The viscoelastic instability of cross-slot flow was first observed experimentally almost half a century ago and reproduced numerically two decades ago, yet its physical origin remains unresolved. We revisit this problem for two-dimensional creeping flow of Oldroyd-B fluid by combining direct numerical simulations, global stability analysis, structural sensitivity analysis and energy-budget analysis. Our simulations reproduce the canonical pitchfork bifurcation, and the stability analysis consistently predicts the threshold and perturbation growth rates. The leading eigenmode consists of a chiral velocity–stress perturbation that tilts and rotates the birefringent strand generated by the extensional flow. Structural sensitivity and energy-budget analyses identify narrow high-extension-rate ridges within the extensional flow as both the spatial core and energetic source of the instability. In these ridges, the stress-based wavemaker colocalizes with large positive disturbance polymeric stress power density, indicating localized transfer of stored elastic energy to the disturbance flow. Analyses of cross-slot variants with rounded corners and with a centred cylinder further reveal that neither sharp corners nor a free central stagnation point is the essential destabilizing ingredient; rather, the instability originates from elastic-energy release in extension-dominated regions characteristic of cross-slot flow.

The mechanisms underlying the generation and evolution of Rayleigh–Taylor instability (RTI) in rarefied gases are an important issue, particularly in many emerging problems where rarefaction effects become significant. Using a graphics processing unit parallelised direct simulation Monte Carlo approach, we simulate RTI across a wide range of Knudsen numbers ($Kn$), and find that instability growth is progressively inhibited as $Kn$ increases, with complete suppression above a critical value $Kn \approx 0.011$. Molecular-scale analysis reveals that rarefaction enhances super-diffusive transport of the molecules, extending the ballistic regime, and favouring the interfacial mixing. This microscale behaviour macroscopically manifests as a rapid decay of the density gradient and the effective Atwood number. To determine whether instability emerges in rarefied conditions, we introduce a stability criterion based on the competition between a diffusive time scale ($\tau _D$) and an instability time scale ($\tau _I$). The transition to instability occurs only when $\tau _I \lt \tau _D$, a condition that fails for $Kn \gtrsim 0.011$. The effect of domain height is interpreted through its influence on the Froude number ($Fr$). Small domains (large $Fr$) hinder the transition from linear to nonlinear growth, while for larger domains, instability suppression is mainly determined by the threshold Knudsen number. The critical threshold $Kn \approx 0.011$ remains consistent for sufficiently small $Fr$. These findings provide a multiscale perspective on flow instability in rarefied gases, and establish a time scale competition framework that quantitatively explains how rarefaction governs the onset of RTI.

The Maxey–Riley–Gatignol equation for the flow around a sphere at low particle Reynolds number tells us that the fluid–particle interaction force decomposes into a contribution from the local flow disturbance caused by the particle’s boundary – consisting of the drag, virtual-mass and history forces, and their Faxén corrections – and another contribution from the stress of the background flow, termed generalized-buoyancy force. There is also a consensus that, for general disperse two-phase flow, the interfacial force density, coupling the average fluid phase and dispersed-phase momentum balances, decomposes in a like manner. However, there has been a long-standing controversy about the physical closure separating the generalized buoyancy from the interfacial force density, especially whether or not pseudostresses, such as the Reynolds stress, should be attributed to the background flow. Furthermore, most existing propositions for this closure involve small-particle approximations. Here, we show that all existing buoyancy closures are inconsistent with particle-resolving numerical simulations and/or at least one of two simple thought experiments designed to determine the roles of pseudostresses and small-particle approximations. We then derive the unique consistent closure. It requires no approximation and implies that all stresses and pseudostresses in the average fluid phase momentum balance, except the Reynolds stress, fully contribute to the background flow responsible for buoyancy. Remarkably, it exhibits a low-pass filter property, attenuating buoyancy at short wavelengths, that prevents it from causing Hadamard instabilities, constituting a first-principles-based solution to the long-standing ill-posedness problem of two-fluid models. When employing the derived closure, even simplistic two-fluid models are linearly well-posed.

We extend the linear stability analysis of electroconvective instability in a quiescent Oldroyd-B viscoelastic fluid subjected to an electric field into the strongly elastic regime, up to Weissenberg numbers ($\textit{Wi}$) of order $10^{3}$. The oscillatory onset is governed by the interplay among destabilising Coulomb forcing, viscous dissipation and elastic stress relaxation. For the quiescent base state, the linearised constitutive relation allows the polymer stress to be represented through a complex effective viscosity. We then apply a single-mode Galerkin projection based on the Newtonian critical eigenfunction, obtaining a reduced dispersion relation and a closed-form prediction for the neutral oscillation frequency. In the limit of large $\textit{Wi}$, this analytical solution yields two asymptotic scaling laws, namely a frequency scaling law $\omega _{c} \propto \sqrt {(1-\beta )/\textit{Wi}}$ ($\omega _{c}$ is the critical frequency and $\beta$ is the solvent to total viscosity ratio) and a threshold scaling law $T_{c} = \beta T_{c}^{N} + a(\beta )/\textit{Wi}$ ($T_{c}$ is the critical electric Rayleigh number, $T_{c}^{N}$ is the Newtonian critical electric Rayleigh number and $a(\beta )$ is the coefficient related to $\beta$), which satisfy a self-consistency relation. Here, $T_{c}$ exhibits a three-stage evolution with $\textit{Wi}$, and a frozen-elasticity limit is identified at large $\textit{Wi}$. With a single geometric constant as input, the values of $\omega _{c}$ and $T_{c}$ obtained from the analytical solution and from high-resolution spectral computations agree to within 0.3 %, providing mutual corroboration.

Self-sustained, flow-induced oscillations of a flexible plate have been identified as a viable mechanism for energy harvesting. However, the efficiency of such devices strongly depends on the onset of aeroelastic instabilities. In this work, we investigate numerically the flutter of a rectangular, three-dimensional, inextensible plate invested by a viscous laminar flow. Through a systematic approach, an unprecedentedly broad range of the parametric space spanned by the mass ratio $M_s$ and bending rigidity $K_B$ is explored. Specifically, three distinct regimes of response can be identified: (i) fixed-point stability, (ii) limit-cycle oscillations and (iii) irregular flutter. We present a high-resolution map of the flapping states in the $M_s$–$K_B$ plane, where the flapping mode of each case is also determined. The instability onset is rationalised through linear stability analysis, accounting for the bending rigidity and the finite aspect ratio of the plate. Across the parameter space investigated, the flapping frequency exhibits only a weak dependence on flapping amplitude and bending rigidity, except within a lock-in regime near the instability threshold at high Reynolds numbers, where it scales with the plate’s natural frequency. In contrast, the flapping frequency is primarily governed by the mass ratio and the flutter mode, with an additional modulation by the Reynolds number. This Reynolds-number dependence is most pronounced at low mass ratios, where different flutter modes emerge as the Reynolds number varies, and progressively weakens with increasing mass ratio. While low-to-moderate mass ratios exhibit predominantly two-dimensional flutter, spanwise modes arise for mass ratios above approximately 2, with amplitudes that increase with both Reynolds number and mass ratio. The optimal flow configurations for energy harvesting are identified in all flow regimes and rationalised by monitoring the effect of $M_s$ and $K_B$ on the corresponding flapping dynamics.

This study identifies a low-frequency, shock-dominated global mode orchestrating thermal choking in confined supersonic flow. The mode exhibits spatial coherence and a bidirectional acoustic–convective feedback loop. Its existence is confirmed – and its inviscid core isolated – through a controlled computational two-dimensional experiment, then validated in fully three-dimensional simulations. Crucially, comparison with one-dimensional Rayleigh flow reveals that choking occurs at a significantly lower heat addition than theoretically predicted, underscoring the dominance of multi-dimensional, mechanically driven choking mechanisms over purely thermal limits. These findings establish a new paradigm for choking transitions and highlight the necessity of multi-dimensional analysis for accurate prediction of supersonic-flow stability.

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.

We perform a long-time Lyapunov analysis of plane Couette turbulence up to friction Reynolds number $h^+=269$. The energy of the leading Lyapunov vector is predominantly concentrated in the buffer layer, with characteristic streamwise and spanwise wavelengths $\lambda _1^+\simeq 100$ and $\lambda _3^+\simeq 50$, respectively. These wavelengths are smaller than the corresponding streamwise and spanwise periods of a minimal flow unit. Through Schmidt–Lyapunov analysis, we find that the first and second Lyapunov exponents are close in value, and that the energy spectra of the first Lyapunov vector and the second Schmidt vector have peaks at similar characteristic wavelengths in the buffer layer. These imply that the fastest orbital instability is not organised by a single minimal-flow-unit-scale perturbation, but by a nearly degenerate set of small-scale buffer-layer perturbation directions, which may be interpreted as a superposition of such perturbations. Coarse graining with a Gaussian low-pass filter further shows that the vortical structures of the local Lyapunov vector tend to appear around near-wall large-scale quasi-streamwise vortices. These results suggest that the orbital instability in wall-bounded turbulence is associated with small-scale buffer-layer perturbations influenced by the strain field around near-wall large-scale structures.

This study combines linear stability analysis (LSA) and direct numerical simulations (DNS) to investigate the coupling mechanisms of rotation, stratification and shear in two flow geometries: spanwise rotating stratified plane Couette flow (between two infinite parallel plates) and sheared annular centrifugal Rayleigh–Bénard convection (between two concentric cylinders). First, LSA reveals a universal scaling law with respect to $(1-\eta )$, where $\eta$ denotes the radius ratio, for the deviation between annular and planar system results, quantitatively establishing the dynamical connection between annular and planar geometries. Building on this, DNS results indicate that the breakdown of the twin-force analogy in the turbulent regime originates from mode competition: buoyancy tends to drive three-dimensional chaotic plumes via baroclinic effects, whereas the Coriolis force tends to sustain quasi-two-dimensional longitudinal vortices. The Prandtl number determines the dominant mechanism in the near-wall region by regulating the relative thicknesses of the thermal and momentum boundary layers, thereby leading to a systematic dissimilarity between heat and momentum transport. Furthermore, this study provides a new theoretical basis for transport prediction in complex flows from the perspectives of enstrophy and mode competition.

This paper concerns the behaviour of reduced models for turbulence–mean-flow interactions, in particular for the modelling of zonostrophic instability, which accounts for the spontaneous generation of zonal flows observed in many planetary and geophysical fluid systems. In modelling, an idealised sea of random waves or turbulent eddies is usually driven by a stochastic body force. However, the stochasticity of the zonal mean flow is often neglected, following an ergodic assumption that the zonal mean is equal to the ensemble mean. In this paper, we examine the stochasticity of the zonal mean flow and its consequences for zonostrophic instability. From large numbers of realisations of an idealised fluid system, we show that the zonal mean flow typically exhibits considerable stochasticity. The actual assumption needed for the theory of zonostrophic instability is a weaker assumption, that the waves and mean flows are statistically decorrelated, and we examine its validity. From the evolution equation for mean flows in stochastic waves, we derive an analytical expression for the weak correlation between waves and mean flow. Results indicate that this correlation becomes stronger and has an impact on growth rates as the instability becomes prominent. We further show that neglecting the correlation will result in an under-estimation of the growth rate, and we derive an improved dispersion relation based on this correlation. Theoretical predictions are verified by numerical simulations.

The stability of a two-degree-of-freedom cube pendulum in an air flow is experimentally investigated at reduced velocities $U^*\gt 20$ and large Reynolds numbers $\textit{Re}\sim 10^5$. Depending on the initial conditions of the cube pendulum at equilibrium (free-stream velocity, yaw and pitch angles), a galloping instability can result in two distinct types of swinging motions: unidirectional cross-flow motions or elliptical motions. An energy-based quasi-steady linear model is developed to analyse the stability of the system and the type of motion resulting from an instability. A good agreement is found between the experimental stability map and the model predictions in situations where the instability develops from rest. The main aerodynamic parameters influencing the stability of the system are identified and linked to the dynamics of flow separation/reattachment on the sides of the cube at the equilibrium state. In addition to the aerodynamic parameters classically highlighted in the Den Hartog criterion regarding cross-flow galloping of bluff bodies, the derivative of the lateral aerodynamic coefficient of the body with respect to its pitch angle is shown to drive the onset of elliptical galloping. Unsteady mechanisms, consisting in delayed dynamics of the flow reattachment line on the sides of the cube, are shown to be involved in the onset of elliptical galloping and highlight the limitations of the quasi-steady approach.

This study experimentally and theoretically investigates the feedthrough between two successive cylindrical water–air interfaces under divergent expansion. Three initial conditions are examined: interfaces initially in-phase and anti-phase, and an initially unperturbed outer interface. Amplitude measurements reveal that the feedthrough depends on the initial perturbation phase difference, thereby altering the instability evolution of both interfaces. A linear theory is developed for cylindrical fluid layers, accounting for surface tension and cylindrically divergent (outward-expanding) geometry, and agrees well with experiments in the near-unity Atwood number regime. Dimensionless feedthrough efficiency factors derived from the linear theory demonstrate that the interface coupling is determined by the radius ratio of two interfaces, interface acceleration, and perturbation wavenumbers. Notably, under extremely high acceleration (as a theoretical limit of our model), the feedthrough is predicted to stabilise both interfaces. This trend, if it persists under compressible conditions, could have implications for inertial confinement fusion. These results provide critical guidance for instability manipulation in both radially convergent and radially divergent geometries.

We report experimental results on the low-frequency dynamics in the wake of a stationary sphere and the corresponding bifurcation. This low frequency is the second frequency appearing past a sphere when the Reynolds number is increased. We measure the velocity field using particle image velocimetry in transverse planes past a sphere placed in a water channel, and obtain the streamwise vorticity. We compute the position of the barycentre of the absolute value of the vorticity and projected the vorticity on azimuthal mode 1. We deduce from periodograms of these signals the Strouhal numbers $\mathit{St}_1$ and ${{\textit{St}}_2}$ associated with the first ($f_1$) and second ($f_2$) frequencies. The frequency ratio $f_2/f_1$ decreases with the Reynolds number, and is close to $2/7$ near the threshold. The two frequencies are generally incommensurate. In the periodograms, many combinations of $f_1$ and $f_2$ are present, in particular $f_1+f_2$ and $f_1-f_2$, consistent with a $T^2$ torus dynamics. We also obtain the squared magnitude associated with $f_1$ and $f_2$. The linear variation of this squared magnitude above the threshold ${\textit{Re}}_{1}$ and the variation of the frequency $f_1$ confirm that this first oscillatory bifurcation is a supercritical Hopf bifurcation. Similar variations for $f_2$ above the corresponding threshold ${\textit{Re}}_{2}$ show that the second oscillatory bifurcation is a supercritical secondary Hopf bifurcation (supercritical Neimark–Sacker bifurcation). The component at $f_2$ of the vorticity field remains significant even $9$ diameters downstream of the sphere.

Absorbing sets are a central organising concept in the long-time analysis of dissipative partial differential equations, especially those arising in fluid mechanics. Roughly, an absorbing set is a bounded region in a suitable function space that eventually contains every trajectory that starts from any bounded set of initial data. Once the dynamics enter this region, they remain controlled in norm, and this ultimate boundedness becomes the gateway to more refined statements: existence of global attractors, finite-dimensional long-time behaviour. A variational framework, based on the background method, is developed to determine an ‘absorbing ball’ in the state space of incompressible shear flows described by the Navier–Stokes equations. This region is defined by the Reynolds–Orr energy identity and is guaranteed to contain all long-term dynamics, including chaotic and non-chaotic attractors. We employ a gradient-based optimisation to find a background flow corresponding to minimal absorbing radius for plane Poiseuille and Couette flows over a range of Reynolds numbers. The optimised background profiles are compared with the turbulent mean flow and they exhibit significantly steeper near-wall gradients than turbulent mean profiles obtained from direct numerical simulations and do not reproduce the universal law of the wall. Despite this quantitative discrepancy, the methodology provides rigorous, provable bounds on the region of state space accessible to the flow dynamics. This offers a novel and promising foundation for improving the global stability limit of the laminar state.

We investigate the effect of the transverse aspect ratio ($B$) of a thermogravitational column (TGC) on the stability of a binary fluid mixture with a negative separation ratio ($\psi$), which generates an adverse vertical density gradient. In extended systems, this configuration is unstable and limits separation. Here we show that confinement fundamentally alters this behaviour and can fully suppress the instability. Combining linear stability analysis of the three-dimensional basic state, three-dimensional simulations and experiments, we identify $B$ as the key control parameter. Linear stability analysis yields a correlation for the critical aspect ratio $B_{\textit{c}}(\psi )$ separating unstable and stable regimes. For $B \lt B_{\textit{c}}$, the system remains in a quiescent state despite the presence of the adverse density stratification. Energy budget analysis shows that confinement enhances transverse diffusive transport, preventing the growth of long-wave instabilities that dominate in extended systems. Direct numerical simulations confirm the predicted transition and reveal the nonlinear evolution towards convection when $B \gt B_{\textit{c}}$. Experiments performed with different mixtures, Tol$|$Ch 0.50$|$0.50, Tol$|$EtOH 0.80$|$0.20 and Tol$|$MeOH 0.8420$|$0.1520, and geometries using two independent $\mu$TGCs ($B$ = 5.88 and $B$ = 1.69) validate the theoretical predictions. These results demonstrate that transverse confinement provides a robust and tunable mechanism for controlling thermogravitational stability.

Highly stretched capillary jets produced by gravity are central to drop generation, micro-thread formation and extensional-rheometry concepts. For Newtonian fluids, the transition from steady jetting to self-excited oscillations in a gravitationally stretched jet is predicted accurately by one-dimensional slender-jet equations that retain the exact interfacial curvature and admit a global eigenvalue analysis (Rubio-Rubio et al. 2013, J. Fluid Mech., vol. 729, pp. 471–483). Separately, weakly viscoelastic jets governed by Oldroyd-B/Giesekus constitutive laws exhibit elastocapillary regimes and beads-on-a-string dynamics that are well captured by one-dimensional free-surface models (Ardekani et al. 2010 J. Fluid Mech., vol. 665, pp. 46–56). Here, we study the global linear stability of a one-dimensional full-curvature model for gravitationally stretched viscoelastic jets in the Oldroyd-B limit. We first benchmark the Newtonian limit, reproducing marginal spectra and base-flow profiles, and then quantify how elasticity shifts the critical jetting–dripping boundary by tracking the leading global Hopf eigenpair across the rheological parametric space. For experimentally relevant moderate elasticity, characterised by order-unity Deborah numbers, polymeric tension modifies both the critical Weber number and the selected oscillation frequency, and endogeneity, i.e. the local contribution of the unperturbed flow dynamics to the selected global eigenvalue, reveals that marginality results from a balance between capillary/kinematic contributions and an additional elastic-stress feedback pathway. To interpret and predict the onset mechanism, we compute wavemakers and receptivity/structural-sensitivity fields from direct–adjoint eigenfunctions, showing that viscoelasticity broadens the sensitivity region downstream while the adjoint remains strongly localised near the inlet, thereby identifying the near-nozzle region as the dominant receptive location.

Turbulent convection is a fundamental transport process that shapes weather and climate, powers flows in planetary interiors and stars, and limits the performance of energy and heat-transfer technologies. Yet how these flows organise simultaneously into large-scale ‘superstructures’ and small-scale near-wall patterns remains unclear. Here, using Rayleigh–Bénard convection with large domain size, we show that both structures are captured by optimal linear modes that maximise transient energy amplification, identified via linear analysis based on turbulent mean profiles. Two amplification regimes emerge with clear scale separation: large-scale modes spanning the full central domain with horizontal wavelengths ${\sim} 6$ times the plate separation and small-scale modes confined near the walls at ${\sim} 11$ times the thermal boundary layer thickness. These results indicate that linear energy amplification plays an important organising role in multiscale turbulent convection and establish a unifying link to analogous non-modal processes in wall-bounded shear turbulence.