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.

We investigate the influence of side-wall wetting on the linear stability of falling liquid films confined in the spanwise direction. A biglobal stability framework is developed, capturing inertia, viscosity, gravity, capillarity and geometric confinement. The base flow exhibits a curved meniscus and a streamwise velocity overshoot near the side walls. Linear stability analysis based on the Navier–Stokes equations is performed in two limiting regimes. In confined channels, where spanwise confinement stabilises moderate-wavenumber perturbations via side-wall boundary layers, wetting weakens this stabilisation; as the contact angle decreases, the neutral curves shift towards the unconfined one-dimensional limit, thus wetting acts as a relative destabilising mechanism. In contrast, in weakly confined channels where side-wall boundary layers do not provide confinement-induced stabilisation, wetting produces a net long-wave stabilisation ($k \rightarrow 0$), significantly increasing the critical Reynolds number. This effect strengthens as the contact angle decreases, indicating a competition between destabilising inertia and stabilising wetting-induced capillary forces. The predicted long-wave stabilisation effect is compared quantitatively with available experimental measurements, showing consistent trends and comparable magnitudes within the accessible parameter range. Perturbation eigenmode structures show that, in confined channels, the relative destabilisation is associated with near-wall vortical structures induced by the meniscus elevation and velocity overshoot, which reduce effective viscous damping. In contrast, in weakly confined channels, stabilisation is consistent with interface tensioning through strong anchoring of the perturbations at the side walls.

We investigate the incompressible flow inside a two-dimensional square cavity, driven by the sliding motion of its four lids, all at the same speed and with facing lids moving in opposite directions. The problem has three symmetries: two mirror symmetries with respect to the diagonals and a $\pi$ rotation invariance about the centre of the cavity. The base flow, a steady state that has all three symmetries, is the unique solution at sufficiently low values of the Reynolds number ($ \textit{Re}$) and acts as a global attractor. At higher $ \textit{Re}$, it has become unstable and shares the phase space with a globally attracting space–time symmetric periodic orbit that, in addition to the rotational invariance, is also invariant under evolution over half a period followed by reflection about either of the diagonals. In between, a wealth of solution branches and intervening bifurcations mediate the transition process. In particular, a pair of steady states that break the mirror symmetries but are mirror-symmetry images of each other regulate the appearance and disappearance of a second space–time symmetric periodic orbit and a pair of asymmetric periodic orbits that are also mirror images of each other. The catalogue of instabilities includes both local (two pitchfork, two Hopf, a saddle-node and a cyclic fold) and global (two heteroclinic and one homoclinic) bifurcations. The sequence of transitions is explained in terms of a one-dimensional path through the parameter space of a codimension-four bifurcation: the double zero bifurcation with Z$_2$ symmetry and degeneracy of the third order terms.

We give evidence of non-modal amplification mechanisms driven by swirl intensity from a bi-global linear analysis of a cold swirling flow representative of a premixed swirl burner: non-uniform, compressible, turbulent, enclosed and subject to vortex breakdown passed the expansion. The monolithic computational approach embeds a realistic axisymmetric swirler model in the computational domain. The amplification mechanisms are identified by stability and resolvent analysis under variations of the length of the annular duct section and combustion chamber, the swirl intensity and the swirler position. While the spectrum is affected by changes in the length only, the gain of the resolvent strongly depends on the swirl intensity. The results suggest an acoustically dominated amplification in the combustion chamber and a non-modal hydrodynamic-dominated process driven by the swirl intensity. Inertial waves carrying swirl fluctuations play a key role in the latter. The results are complemented by a resolvent sensitivity analysis that identifies the tip of the inner recirculation region and the surrounding shear layer as a wavemaker region that drives at high swirl numbers the non-modal amplification. The sensitivity of that region also enables the transfer of azimuthal momentum perturbations to axial momentum, hence activating a longitudinal acoustic resonance from azimuthal fluctuations.

Spatial linear instability analysis is employed to investigate the instability of a viscoelastic liquid jet in a co-flowing gas stream. The theoretical model incorporates a non-uniform axial base profile represented by a hyperbolic tangent, capturing the shear layer. The Oldroyd-B model discretised with Chebyshev polynomials is employed, and energy budget analysis is used to interpret underlying mechanisms. At low Weber numbers, the jet evolves axisymmetrically and the instability is governed by interfacial gas-pressure fluctuations; as the Weber number increases, the growing inertia drives a transition of the predominant mode from axisymmetric to helical. At weak elasticity, the instability is also primarily governed by gas-pressure fluctuations. As elasticity increases, the predominant mode transitions from axisymmetric to helical. This transition is accompanied by a migration of disturbance structures from the interface toward the jet interior and an enhanced coupling between velocity perturbation and the basic flow. These trends reveal a new predominant instability mechanism – the elasticity-enhanced shear-driven instability – which is distinct from capillary or Kelvin–Helmholtz instabilities in Newtonian jets. A $\textit{We}$–$El$ phase diagram delineates the boundary between predominant modes and experimental results obtained in a flow-focusing configuration validate the theoretical predictions. Compared with temporal stability results, the spatial framework – by directly resolving the convective downstream amplification of disturbances – achieves quantitative agreement with experiments and highlights the superiority of spatial instability analysis in capturing the dynamics of strongly convective, non-parallel jet flows. These findings provide mechanistic insight into viscoelastic jet instabilities and offer guidance for applications involving droplet and fibre formation in co-flow systems.