The continental plates of Earth are known to drift over a geophysical time scale, and their interactions have led to some of the most spectacular geoformations of our planet while also causing natural disasters such as earthquakes and volcanic activity. Understanding the dynamics of interacting continental plates is thus significant. In this work, we present a fluid mechanical investigation of the plate motion, interaction and dynamics. Through numerical experiments, we examine the coupling between a convective fluid and plates floating on top of it. With physical modelling, we show the coupling is both mechanical and thermal, leading to the thermal blanket effect: the floating plate is not only transported by the fluid flow beneath, it also prevents the heat from leaving the fluid, leading to a convective flow that further affects the plate motion. By adding several plates to such a coupled fluid–structure interaction, we also investigate how floating plates interact with each other, and show that under proper conditions, small plates can converge into a supercontinent.

In this contribution, we develop a versatile formalism to derive unified two-phase models describing both the separated and disperse regimes as introduced by Loison et al. (Intl J. Multiphase Flow, vol. 177, 2024, 104857). It relies on the stationary action principle and interface geometric variables. This contribution provides a novel method to derive small-scale models for the dynamics of the interface geometry. They are introduced here on a simplified case where all the scales and phases have the same velocity and that does not take into account large-scale capillary forces. The derivation tools yield a proper mathematical framework through hyperbolicity and signed entropy evolution. The formalism encompasses a hierarchy of small-scale reduced-order models based on a statistical description at a mesoscopic kinetic level and is naturally able to include the description of a disperse phase with polydispersity in size. This hierarchy includes both a cloud of spherical droplets and non-spherical droplets experiencing a dynamical behaviour through incompressible oscillations. The associated small-scale variables are moments of a number density function resulting from the geometric method of moments (GeoMOM). This method selects moments as small-scale geometric variables compatible with the structure and dynamics of the interface; they are defined independently of the flow topology and, therefore, this model allows the coupling of the two-scale flow with an inter-scale transfer. It is shown, in particular, that the resulting dynamics provides partial closures for the interface area density equation obtained from the averaging approach.

Large-eddy simulations (LES) of a hypersonic boundary layer on a $7^\circ$-half-angle cone are performed to investigate the effects of highly cooled walls (wall-to-recovery temperature ratio of $T_w / T_r \sim 0.1$) on fully developed turbulence and to validate a newly developed rescaling method based on volumetric flow extraction. Two Reynolds numbers are considered, $Re_m = 4.1 \times 10^6\ \text {m}^{-1}$ and $6.4 \times 10^6\ \text {m}^{-1}$, at free-stream Mach numbers of $M_\infty = 7.4$. A comparison with a reference laminar-to-turbulent simulation, capturing the full history of the transitional flow dynamics, reveals that the volumetric rescaling method can generate a synthetic turbulent inflow that preserves the structure of the fluctuations. Equilibrium conditions are recovered after approximately 40 inlet boundary layer thicknesses. Numerical trials show that a longer streamwise extent of the rescaling box increases numerical stability. Analyses of turbulent statistics and flow visualizations reveal strong pressure oscillations, up to $50\,\%$ of local mean pressure near the wall, and two-dimensional longitudinal wave structures resembling second-mode waves, with wavelengths up to 50 % of the boundary layer thickness, and convective Mach numbers of $M_c \simeq 4.5$. It is shown that their quasi-periodic recurrence in the flow is not an artefact of the rescaling method. Strong and localized temperature fluctuations and spikes in the wall-heat flux are associated with such waves. Very high values of temperature variance near the wall result in oscillations of the wall-heat flux exceeding its average. Instances of near-wall temperature falling below the imposed wall temperature of $T_w=300$ K result in pockets of instantaneous heat flux oriented against the statistical mean direction.

Ciliated microorganisms near the base of the aquatic food chain either swim to encounter prey or attach at a substrate and generate feeding currents to capture passing particles. Here, we represent attached and swimming ciliates using a popular spherical model in viscous fluid with slip surface velocity that affords analytical expressions of ciliary flows. We solve an advection–diffusion equation for the concentration of dissolved nutrients, where the Péclet number ($Pe$) reflects the ratio of diffusive to advective time scales. For a fixed hydrodynamic power expenditure, we ask what ciliary surface velocities maximize nutrient flux at the microorganism's surface. We find that surface motions that optimize feeding depend on $Pe$. For freely swimming microorganisms at finite $Pe$, it is optimal to swim by employing a ‘treadmill’ surface motion, but in the limit of large $Pe$, there is no difference between this treadmill solution and a symmetric dipolar surface velocity that keeps the organism stationary. For attached microorganisms, the treadmill solution is optimal for feeding at $Pe$ below a critical value, but at larger $Pe$ values, the dipolar surface motion is optimal. We verified these results in open-loop numerical simulations and asymptotic analysis, and using an adjoint-based optimization method. Our findings challenge existing claims that optimal feeding is optimal swimming across all Péclet numbers, and provide new insights into the prevalence of both attached and swimming solutions in oceanic microorganisms.

Even though liquid foams are ubiquitous in everyday life and industrial processes, their ageing and eventual destruction remain a puzzling problem. Soap films are known to drain through marginal regeneration, which depends upon periodic patterns of film thickness along the rim of the film. The origin of these patterns in horizontal films (i.e. neglecting gravity) still resists theoretical modelling. In this work, we theoretically address the case of a flat horizontal film with a thickness perturbation, either positive (a bump) or negative (a groove), which is initially invariant under translation along one direction. This pattern relaxes towards a flat film by capillarity. By performing a linear stability analysis on this evolving pattern, we demonstrate that the invariance is spontaneously broken, causing the elongated thickness perturbation pattern to destabilise into a necklace of circular spots. The unstable and stable modes are derived analytically in well-defined limits, and the full evolution of the thickness profile is characterised. The original destabilisation process we identify may be relevant to explain the appearance of the marginal regeneration patterns near a meniscus and thus shed new light on soap-film drainage.

A phenomenological description is presented to explain the intermediate and low-frequency/large-scale contributions to the wall-shear-stress (${\tau }_w$) and wall-pressure ($\,{p}_w$) spectra of canonical turbulent boundary layers, both of which are well known to increase with Reynolds number, albeit in a distinct manner. The explanation is based on the concept of active and inactive motions (Townsend, J. Fluid Mech., vol. 11, issue 1, 1961, pp. 97–120) associated with the attached-eddy hypothesis. Unique data sets of simultaneously acquired ${\tau }_w$, ${p}_w$ and velocity-fluctuation time series in the log region are considered, across a friction-Reynolds-number ($Re_{\tau }$) range of $ {O}(10^3) \lesssim Re_{\tau } \lesssim {O}(10^6)$. A recently proposed energy-decomposition methodology (Deshpande et al., J. Fluid Mech., vol. 914, 2021, A5) is implemented to reveal the active and inactive contributions to the ${\tau }_w$- and $p_w$-spectra. Empirical evidence is provided in support of Bradshaw's (J. Fluid Mech., vol. 30, issue 2, 1967, pp. 241–258) hypothesis that the inactive motions are responsible for the non-local wall-ward transport of the large-scale inertia-dominated energy, which is produced in the log region by active motions. This explains the large-scale signatures in the ${\tau }_w$-spectrum, which grow with $Re_{\tau }$ despite the statistically weak signature of large-scale turbulence production, in the near-wall region. For wall pressure, active and inactive motions respectively contribute to the intermediate and large scales of the $p_w$-spectrum. Both these contributions are found to increase with increasing $Re_{\tau }$ owing to the broadening and energization of the wall-scaled (attached) eddy hierarchy. This potentially explains the rapid $Re_{\tau }$-growth of the $p_w$-spectra relative to ${\tau }_w$, given the dependence of the latter only on the inactive contributions.

We seek the conditions in which Alfvén waves (AW) can be produced in laboratory-scale liquid metal experiments, i.e. at low magnetic Reynolds Number ($Rm$). Alfvén waves are incompressible waves propagating along magnetic fields typically found in geophysical and astrophysical systems. Despite the high values of $Rm$ in these flows, AW can undergo high dissipation in thin regions, for example in the solar corona where anomalous heating occurs (Davila, Astrophys. J., vol. 317, 1987, p. 514; Singh & Subramanian, Sol. Phys., vol. 243, 2007, pp. 163–169). Understanding how AW dissipate energy and studying their nonlinear regime in controlled laboratory conditions may thus offer a convenient alternative to observations to understand these mechanisms at a fundamental level. Until now, however, only linear waves have been experimentally produced in liquid metals because of the large magnetic dissipation they undergo when $Rm\ll 1$ and the conditions of their existence at low $Rm$ are not understood. To address these questions, we force AW with an alternating electric current in a liquid metal in a transverse magnetic field. We provide the first mathematical derivation of a wave-bearing extension of the usual low-$Rm$ magnetohydrodynamics (MHD) approximation to identify two linear regimes: the purely diffusive regime exists when $N_{\omega }$, the ratio of the oscillation period to the time scale of diffusive two-dimensionalisation by the Lorentz force, is small; the propagative regime is governed by the ratio of the forcing period to the AW propagation time scale, which we call the Jameson number $Ja$ after (Jameson, J. Fluid Mech., vol. 19, issue 4, 1964, pp. 513–527). In this regime, AW are dissipative and dispersive as they propagate more slowly where transverse velocity gradients are higher. Both regimes are recovered in the FlowCube experiment (Pothérat & Klein, J. Fluid Mech., vol. 761, 2014, pp. 168–205), in excellent agreement with the model up to $Ja \lesssim 0.85$ but near the $Ja=1$ resonance, high amplitude waves become clearly nonlinear. Hence, in electrically driving AW, we identified the purely diffusive MHD regime, the regime where linear, dispersive AW propagate, and the regime of nonlinear propagation.

Poiseuille flow is a fundamental flow in fluid mechanics and is driven by a pressure gradient in a channel. Although the rheology of active particle suspensions has been investigated extensively, knowledge of the Poiseuille flow of such suspensions is lacking. In this study, dynamic simulations of a suspension of active particles in Poiseuille flow, situated between two parallel walls, were conducted by Stokesian dynamics assuming negligible inertia. Active particles were modelled as spherical squirmers. In the case of inert spheres in Poiseuille flow, the distribution of spheres between the walls was layered. In the case of non-bottom-heavy squirmers, on the other hand, the layers collapsed and the distribution became more uniform. This led to a much larger pressure drop for the squirmers than for the inert spheres. The effects of volume fraction, swimming mode, swimming speed and the wall separation on the pressure drop were investigated. When the squirmers were bottom heavy, they accumulated at the channel centre in downflow, whereas they accumulated near the walls in upflow, as observed in former experiments. The difference in squirmer configuration alters the hydrodynamic force on the wall and hence the pressure drop and effective viscosity. In upflow, pusher squirmers induced a considerably larger pressure drop, while neutral and puller squirmers could even generate negative pressure drops, i.e. spontaneous flow could occur. While previous studies have reported negative viscosity of pusher suspensions, this study shows that the effective viscosity of bottom-heavy puller suspensions can be negative for Poiseuille upflow, which is a new finding. The knowledge obtained is important for understanding channel flow of active suspensions.

This study investigates experimentally the pressure fluctuations of liquids in a column under short-time acceleration. It demonstrates that the Strouhal number $St=L/(c\,\Delta t)$, where $L$, $c$ and $\Delta t$ are the liquid column length, speed of sound, and acceleration duration, respectively, provides a measure of the pressure fluctuations for intermediate $St$ values. On the one hand, the incompressible fluid theory implies that the magnitude of the averaged pressure fluctuation $\bar {P}$ becomes negligible for $St\ll 1$. On the other hand, the water hammer theory predicts that the pressure tends to $\rho cu_0$ (where $u_0$ is the change in the liquid velocity) for $St\geq O(1)$. For intermediate $St$ values, there is no consensus on the value of $\bar {P}$. In our experiments, $L$, $c$ and $\Delta t$ are varied so that $0.02 \leq St \leq 2.2$. The results suggest that the incompressible fluid theory holds only up to $St\sim 0.2$, and that $St$ governs the pressure fluctuations under different experimental conditions for higher $St$ values. The data relating to a hydrogel also tend to collapse to a unified trend. The inception of cavitation in the liquid starts at $St\sim 0.2$ for various $\Delta t$, indicating that the liquid pressure goes lower than the liquid vapour pressure. To understand this mechanism, we employ a one-dimensional wave propagation model with a pressure wavefront of finite thickness that scales with $\Delta t$. The model provides a reasonable description of the experimental results as a function of $St$.

Vesicles are important surrogate structures made up of multiple phospholipids and cholesterol distributed in the form of a lipid bilayer. Tubular vesicles can undergo pearling – i.e. formation of beads on the liquid thread akin to the Rayleigh–Plateau instability. Previous studies have inspected the effects of surface tension on the pearling instabilities of single-component vesicles. In this study, we perform a linear stability analysis on a multicomponent cylindrical vesicle. We solve the Stokes equations along with the Cahn–Hilliard equation to develop the linearized dynamic equations governing the vesicle shape and surface concentration fields. This helps us to show that multicomponent vesicles can undergo pearling, buckling and wrinkling even in the absence of surface tension, which is a significantly different result from studies on single-component vesicles. This behaviour arises due to the competition between the free energies of phase separation, line tension and bending for this multi-phospholipid system. We determine the conditions under which axisymmetric and non-axisymmetric modes are dominant, and supplement our results with an energy analysis that shows the sources for these instabilities. Lastly, we delve into a weakly nonlinear analysis where we solve the nonlinear Cahn–Hilliard equation in the weak deformation limit to understand how mode-mixing alters the late time dynamics of coarsening. We show that in many situations, the trends from our simulations qualitatively match recent experiments (Yanagisawa et al., Phys. Rev. E, vol. 82, 2010, p. 051928).

Bioconvection is the prototypical active matter system for hydrodynamic instabilities and pattern formation in suspensions of biased swimming microorganisms, particularly at the dilute end of the concentration spectrum where direct cell–cell interactions are less relevant. Confinement is an inherent characteristic of such systems, including those that are naturally occurring or industrially exploited, so it is important to understand the impact of boundaries on the hydrodynamic instabilities. Despite recent interest in this area, we note that commonly adopted symmetry assumptions in the literature, such as for a vertical channel or pipe, are uncorroborated and potentially unjustified. Therefore, by employing a combination of analytical and numerical techniques, we investigate whether confinement itself can drive asymmetric plume formation in a suspension of bottom-heavy swimming microorganisms (gyrotactic cells). For a class of solutions in a vertical channel, we establish the existence of a first integral of motion, and reveal that asymptotic asymmetry is plausible. Furthermore, numerical simulations from both Lagrangian and Eulerian perspectives demonstrate with remarkable agreement that asymmetric solutions can indeed be more stable than symmetric; asymmetric solutions are, in fact, dominant for a large, practically important region of parameter space. In addition, we verify the presence of blip and varicose instabilities for an experimentally accessible parameter range. Finally, we extend our study to a vertical Hele-Shaw geometry to explore whether a simple linear drag approximation can be justified. We find that although two-dimensional bioconvective structures and associated bulk properties have some similarities with experimental observations, approximating near-wall physics in even the simplest confined systems remains challenging.

Direct numerical simulations of temporally developing compressible mixing layers have been performed to investigate the effects of large-scale structures (LSSs) on turbulent kinetic energy (TKE) budgets at convective Mach numbers ranging from $M_c=0.2$ to $1.8$ and at Taylor Reynolds numbers up to 290. In the core region of mixing layers, the volume fraction of low-speed LSSs decreases linearly with respect to the vertical distance at a Mach-number-independent rate. The contributions of low-speed LSSs to TKE, and its budget, including production, dissipation, pressure-strain and spatial diffusion terms, are primarily concentrated in the upper region of mixing layer. The streamwise and vertical mass flux coupling terms mainly transport TKE downwards in low-speed LSSs, and their magnitudes are comparable to the other dominant terms. Near the edges of LSSs, the sources and losses of all three components of TKE are completely different to each other, and dominated by turbulent diffusion, pressure diffusion, pressure-strain and dissipation terms. The TKE, their total variation and dissipation are significantly amplified at edges of low-speed LSSs, especially at the upper edge. This observation supports the existence of amplitude modulation exerted by the LSSs onto the near-edge small-scale structures in mixing layers. The level of amplitude modulation is strongest for the vertical velocity, followed by the streamwise velocity, and weakest for the spanwise velocity. Additionally, the amplitude modulation effect decreases significantly with increasing convective Mach number. The results on the amplitude modulation effect is helpful for developing predictive models of budget terms of TKE in mixing layers.

Active suspensions encompass a wide range of complex fluids containing microscale energy-injecting particles, such as cells, bacteria or artificially powered active colloids. Because they are intrinsically non-equilibrium, active suspensions can display a number of fascinating phenomena, including turbulent-like large-scale coherent motion and enhanced diffusion. Here, using a recently developed active fast Stokesian dynamics method, we present a detailed numerical study of the hydrodynamic diffusion in apolar active suspensions of squirmers. Specifically, we simulate suspensions of active but non-self-propelling spherical squirmers (or ‘shakers’), of either puller type or pusher type, at volume fractions from 0.5 % to 55 %. Our results show little difference between pulling and pushing shakers in their instantaneous and long-time dynamics, where the translational dynamics varies non-monotonically with the volume fraction, with a peak diffusivity at around 10 % to 20 %, in stark contrast to suspensions of self-propelling particles. On the other hand, the rotational dynamics tends to increase with the volume fraction as is the case for self-propelling particles. To explain these dynamics, we provide detailed scaling and statistical analyses based on the activity-induced hydrodynamic interactions and the observed microstructural correlations, which display a weak local order. Overall, these results elucidate and highlight the different effects of particle activity versus motility on the collective dynamics and transport phenomena in active fluids.

Gravito–capillary waves at free surfaces are ubiquitous in several natural and industrial processes involving quiescent liquid pools bounded by cylindrical walls. These waves emanate from the relaxation of initial interface distortions, which often take the form of a cavity (depression) centred on the symmetry axis of the container. The surface waves reflect from the container walls leading to a radially inward propagating wavetrain converging (focussing) onto the symmetry axis. Under the inviscid approximation and for sufficiently shallow cavities, the relaxation is well-described by the linearised potential-flow equations. Naturally, adding viscosity to such a system introduces viscous dissipation that enervates energy and dampens the oscillations at the symmetry axis. However, for viscous liquids and deeper cavities, these equations are qualitatively inaccurate. In this study, we decompose the initial localised interface distortion into several Bessel functions and study their time evolution governing the propagation of concentric gravito–capillary waves on a free surface. This is carried out for inviscid as well as viscous liquids. For a sufficiently deep cavity, the inward focussing of waves results in large interfacial oscillations at the axis, necessitating a second-order nonlinear theory. We demonstrate that this theory effectively models the interfacial behaviour and highlights the crucial role of nonlinearity near the symmetry axis. This is rationalised via demonstration of the contribution of bound wave components to the interface displacement at the symmetry axis Contrary to expectations, the addition of slight viscosity further intensifies the oscillations at the symmetry axis although the mechanism of wavetrain generation here is quite different compared with bubble bursting where such behaviour is well known (Duchemin et al., Phys. Fluids, vol. 14, issue 9, 2002, pp. 3000–3008). This finding underscores the limitations of the potential flow model and suggests avenues for more accurate modelling of such complex free-surface flows.

This study employs direct numerical simulations to examine the effects of varying backpressure conditions on the turbulent atomisation of impinging liquid jets. Using the incompressible Navier–Stokes equations, and a volume-of-fluid approach enhanced by adaptive mesh refinement and an isoface-based interface reconstruction algorithm, we analyse spray characteristics in the environments with ambient gas densities ranging from 1 to 40 times the atmospheric pressure under five different backpressure scenarios. We investigate the behaviour of turbulent jets, incorporate realistic orifice geometries and identify significant variations in the atomisation patterns depending on backpressure. Two distinct atomisation types emerge, namely jet-sheet-ligament-droplet at lower backpressures and jet-sheet-fragment-droplet at higher ones, alongside a transition from dilute to dense spray patterns. This variation affects the droplet size distribution and spray dynamics, with increased backpressure reducing the spray's spreading angle and breakup length, while increasing the droplet size variation. Furthermore, these conditions promote distributions that induce rapid, nonlinear wavy motion in liquid sheets. Topological analysis of the atomisation field using velocity-gradient tensor invariants reveals significant variations in topology volume fractions across different regions. Downstream, the droplet Sauter mean diameter increases and then stabilises, reflecting the continuous breakup and coalescence processes, notably under higher backpressures. This research underscores the substantial impact of backpressure on impinging-jet atomisation and provides essential insights for nozzle design to optimise droplet distributions.

We develop a time-dependent conformal method to study the effect of viscosity on steep surface waves. When the effect of surface tension is included, numerical solutions are found that contain highly oscillatory parasitic capillary ripples. These small-amplitude ripples are associated with the high curvature at the crest of the underlying viscous-gravity wave, and display asymmetry about the wave crest. Previous inviscid studies of steep surface waves have calculated intricate bifurcation structures that appear for small surface tension. We show numerically that viscosity suppresses these. While the discrete solution branches still appear, they collapse to form a single smooth branch in the limit of small surface tension. These solutions are shown to be temporally stable, both to small superharmonic perturbations in a linear stability analysis, and to some larger amplitude perturbations in different initial-value problems. Our work provides a convenient method for the numerical computation and analysis of water waves with viscosity, without evaluating the free-boundary problem for the full Navier–Stokes equations, which becomes increasingly challenging at larger Reynolds numbers.

The Hele-Shaw–Cahn–Hilliard model, coupled with phase separation, is numerically simulated to demonstrate the formation of anomalous fingering patterns in a radial displacement of a partially miscible binary-fluid system. The composition of injected fluid is set to be less viscous than the displaced fluid and within the spinodal or metastable phase-separated region, in which the second derivative of the free energy is negative or positive, respectively. Because of phase separation, concentration evolves non-monotonically between the injected and displaced fluids. The simulations reveal four areas of the concentration distribution between the fluids: the inner core; the low-concentration grooves/high-concentration ridges; the isolated fluid fragments or droplets; the mixing zone. The grooves/ridges and the fragments/droplets, which are the unique features of phase separation, form in the spinodal and metastable regions. Four typical types of patterns are categorized: core separation (CS); fingering separation (FS); separation fingering (SF); lollipop fingering, in the order of the dominance of phase separation, respectively. For the patterns of CS and FS, isolated fluid fragments or droplets around the inner core are the main features. Fingering formation is better maintained with droplets in the SF pattern if the phase separation is relatively weaker than viscous fingering (VF). Even continuous fingers are well preserved in the case of dominant VF; phase separation results in lollipop-shaped fingers. The evolving trend of the patterns is in line with the experiments. These patterns are summarized in a pattern diagram, mainly by the magnitude of the second derivative of the free energy profile.

We have investigated the dynamics of floating tracer in an idealised turbulent quasi-geostrophic ocean by advecting Lagrangian particles in a high-resolution velocity field enhanced by the potential flow associated with vortex stretching. At first order in the Rossby number expansion, this component of the ageostrophic circulation can be derived through a diagnostic equation in terms of the geostrophic velocities. Borrowing methods from the theory of Lagrangian coherent structures, we identify coherent material loops around strong vortex cores using the Lagrangian averaged vorticity deviation (LAVD). Building on studies of clustering in kinematic, stochastic velocity fields, we utilise methods from statistical topography to show that the coherent vortices dominate the distribution of extreme values of the concentration field. We find that the presence of clusters and voids in a coherent vortex depends on more than just the sense of rotation, but also on the full evolution of the vorticity over its lifecycle. We identify the mechanism behind the cluster formation that respects the symmetries of the quasi-geostrophic equations but can be expected to hold robustly in more complicated regimes, due to the simple physical description. The association of cluster formation with vortex stretching implies that LAVD is a particularly relevant metric for floating tracer dynamics. The detection of intense clustering also has implications for reaction rates between ocean-borne flotsam, meaning that our results are relevant to understanding the general risk of floating microplastics and marine biological populations.

The manipulation of the Richtmyer–Meshkov instability growth at a heavy–light interface via successive shocks is theoretically analysed and experimentally realized in a specific shock-tube facility. An analytical model is developed to forecast the interface evolution before and after the second shock impact, and the possibilities for the amplitude evolution pattern are systematically discussed. Based on the model, the parameter conditions for each scenario are designed, and all possibilities are experimentally realized by altering the time interval between two shock impacts. These findings may enhance the understanding of how successive shocks influence hydrodynamic instabilities in practical applications.