This work provides new lower bounds on the global (nonlinear) stability limit of pressure-driven two-dimensional plane Poiseuille flow, improving on the energy stability limit $\textit{Re}_E$ originally computed by Orr in 1907. Using a computer, we carefully construct quartic Lyapunov functionals of the velocity perturbations about the laminar profile, which certify the nonlinear stability of the flow to arbitrary perturbations. The formulation combines a decomposition of the velocity into finitely many energy eigenmodes, referred to as a ‘mode set’, and an infinite-dimensional ‘tail’, together with explicit bounds that recast the Lyapunov inequality conditions as semidefinite programs, whose feasibility is tested. Over the streamwise lengths considered, the certified stability limit exceeds the classical energy bound. In particular, at the critical energy-stable streamwise length, where $\textit{Re}_E\approx 87.59$, the flow is found to be globally stable up to $\textit{Re} \approx 106.8$ (representing a $22\,\%$ improvement). Various modestly sized mode sets, capable of capturing sufficient features of the nonlinear dynamics of energy growth and subsequent decay, are proposed and found to be successful in producing improved bounds, with the simplest one involving only five modes.

Recent experiments have shown that microparticles subjected simultaneously to electrophoresis and pressure-driven flow undergo lateral migration. Although fluid inertia has been proposed to explain this phenomenon, it remains negligible in these regimes, leaving the underlying mechanism an open question. In this work, we demonstrate that concentration polarisation alone can account for this effect. By modelling a dielectric particle with finite surface conductance in a shear flow, we show that the flow breaks the symmetry of the ionic cloud in the direction perpendicular to the applied electric field. The resulting asymmetry induces lateral migration through coupled electrophoretic and diffusiophoretic mechanisms. Our analytical expression predicts migration velocities of the order of ${\unicode{x03BC}}\mathrm{m\,s}^{-1}$ and reveals a reversal in the migration direction for Dukhin numbers of order unity.

Bubble breakup in turbulence is driven by velocity increments acting on the bubble interface. Traditionally, these instantaneous increments are replaced by the inertial-range second-order moment scaling, but whether this averaged description captures the kinematics leading to breakup remains unclear. Here, we experimentally investigate how bubble deformation preceding breakup relates to carrier-phase statistics. The experiments are conducted in nearly homogeneous isotropic turbulence with $\textit{Re}_\lambda =635$. Five high-speed cameras are used to reconstruct both the three-dimensional bubble topology and the trajectories of flow tracers in the carrier phase. A two-point description of bubble deformation is used to enable a direct comparison with particle pair dispersion theory and velocity increments. We find that the pre-breakup deformation stage can be described as a ballistic-dispersion problem, mirroring the short-time behaviour of particle pairs. For bubbles with $\textit{We} = \mathcal{O}(1)$, corresponding here to sizes close to the Hinze scale, the deformation is biased towards extensional events in which positive longitudinal velocity increments separate the daughter-associated subvolumes. Using a multifractal representation of the carrier-phase longitudinal-increment probability density function, conditioned by a capillary threshold, our results suggest that the measured deformation increments are consistent with stronger-than-average extensional increments, rather than with the inertial-range estimate.

Two distinct regimes arise during the early development of flow near the trailing edge of a flat plate impulsively started at small angles of attack. The first regime concerns the self-similar growth of a starting vortex at the edge; the second regime involves the advection of the starting vortex away from the edge, as described by Wagner’s classical model, which omits the first regime. In this work, we investigate the transition between these two regimes using a computational inviscid vortex-sheet method and an analytical point-vortex model. We show that this transition is triggered by vortex detachment, defined here as the instant at which two stagnation points on the plate surface annihilate each other and move off the plate. This vortex detachment is found to occur at the time, $t \approx 0.019 \, \alpha ^2 c/U$, where $\alpha$ is the angle of attack, $c$ is the plate chord and $U$ is the plate speed. We also demonstrate that this detachment process is independent of the angle of attack when the distance between the edge and the initial stagnation point is used as the characteristic length scale. These results provide insight into the early-time dynamics of vortex generation, motivating experimental validation and verification through numerical simulation of the Navier–Stokes equations.

Large regions of giant planets are thought to possess unstable thermal gradients stabilised by gradients of heavy-element composition. Such a fluid can develop semi-convection, a double-diffusive instability driven by the unequal molecular diffusivities of heat and composition. While previous studies have focused mainly on local Cartesian models, we investigate semi-convection in rotating spherical shells, the geometry relevant to planetary interiors, using direct numerical simulations. In a first nonlinear phase, the flow spontaneously forms concentric density staircases composed of well-mixed layers separated by thin, strongly stratified interfaces. We propose scalings for both the thickness of these layers and their survival time in terms of the rotation rate and stratification. Over longer time scales, layers merge to produce statistically steady states consisting either of a fully convective shell, or of a convective interior overlain by a persistent stably stratified layer (SSL), depending on the balance between stratification and rotation. Dynamo simulations show that flow within the convective region can generate a self-sustained magnetic field, which is filtered by zonal flows in the overlying SSL. This results in a strongly dipolar and axisymmetric external field, in encouraging agreement with Saturn’s magnetic field. Across the explored parameter range, both the Rossby number and the thickness of the SSL are governed by a single combination of control parameters. This enables identification of a regime favourable to planetary-like dynamos.

Liquid bridges suspended between two parallel horizontal cylinders are studied using experiments, reduced-order mathematical modelling and numerical simulations. Both non-electrified and electrified configurations are considered, with the cylinders acting as electrodes between which a potential difference can be applied. The initial focus is on equilibrium bridge shapes, while the dynamics is examined only in the non-electrified case. In the experiments, transformer-oil bridges are investigated and modelled as perfect dielectrics. The electric field counteracts the gravitational effects and causes the bridges to rise and become flatter, with shapes that are well described by Young–Laplace-type equations containing non-local electric-field contributions evaluated using a boundary-element method. Bifurcation diagrams of these solutions are constructed by pseudo-arclength continuation to characterise the dependence of bridge shapes on liquid volume and electric-field strength. In the absence of an electric field, the transient relaxation to equilibrium is analysed using a reduced-order model developed using Onsager’s variational principle, with comparison to direct numerical simulations. The steady states predicted by the reduced-order model agree closely with those of the full formulation over a wide parameter range, and the dynamics is well captured in the overdamped regime.

The present study examines the evolution of the phase velocity of surface gravity waves propagating against an opposing current in the vicinity of wave-blocking conditions, defined as flow regimes in which the wave-energy flux can no longer propagate upstream relative to an adverse background current. We adopted a novel experimental technique by deploying passive devices (‘flow conditioners’), which offer robust and repeatable control over the current profile to represent several shear structures. The classical evaluation of phase velocity through cross-correlation of wave-gauge signals is used as a reference for testing a novel experimental method based on high-resolution velocity fields obtained by particle image velocimetry. The two experimental estimates show generally good agreement. The experimental data are compared with theoretical predictions obtained from eigenvalue-based approximation schemes for the governing problem and with the full numerical solution of the classical Rayleigh eigenvalue problem. Second-order approximation models give the best theoretical estimates, while results from the full numerical solution are slightly more dispersed. The hypothesis of a uniform current largely underestimates the experimental values and breaks down systematically. Our results indicate that bulk current strength alone is insufficient to characterise the wave kinematics and that the vertical distribution of the adverse current within the wave-active layer exerts a significant additional influence. This helps explain why shear-weighted dispersion approximations perform well, contrary to the uniform-current assumption. This work is particularly relevant for the full-scale characterisation and prediction of compound flooding events arising from the combination of river flood discharges with intense marine storm conditions, offering quantitative considerations on the best predictive models that can be used in practical situations.

We numerically investigate the coalescence dynamics of two equal-sized droplets coated with solid particles using a lattice Boltzmann colour-gradient model coupled with Lagrangian particle tracking. It is shown that the particle coverage critically determines the final morphology of coalescing droplets, giving rise to three distinct regimes: total coalescence, arrested coalescence and total stability. Arrested coalescence arises from dynamic particle jamming at the shrinking interface, where increasing particle coverage generates mechanical resistance against further capillary-driven shape relaxation. A theoretical criterion for the onset of arrested coalescence is derived and validated against a phase diagram constructed from 110 simulation cases. Furthermore, particle wettability governs the formation of inter-droplet bridges: only particles preferentially wetted by the ambient fluid can establish stable bridges. While single-particle bridges are susceptible to flow perturbations, multi-particle bridges enhance stability by entrapping ambient fluid and resisting film drainage. The study is extended to a droplet group under an initial instantaneous shear, where neutrally wetting particles at low coverages are found to promote coalescence, whereas higher coverages lead to arrested coalescence and complex collective morphologies. At moderate coverages, two stabilisation mechanisms are identified: (i) steric hindrance from dense particle layers, and (ii) capillary-force-mediated assembly of droplet chains via multi-particle bridging. These insights provide guidelines for tailoring the morphology and stability of Pickering emulsions in material synthesis, enhanced oil recovery and pharmaceutical applications.

The locomotion of catalytic micromotors is strongly influenced by nearby boundaries, where hydrodynamic and chemical interactions collectively govern their dynamics. By incorporating hydrodynamic slip into the theoretical framework, we capture the effects of nearby surface wettability on the dynamics of a self-diffusiophoretic micromotor. The hydrodynamic slip alters the balance between phoretic propulsion and wall-induced resistance, resulting in distinct dynamical responses across the space of catalytic coverage and slip length ($l_s$). The slip-induced downward attraction, combined with the reduced reorienting torque, disrupts the near-wall ‘cushioning effect’ in no-slip conditions, resulting in the enhanced trapping of micromotors near the boundary. For intermediate catalytic coverages, the micromotor undergoes a sharp transition from escaping trajectories to either wall-collision or stable sliding states. Further, the wall slip lowers the activity threshold required to achieve stable near-wall trapping, thereby expanding the parameter space for controlled transport. The slip-induced changes in hydrodynamic resistance and phoretic thrust result in a counterintuitive reduction in longitudinal velocity as slip increases. In addition, increasing $l_s$ systematically shifts the hovering height to larger wall–particle separations. The slip-controlled regimes identified here provide a compact design strategy for manipulating active matter near boundaries. By choosing substrate coating, microtopography and stimulus-responsive materials, one can steer, trap or release micromotors in microfluidic and biological environments without redesigning the swimmers themselves.

Starting from the Navier–Stokes equations in rotating spherical coordinates with depth-varying density and eddy viscosity, we derive an asymptotic model describing non-equatorial wind-generated ocean drift currents. Our approach allows for large-scale flows that cannot be captured by classical tangent-plane approximations. The strategy is to perform a careful scaling and to perform a double asymptotic expansion with respect to two small parameters arising from the scaling: the Rossby number and the ratio between the Ekman depth and the Earth’s radius. We obtain a system of linear ordinary differential equations with nonlinear boundary conditions governing the leading-order dynamics, highlighting that the dynamics is governed by the linear terms, whereas the nonlinear ones, related to the injection and dissipation of kinetic energy, appear only at higher order. We use the leading-order equations to compare our model with the simplest theory of ocean circulation due to Sverdrup and note that, even at this level of simplification, our equations have the potential to provide deeper insight. Subsequently, focusing on Ekman flows, we prove existence and uniqueness of the leading-order solution, which retains the classical Ekman spiral structure for arbitrary eddy-viscosity profiles. Finally, we compute the surface deflection angle of the wind-driven current for three explicit eddy-viscosity profiles, obtaining results consistent with observations. In addition, we derive the governing equations for the first-order correction with respect to the Rossby number and provide a priori bounds for their solution.

A new mode of flame propagation from a wall in a mixture below the flammability limit is described. We consider a simple chemistry model in which the reaction rate vanishes for temperatures below some critical cutoff value. Two models for flame initiation by the wall are considered. The first model assumes heat release by the wall over a finite time, after which it becomes adiabatic. The second model assumes that the wall temperature changes according to a given law. Unlike the standard flame propagation mode, in which the flame propagates at a constant velocity through the unburned mixture, and which is not possible under the considered conditions, in the new mode the flame moves at a speed inversely proportional to the square root of time. It is found that the value of the fuel Lewis number, $\textit{Le}$, is determinant: the new mode of propagation exists only if $\textit{Le}\lt 1$. The influence of thermal expansion on the existence of this new propagation regime is investigated. It is interesting to note that the flame structure resembles the classical Stefan solution for phase-transition front propagation. However, the fundamental difference is that in the classical phase-transition front propagation problem the Lewis number is infinite, in our nomenclature, while the new flame propagation regime arises at $\textit{Le}\lt 1$.

Direct numerical simulations of spatially developing high-enthalpy hypersonic turbulent boundary layers are performed under non-catalytic wall-cooling conditions. Two distinct levels of chemical reactivity are considered, together with a chemically frozen case for reference. The main objective is to elucidate the underlying mechanisms of high-enthalpy turbulence–chemistry interactions and to develop reliable closure models for high-enthalpy hypersonic turbulent boundary layers. It is well established that high-enthalpy reactions have a negligible influence on the turbulent velocity statistics, and it is further shown here that this behaviour arises from the compensating effects. Specifically, the mean temperature is reduced by endothermic reactions, accompanied by a decrease in mixture molar mass, leading to an approximately unchanged mean density. Furthermore, the chemical time scale is found to be much larger than the large-eddy turnover time, resulting in strong linear correlations between the temperature and species mass fraction fluctuations. Consequently, the refined strong Reynolds analogy (RSRA) can be effectively extended from enthalpy to temperature and species mass fraction fluctuations in high-enthalpy hypersonic turbulent boundary layers. A correlation correction for RSRA is proposed, which corrects the breakdown of the original RSRA in the region near the mean temperature peak, while retaining good performance in the outer layer. An isobaric presumed probability density function model incorporating the correlation-corrected RSRA is further developed, which extends the applicability of the temperature-based presumed probability density function approach to stronger high-enthalpy conditions. This work contributes to the development of engineering models for turbulence–chemistry interactions in high-enthalpy hypersonic turbulent boundary layers.

This study investigates the flow field in the flap cove of a two-element high-lift airfoil, focusing on mechanisms responsible for noise generation. The main-element trailing-edge wake mixes weakly with the flap-cove shear layer, producing a persistent velocity deficit along the flap suction surface. This contrasts with the slat region, where stronger mixing between the shear layer and slat wake at low angles of attack leads to higher broadband noise levels. Within the flap cove, a turbulent separation bubble (TSB) bounded by a shear layer is identified. Strong correlations between flap leading-edge (LE) pressure fluctuations and cove flow support the use of LE noise models. However, significant correlation near the shear-layer reattachment point indicates additional acoustic scattering from the main element, suggesting that LE noise is not the sole contributor. The shear layer reattaches on the main element approximately three step heights downstream, where the step corresponds to the separation lip at the main-element cusp, and the step height represents the characteristic depth of the cove. This reattachment is driven by high-speed flow through the flap gap, shortening the TSB relative to canonical backward-facing step flows. Spectral analysis of hot-wire and wall-pressure data reveals dominant mid- and low-frequency peaks associated with shear-layer vortex shedding and TSB breathing. The shedding frequency agrees with predictions of an analytical slat-noise model, despite the absence of strong flow–acoustic coupling. Proper orthogonal decomposition shows that low-frequency breathing drives large-scale oscillations of the stagnation line and reattachment location, likely triggered by Görtler-type centrifugal instabilities.

We study temporal instabilities of an inviscid, compressible, hyperbolic tangent shear layer. For this, we use a newly developed algorithm to avoid spurious modes based on remapping of the physical transverse coordinate using the analytic flow profile and connecting Frobenius-series solutions in this remapped space. The transformation maps the infinitely many singularities of the Pridmore-Brown equation to four singularities. Our algorithm is combined with symmetry methods that facilitate equivalence transformations, enabling the extension of the two-dimensional (2-D) isothermal eigenvalue problem to the 3-D and non-isothermal one. We provide a comprehensive overview of unstable 2-D solutions to the temporal stability problem, revealing Mach-number regions where only the first or second modes exist, or a combination of both. Extending to the 3-D, non-isothermal case shows how this solution space is constructed from the 2-D propagating modes in an isothermal shear layer. We find that fundamental effects, such as the occurrence of the first and second modes, are preserved but adjusted. The consideration of 3-D propagating modes shows that above Mach number $M\approx 2$, all types of modes occur, with some only occurring in a purely 3-D propagation pattern. Above $M\approx 1.3$, the 3-D propagating modes show larger growth rates. Temperature gradients break the fundamental symmetry properties observed in the isothermal case. A temperature gradient enables modes that only existed in the supersonic isothermal case to also exist at subsonic Mach numbers. This work demonstrates how analytical solutions and symmetry methods are combined to solve singular eigenvalue problems, providing an alternative to classical numerical methods.

The properties of the solution to the adjoint two-dimensional boundary layer equations on a flat plate are investigated from the viewpoint of Libby–Fox theory, which describes the algebraic perturbations to the Blasius boundary layer. The adjoint solution is obtained from the Green’s function of the perturbation equation as a sum over the infinite perturbation modes of the Blasius solution. The explicit representation of the adjoint solution allows us to derive constraints on the eigenvalues and eigenfunctions, explicitly compute the adjoint transpose convection term, and evaluate flow sensitivities for shape design, initial-value perturbations and active flow control. The extension of the analysis to the case with non-zero pressure gradient, corresponding to the Falkner–Skan solution, is also briefly discussed.

We investigate the effect of harmonic acoustic forcing on turbulent jets using cyclostationary analysis, a statistical framework that accounts for the phase-dependent second-order statistics excluded by a statistically stationary description. Large-eddy simulations of a Mach 0.4 jet are performed with axisymmetric acoustic forcing at Strouhal $0.3$ and $1.5$, and amplitudes up to $10\,\%$ of the jet velocity. Although forcing generates a strong tonal response and phase-dependent flow structures, its influence on the period-averaged mean flow and turbulence statistics is comparatively modest. Low-frequency forcing modifies the jet globally, whereas high-frequency forcing remains largely confined to the near-nozzle region. Only the strongest forcing produces substantial phase-dependent modulation of the local energy transfer between the mean and turbulent fields, yet the spatially integrated energy transfer varies only weakly over the forcing cycle. Coherent structures at low azimuthal mode order are analysed using cyclostationary spectral proper orthogonal decomposition, which enables the phase-dependent component of the turbulent statistics to be isolated and characterised. The dominant structures retain the wavepacket characteristics of the natural jet but acquire a phase dependence reflecting the periodic modulation of the mean flow. Overall, the results indicate that harmonic forcing modifies the coherent structures primarily through its influence on the underlying mean flow, while introducing only comparatively modest changes to the broadband turbulent dynamics, despite the strong tonal response and substantial local phase-dependent modulation observed at the highest forcing amplitude.

This paper introduces the solubility of heat, a new parameter useful in the modelling of the thermo-mechanical behaviour of two-phase vapour–liquid processes. Unlike other parameters in current use, the solubility of heat is dimensionless and is sensitive to both thermal and mechanical aspects of the process of interest. Large values of the parameter correspond to situations in which the latent heat associated with the generation and condensation of vapour is small so that the system’s behaviour is dominated by mechanical factors. Conversely, smaller values of the parameter indicate a strong effect of the energy cost of phase-change phenomena on the mechanics of the system. These statements are illustrated with a number of examples featuring the thermo-mechanics of vapour cavities in one, two and three spatial dimensions including growth, forced oscillations, condensation and other problems. These examples are based on a generalised formulation of the pertinent equations valid in any number of space dimensions, with the vapour cavity surrounded by a finite or infinite mass of liquid. The general case of a three-dimensional cavity of unspecified shape is also briefly considered.

Eddy resolving simulations are used to describe the mean flow and turbulence structure inside and around a square array of width D containing emerged (i.e., cylinder extending up, or above, the free surface), rigid cylinders of width d. The array is placed at the centre of a long open channel of width W. The study describes how the structure of the wake and shear layers forming along the lateral sides of the array, the wake spectral content and shedding modes, the bleeding velocity, the lengths of the steady and total wake regions, the coherence of the horseshoe vortices forming around the frontal face of the array, the bed shear stresses and the sediment entrainment capacity of the flow inside and outside the array change with the main geometrical parameters defining the array (i.e. the solid volume fraction, SVF, the non-dimensional frontal area per unit volume of the array, $\textit{aD}$, the shape of the cylinders and $d/D$) and the blockage ratio, $W/D$. The effects of varying the same geometrical parameters on the total non-dimensional streamwise drag force acting on the array and on the streamwise distribution of the drag force inside the array are quantified. The mean flow and turbulence inside the array are analysed based on the streamwise distributions of the double-averaged (i.e. over time, D and the flow depth, H) streamwise velocity, $\overline{\overline{U}}$, and turbulent kinetic energy. Contrary to wakes generated behind circular arrays of emerged cylinders, several streamwise-oriented vortices are generated inside the wake of square arrays for a limited range of $\textit{aD}$. These vortices increase flow three-dimensionality and vertical transport behind square arrays. Despite the relatively large values of the array and channel Reynolds numbers (i.e. ReD = 40 000, ReH = 14 000), the mean (time and spanwise-averaged) cylinder drag coefficient, ${C}_{d}^{\prime}$(x), defined with d, H and $\overline{\overline{U}}$ is not close to constant over the whole length of the array in some of the dense array simulations. In these cases, ${C}_{d}^{\prime}$ starts increasing monotonically over the downstream part of the array, as the flow transitions from the quadratic drag regime to the linear drag regime. When plotted versus the cylinder Reynolds number $\overline{\textit{Re}_{d}} = \overline{\overline{U}}d/$ν (ν is the viscosity), ${C}_{d}^{\prime}$ = 2α0/$\overline{\textit{Re}_{d}}$ over the linear drag regime. The parameter α0 and the critical value of $\overline{\textit{Re}_{d}}$ corresponding to the start of the linear drag regime are strongly influenced by the main geometrical parameters of the array and the blockage ratio. Simulation results show that the close-to-constant value of the cylinder drag coefficient characterising the quadratic drag regime, $\overline{{C}_{d}^{\prime}},$ increases with $\textit{SVF}$ and $\textit{aD}$ for constant $d/D$, cylinder shape and $W/D$. Moreover, $\overline{{C}_{d}^{\prime}}$ increases with $d/D$ for constant $\textit{aD}$, cylinder shape and $W/D$, and decreases with $W/D$ for constant $\textit{aD}$, cylinder shape and $d/D$.

In this study, direct numerical simulations were conducted to investigate turbulent channel flow under the influence of a spanwise magnetic field, at a friction Reynolds number $ \textit{Re}_{\tau } = 180$ and Hartmann numbers $\textit{Ha}$ ranging from 0 to 15. The study elucidates how the spanwise magnetic field modulates energy spectra, near-wall streaks and energy transfer. With increasing $\textit{Ha}$, near-wall streamwise streaks elongate markedly and energetic spectral peaks move away from the wall, whereas the spanwise length of streak enlarges only marginally. At $\textit{Ha}=15$, dual spectral peaks appear near the wall, reflecting coexisting high- and low-momentum regions that alternate to form a quasi-steady organisation. Concurrently, an additional very large-scale peak arises in the outer layer ($y\gt 0.3h$), indicating a very large-scale structure that abruptly amplifies the streamwise normal stress there beyond the zero-magnetic-field level. Temporal statistics show that stronger fields reduce streak density but substantially prolong streak lifetimes, attributable to suppression of the nonlinear energy cascade and the attendant decrease in turbulent decay rates. Budget analyses further reveal that, as the field intensifies, the influence of outer-layer large scales on the viscous sublayer weakens; the dominant contributors to wall-normal energy transport shift towards larger scales; and principal energy-transfer pathways migrate away from the near-wall region. Finally, the streamwise elongation of streaks is primarily driven by the wall-normal gradient of the electric potential, while their energetic amplification is effectively damped by Joule dissipation.

We present an experimental and theoretical study of the coiling dynamics of a viscous fluid thread falling on a rotating surface. The experiments conducted for two distinct flow rates explore the rich variety of behaviours governed by three control parameters: the nozzle height above the rotating plate, the radial distance of the point of impact, and the angular velocity of the table. To contextualise the observations, we compare our results with the classical fluid-mechanical sewing machine (FMSM), where a viscous thread falls onto a translating belt moving at constant speed. However, in the rotating system, there is an additional degree of freedom, allowing a systematic investigation of the coupled influence of radial distance and the angular velocity, and manifesting itself in rotational effects. At increasing linear speeds, the prominent ‘stitching’ patterns follow the canonical FMSM sequence – here termed the main sequence – but with radial deformations dependent on the relative directions of coiling and surface rotation. We propose a simplified ‘geometrical’ model that captures these qualitative trends and accurately predicts the critical speeds for pattern transitions within the main sequence. The findings highlight how rotation enriches the dynamics of viscous coiling beyond the classical FMSM behaviour, with multiple new forms of fluid-dynamical stitching.