectric phenomena, an innovative approach is proposed to actively modulate heat transfer efficiency. The core concept lies in modulating the intensity of large-scale circulation (LSC) in VC systems through the torque generated by the interaction between thermoelectric currents and magnetic fields via Lorentz forces. The findings reveal that when the torque aligns with the direction of LSC induced by pure buoyancy, both momentum and heat transfer are enhanced. However, due to the magnetic damping itself, this enhancement is not sustained indefinitely, resulting in a trend of initial increase followed by decline in both momentum and heat transfer efficiency. Conversely, when the magnetic field direction is reversed, causing the Lorentz force torque to oppose the buoyancy-driven circulation, both momentum and heat transfer efficiency diminish until the flow reverses. By varying the magnetic field intensity, three distinct flow regimes are identified: the buoyancy-dominated regime, the thermoelectric-dominated regime and the magnetic-damping-dominated regime. The transition between the buoyancy-dominated regime and thermoelectric-dominated regime – specifically, the onset of flow reversal – is analysed through a boundary-layer–bulk–boundary-layer coupling model. This model enables precise prediction of the critical $Ha$ based on the torque balance between buoyancy forces and thermoelectrically induced Lorentz forces, and demonstrates close agreement with numerical simulations.

Ocean submesoscales, flows with characteristic size $10\,\text{m}{-}10\,\text{km}$, are transitional between the larger, rotationally constrained mesoscale and three-dimensional turbulence. In this paper, we present simulations of a submesoscale ocean filament. In our case, the filament is strongly sheared in both vertical and cross-filament directions, and is unstable. Instability indeed dominates the early behaviour with a fast extraction of kinetic energy from the vertically sheared thermal wind. However, the instability that emerges does not exhibit characteristics that match the perhaps expected symmetric or Kelvin–Helmholtz instabilities, and appears to be non-normal in nature. The prominence of the transient response depends on the initial noise, and for large initial noise amplitudes, saturates before symmetric instability normal modes are able to develop. The action of the instability is sufficiently rapid – with energy extraction from the mean flow emerging and peaking within the first inertial period ($\sim\! 18\ \text{h}$) – that the filament does not respond in a geostrophically balanced sense. Instead, at all initial noise levels, it later exhibits vertically sheared near-inertial oscillations with higher amplitude as the initial minimum Richardson number decreases. Horizontal gradients strengthen only briefly as the fronts restratify. These unstable filaments can be generated by strong mixing events at pre-existing stable structures; we also caution against inadvertently triggering this response in idealised studies that start in a very unstable state.

Bubbles entrained by breaking waves rise to the ocean surface, where they cluster before bursting and release droplets into the atmosphere. The ejected drops and dry aerosol particles, left behind after the liquid drop evaporates, affect the radiative balance of the atmosphere and can act as cloud condensation nuclei. The remaining uncertainties surrounding the sea spray emissions function motivate controlled laboratory experiments that directly measure and link collective bursting bubbles and the associated drops and sea salt aerosols. We perform experiments in artificial seawater for a wide range of bubble size distributions, measuring both bulk and surface bubble distributions (measured radii from $30\,\unicode{x03BC} \mathrm{m}$ to $5\,\mathrm{mm}$), together with the associated drop size distribution (salt aerosols and drops of measured radii from $50\,\mathrm{nm}$ to $500\,\unicode{x03BC} \mathrm{m}$) to quantify the link between emitted drops and bursting surface bubbles. We evaluate how well the individual bubble bursting scaling laws describe our data across all scales and demonstrate that the measured drop production by collective bubble bursting can be represented by a single framework integrating individual bursting scaling laws over the various bubble sizes present in our experiments. We show that film drop production by bubbles between $100\,\unicode{x03BC} \mathrm{m}$ and $1\,\mathrm{mm}$ describes the submicron drop production, while jet drop production by bubbles from $30\,\unicode{x03BC} \mathrm{m}$ to $2\,\mathrm{mm}$ describes the production of drops larger than $1\,\unicode{x03BC} \mathrm{m}$. Our work confirms that sea spray emission functions based on individual bursting processes are reasonably accurate as long as the surface bursting bubble size distribution is known.

We present the results of a theoretical investigation of orbital stability in pilot-wave hydrodynamics, wherein a droplet bounces and self-propels across the surface of a vertically vibrating liquid bath. A critical notion in pilot-wave hydrodynamics is that the bath plays the role of the system memory, recording the history of the droplet in its wave field. Quantised orbital motion may arise when the droplet is confined by either an axisymmetric potential or the Coriolis force induced by system rotation. We here elucidate the dependence of the stability of circular orbits on both the form of the confining force and the system memory. We first provide physical insight by distinguishing between potential- and wave-driven instabilities. We demonstrate that the former are a generic feature of classical orbital dynamics at constant speed, while the latter are peculiar to pilot-wave systems. The wave-driven instabilities are marked by radial perturbations that either grow monotonically or oscillate at an integer multiple of the orbital frequency, in which case they are said to be resonant. Conversely, for potential-driven wobbling, the instability frequency may be resonant or non-resonant according to the form of the applied potential. Asymptotic analysis rationalises the different stability characteristics for linear-spring and Coriolis forces, the two cases that have been explored experimentally. Our results are generalised to consider other potentials of interest in pilot-wave hydrodynamics, and elucidate the distinct roles of wave- and potential-driven instabilities. Our study highlights the limitations of prior heuristic arguments for predicting the onset of orbital instability.

The Leidenfrost effect occurs when drops are deposited on a highly superheated solid surface, creating a thin vapour film through rapid evaporation that levitates the drops. For drop with a radius exceeding a critical value, a vapour bubble forms and bursts from its bottom centre, a phenomenon known as chimney instability. Despite extensive investigation, the impact of Leidenfrost drop’s rotation on its chimney instability has remained unexplored. This study addresses this gap by providing both numerical and approximate solutions to the theoretical models. We identify two distinct regimes where either gravitational force or centrifugal force is the primary driver of chimney instability. These regimes are characterised by a non-dimensional rotation number, Ro, which represents the ratio between centrifugal force and gravitational force. Our findings reveal clear scaling laws that relate the critical geometrical parameters (radius, volume and height of the drop) for chimney instability to Ro, demonstrating that rotation can induce chimney instability in smaller drops. The scaling laws are elucidated through pressure perturbation analyses under a virtual perturbation to the drop profile at the critical state for chimney instability. Additionally, by varying the evaporation number $Ev$, we demonstrate that while increased superheat reduces the critical radius in the absence of rotation, the scaling laws related to Ro for a rotating drop remain unaffected. Building on these insights, we present a master curve in a simplified form that accurately predicts the critical state for chimney instability under various angular velocities, gravitational accelerations and superheat conditions.

The existing studies on vortex rings have concentrated on non-zero circulation. However, the cases of zero circulation may also be significantly noteworthy on both theoretical and practical grounds. As the first attempt on this subject, in this paper a family of viscous laminar vortex rings with zero circulation and a moderate ratio of core radius to ring radius is studied using numerical simulations of the incompressible Navier–Stokes equations. This unusual zero circulation is achieved by assigning a special layered vorticity distribution with alternate signs to the vortex core. At the initial moment, the ring is axisymmetric, swirl-free and of a circular cross-section. It is found that the axial symmetry and the non-swirl nature of the vortex ring are preserved during the evolution, and the vortex ring endures a transition from the initial layered structure to a shell structure, then degenerates to an ordinary vortex ring with non-zero circulation at last. Significant vorticity cancellation is observed due to the interactions among the layered structures. A new Reynolds number, based on the absolute value of vorticity, is applied to the zero-circulation vortex rings in the present work. For such vortex rings, cases of both zero and non-zero vortical impulse can happen, unlike the ordinary ones with only non-zero vortical impulse. Additionally, it is found that the vortical impulse can be irrelevant to the ring diameter. The study may shed light on modelling certain real flows characterised by distinct vortex structures or configurations.

Volcanic fissure eruptions typically start with the opening of a linear fissure that erupts along its entire length, following which, activity localises to one or more isolated vents within a few hours or days. Localisation is important because it influences the spatiotemporal evolution of the hazard posed by the eruption. Previous work has proposed that localisation can arise through a thermoviscous fingering instability driven by the strongly temperature dependent viscosity of the rising magma. Here, we explore how thermoviscous localisation is influenced by the irregular geometry of natural volcanic fissures. We model the pressure-driven flow of a viscous fluid with temperature-dependent viscosity through a narrow fissure with either sinusoidal or randomised deviations from a uniform width. We identify steady states, determine their stability and quantify the degree of flow enhancement associated with localised flow. We find that, even for relatively modest variations of the fissure width (${\lt } 10$ %), the non-planar geometry supports strongly localised steady states, in which the wider parts of the fissure host faster, hotter flow, and the narrower parts of the fissure host slower, cooler flow. This geometrically driven localisation differs from the spontaneous thermoviscous fingering observed in planar geometries and can strongly impact the localisation process. We delineate the regions of parameter space under which geometrically driven localisation is significant, showing that it is a viable mechanism for the observed localisation under conditions typical of basaltic eruptions, and that it has the potential to dominate the effects of spontaneous thermoviscous fingering in these cases.

The present work experimentally investigates the interaction of a buoyant (rigid) spherical particle with a single translating (water) vortex ring, focusing on the effects of particle-to-vortex core size ratio ($D_p/D_{c,o}$) on both the particle dynamics and ring dynamics ($D_p$ = particle diameter, $D_{c,o}$ = vortex core diameter). These interactions are studied for $D_p/D_{c,o}$ = 0.6–1.7, over ring Reynolds numbers ($Re={\varGamma }/{\nu }$; $\varGamma$ = ring circulation) of 6000–67 300. As the buoyant particle comes close to the ring, it gets captured into the low-pressure vortex core, and the interaction begins. The particle within the core undergoes radial oscillation, spins and translates along the ring’s azimuthal axis. As $D_p/D_{c,o}$ increases, the particle undergoes higher-amplitude radial oscillation and a relatively shorter azimuthal translation. The differences in the particle size and its motion within the ring lead to large differences in the ring’s dynamics. A larger particle is seen to lead to a higher ring disruption, substantially reducing the ring’s convection speed and azimuthal enstrophy, which are seen to scale as $(D_p/D_{c,o})^{2.3}Re^{-0.37}$ and $(D_p/D_{c,o})^{1.3}Re^{-0.25}$, respectively. The ring disruption is significant above $D_p/D_{c,o}\approx$ 1.0, beyond which the ring fragments, with up to 60 % drop in convection speed and 90 % drop in enstrophy, at low $Re$, as compared with the base ring. These results for the rigid particle size effects on the vortex ring dynamics are more dramatic than for a deforming bubble. Our results could help to better understand and model buoyant particle (and bubble) interactions with coherent structures in turbulence.

Experimental investigation of the Rayleigh–Taylor instability (RTI) and its dependence on initial conditions has been challenging, primarily due to the difficulty of creating a well-defined gaseous interface. To address this, a novel soap film technique was developed to create a discontinuous two-dimensional SF$_6$air interface with precisely controlled initial conditions. High-order modes were superimposed on a long-wavelength perturbation to study the influence of initial conditions on RTI evolution. Experiments conducted at Atwood numbers ranging from 0.26 to 0.66 revealed that bubble growth shows a weak dependence on both initial conditions and Atwood numbers, whereas spike growth is more influenced by these factors. Spike growth accelerates as the wavenumber of the imposed high-order modes decreases and/or the Atwood number increases. To quantify these effects, a variation on the previously developed potential flow model was applied, capturing the suppression of high-order modes and Atwood number dependence on RTI growth. In turbulent flow, the self-similar factors of bubbles and spikes exhibit minimal sensitivity to initial conditions. However, in relation to the Atwood number, the self-similar factors of bubbles (or spikes) demonstrate negligible (or significant) dependence. Comparisons with literature revealed that two-dimensional flows yield lower self-similar factors than three-dimensional flows. Furthermore, the discontinuity of the initial interface in this study, achieved through the soap film technique, results in faster spike growth compared with previous studies involving a diffusive initial interface. These findings provide critical insights into the nonlinear dynamics of RTI and underscore the importance of well-characterised initial conditions in experimental studies.

Non-spherical bubble collapses near solid boundaries, generating water hammer pressures and shock waves, were recognized as key mechanisms for cavitation erosion. However, there is no agreement on local erosion patterns, and cavitation erosion damage lacks quantitative analysis. In our experiments, five distinct local erosion patterns were identified on aluminium sample surfaces, resulting from the collapse of laser-induced cavitation bubbles at moderate stand-off distances of $0.4\leqslant \gamma \leqslant 2.2$, namely bipolar, monopolar, annular, solar-halo and central. Among them, the bipolar and monopolar patterns exhibit the most severe cavitation erosion when the toroidal bubbles undergo asymmetrical collapse along the circumferential direction during the second cycle. Shadowgraphy visualization revealed that asymmetrical collapse caused shockwave focusing through head-on collision and oblique superposition of wavefronts. This led to the variations in toroidal bubble radii and the positions of maximum erosion depth not matching at certain stand-off distances. Both initial plasma asymmetry and bubble–wall stand-off distance were critical in determining circumferential asymmetrical collapse behaviours. At large initial aspect ratios, the elliptical jet tips form during the contraction process, resulting in the toroidal bubble collapsing from regions with smaller curvature radii, ultimately converging to the colliding point along the circumferential direction. Our three-dimensional simulations using OpenFOAM successfully reproduce the key features of circumferentially asymmetrical bubble collapse. This study provides new insights into the non-spherical near-wall bubble collapse dynamics and provides a foundation for developing predictive models for cavitation erosion.

We investigate enstrophy variations by collapse of point vortices in an inviscid flow and, in particular, focus on the enstrophy dissipation that is a significant property characterising two-dimensional (2-D) turbulent flows. To reveal the vortex dynamics causing the enstrophy dissipation, we consider the dynamics of point vortices, whose vorticity is concentrated on points and dynamics on the inviscid flow, governed by the point-vortex system. The point-vortex system has self-similar collapsing solutions, which are expected to be a key to understand the enstrophy dissipation, but the collapsing process cannot be described by solutions to the 2-D Euler equations. We thus consider the 2-D filtered-Euler equations, which are a regularised model of the 2-D Euler equations, and their point-vortex solutions. The preceding studies (Gotoda and Sakajo, J. Nonlinear Sci. 2016, vol. 26, pp. 1525–1570, Gotoda and Sakajo, SIAM J. Appl. Math. 2018, vol. 78, 2105–2128) have proven that there exist three point-vortex solutions to the filtered model such that they converge to self-similar collapsing orbits in the three point-vortex system and dissipate the enstrophy at the event of collapse in the zero limit of the filter parameter. In this study, we numerically show that the enstrophy dissipation by the collapse of point vortices could occur for the four and five vortex problems in a filtered model. Moreover, we show the detailed convergence process of the point vortices for gradually decreasing filter parameters, which provides a new insight for the three vortex problem. In addition, numerical computations suggest that the enstrophy dissipation is caused by collapse of separated point vortices with the negative interactive energy.

In this work, we numerically investigate heat transfer in low-Prandtl-number drop-laden wall-bounded turbulence. These flows are characteristic of nuclear and fusion technologies, where liquid metals – known for their high thermal conductivity – are laden with drops or bubbles of another liquid or pressurised gas. To this end, we consider forced convection turbulence between two differentially heated parallel plates. The carrier phase (i.e. liquid metal) is characterised by a low Prandtl number $Pr_c=0.013$, while for the dispersed phase, we explore a range of Prandtl numbers from $Pr_d=0.013$ (matched case) to $Pr_d=7$ (super-unitary Prandtl number in the dispersed phase). Simulations are conducted at constant friction Reynolds number $Re_\tau =300$, and for each dispersed phase Prandtl number, two volume fractions are examined: $\alpha =5.4\,\%$ and $\alpha =10.6\,\%$. The simulation framework relies on direct numerical simulation of the Navier–Stokes equations, coupled with a phase-field method and the energy equation. Results show that an increase of the dispersed phase Prandtl number reduces heat transfer, leading to a lower Nusselt number for both volume fractions. To explain this behaviour, we analyse how the drops modify the temperature field, and demonstrate that the heat transfer reduction stems from a decreased diffusive heat flux within the dispersed phase. Finally, we propose a phenomenological model to predict the Nusselt number as a function of both the dispersed phase volume fraction and Prandtl number.

A knowledge gap exists for flows and transport phenomena at the angstrom scale when the Poisson–Nernst–Planck equation based on the concept of the electrical double layer fails. We discovered that streaming conductance becomes pressure-dependent in angstrom channels using latent-track membranes. The streaming current emerges only when the applied pressure exceeds a threshold value, which is inconsistent with the existing knowledge as a constant. With increasing channel size, we found that the pressure-dependent streaming conductance phenomenon weakens and vanishes into a constant streaming conductance regime when the mean channel radius exceeds $\sim$2 nm. The effective surface potential derived from the streaming conductance that divides conduction anomalously increases as the channel narrows. We suspect that the pressure-dependent streaming current is due to the reinforced Coulomb interaction between counterions and deprotonated carboxyl groups at the surface, which is close to the ion channel but different from that of electrified two-dimensional materials. The streaming current emerged due to hydrodynamic friction when the counterions were released from the surface. We approximated the stochastic process of counterion dissociation by a one-dimensional Kramer escape theory framework and defined the Damk$\ddot {\mathrm{o}}$hler number to describe the transition from the nonlinear streaming conductance regime to the linear regime as functions of applied pressure and channel radius and well explained the enhanced effective surface potential in confinement.

The connection between the drag and vorticity dynamics for viscous flow over a bluff body is explored using the Josephson–Anderson (J–A) relation for classical fluids. The instantaneous rate of work on the fluid, associated with the drag force, is related to the vorticity flux across the streamlines of a background potential flow. The vorticity transport itself is examined by aid of the Huggins vorticity-flux tensor. The analysis is performed for three flows: flow over a sphere at Reynolds numbers $Re=200,3700$, and flow over a prolate spheroid at $Re=3000$ and $20^{\circ }$ incidence. In these flows, the vorticity transport shifts the flow away from and towards the ideal potential flow, with a net balance towards the former effect thus making an appreciable contribution to the drag. The J–A relation is first demonstrated for the flow over a sphere at $Re=200$. The drag power injection is related to the viscous flux of azimuthal vorticity from the wall into the fluid, and the advection of vorticity by the detached shear layer. In the wake, the azimuthal vorticity is advected towards the wake centreline and is annihilated by viscous effects, which contributes a reduction in drag. The analysis of the flow over a sphere at $Re=3700$ is reported for the impulsively started and stationary stages, with emphasis on the effects of unsteady two-dimensional separation and turbulent transport in the transitional wake. The turbulent flux in the wake enhances the transport of mean azimuthal vorticity towards the wake centreline, and is the main driver of the recovery of enthalpy between the rear point of the sphere and far downstream. The rate of work on the fluid by the drag force for a prolate spheroid is mostly due to the transport of vorticity along the separated boundary layers. Primary and secondary separation contribute oppositely to the power injection by the drag force, while the large-scale vortices only re-distribute vorticity without a net contribution. A mechanism for secondary separation is proposed based on the theory of vortex-induced separation.