We investigate the dynamics of a rigid spherical particle in plane Couette flow with a viscosity that varies linearly across the channel, from the Stokes limit to non-zero Reynolds numbers, using analytical theory and immersed-boundary lattice Boltzmann simulations. In the Stokes limit, we find that the particle develops a constant, viscosity-gradient-induced translational slip, defined as the difference between the particle velocity and the undisturbed fluid velocity at its centre. The particle leads the flow when the viscosity increases towards the moving wall and lags when it decreases. Its angular velocity equals half the local, viscosity-gradient-modified fluid vorticity, experiencing no additional rotational slip. Furthermore, we show that cross-stream particle migration is strictly prohibited at any confinement ratio or gradient magnitude. At non-zero Reynolds numbers in a constant-viscosity fluid, increasing the channel Reynolds number beyond a critical value induces a supercritical pitchfork bifurcation that splits the stable centreline equilibrium into two symmetric off-centre equilibria. With a viscosity gradient, we find that for Reynolds numbers below this bifurcation point, the particle consistently settles at a single off-centre equilibrium in the less-viscous half. For Reynolds numbers exceeding this point, however, we identify a critical viscosity gradient threshold: below this limit, the particle can stabilise in either channel half depending on its initial placement; above it, the particle focuses exclusively in the less-viscous half regardless of its initial position. These findings enable predictive particle manipulation in viscosity-stratified microflows.

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.

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.

A pressure disturbance in steady rectilinear translation applied at a capillary–gravity interface radiates waves only above a minimum speed $c_{\textit{min}}$, where it experiences a wave resistance; below this threshold it generates no waves, and the response is merely a localised deformation of the interface around the source, free of wave resistance. Modulating the source amplitude at a frequency $\varOmega$ shifts the forcing to the sideband pair $kV\pm \varOmega$, where $k$ is the wavenumber and $V$ is the translation speed, opening resonance conditions absent from the steady problem; radiation then sets in below the steady threshold $c_{\textit{min}}$. Here, we derive the mean wave resistance $\langle R\rangle$ of a line source in closed form, and show it to be strictly positive at every non-zero speed $V$: the drag-free below-threshold regime disappears for any $\varOmega \neq 0$. Crossing the radiation transitions – where the group velocity matches the source speed – produces integrable resonant peaks in $\langle R\rangle$, whose exponents we derive, including at the cusp where two transition branches merge. Below threshold, where a steadily moving source feels no drag, $\langle R\rangle$ rises from zero as $V\varOmega ^{5}$. Throughout, surface tension is essential to $\langle R\rangle$: because it controls the short, dominant capillary-scale waves, discarding it misrepresents $\langle R\rangle$ over almost the entire speed–frequency plane.

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.

Horizontal convection (HC), the flow in a box heated and cooled from the bottom (or top), is a canonical model of buoyancy-driven transport with direct relevance to subglacial lakes, ocean circulation and climate in general. While HC is commonly characterised by the Rayleigh number $Ra$ and Prandtl number $Pr$, the role of the domain aspect ratio $\varGamma$ (defined as horizontal extension over height) has received less attention. Here, we perform three-dimensional direct numerical simulations over a broad range of Rayleigh numbers and aspect ratios at fixed Prandtl number in order to quantify how geometry affects flow organisation and heat transport. We find that $\varGamma$ strongly influences both the development of the overturning circulation and the scaling of heat transfer (i.e. in dimensionless form, the Nusselt number $Nu$). As the aspect ratio increases, the emergence of an organised overturning flow is shifted to higher Rayleigh numbers. Moreover, with increasing $\varGamma$ the Nusselt number shows a stronger dependence on the Rayleigh number. The results clarify the gradual cross-over from a finite intermediate range compatible with the Rossby-type scaling $\textit{Nu}\sim \textit{Ra}^{1/5}$ (Rossby 1965 Deep-Sea Res. Oceanogr. Abstr. vol. 12, pp. 9–16) towards the $\textit{Nu}\sim \textit{Ra}^{1/4}$ behaviour suggested by Shishkina, Grossmann & Lohse (2016 Geophys. Res. Lett. vol. 43, pp. 1219–1225). It reflects the progressively decreasing vertical confinement of the kinetic boundary layer with increasing aspect ratio. These results show that the aspect ratio is an important control parameter in laminar HC and should be incorporated explicitly into scaling arguments for geophysical flows.

A physics-based model is developed to predict the spatially varying sensitivity of turbulence to roughness geometry and the corresponding flow modifications. Resolvent analysis with a drag-normalized, Reynolds-decomposed volume penalization is formulated to efficiently link the mean flow profile to predictions of the drag force, wake field fluctuations, and dispersive stresses for broadband (sandgrain) surface roughness. An eddy viscosity approach models the modification of the stochastic stress arising due to the attenuation of convecting fluctuations within the roughness sublayer. Crucially, these components link the mean-flow profile to the force profiles and stresses which determine its shape, so that the model can be iteratively solved for a given bulk flow over a surface roughness, starting from an initial corresponding smooth wall flow condition. For engineering-relevant surface geometries in flows which respect outer similarity, the model output estimates turbulent-drag response, namely equivalent sandgrain roughness, Hama roughness-function and spatial variation of roughness sublayer (wake field) turbulent fluctuations. The approach is demonstrated on a channel flow with sandgrain roughness at an engineering-relevant Reynolds number.

Physics-based eddy-viscosity and backscattering closures are widely used for large-eddy simulation (LES) of geophysical turbulence, but their key parameters are often chosen empirically. Here, we develop a semi-analytical framework for estimating these parameters in two-dimensional (2-D) geophysical turbulence by adapting a Lilly-type scaling argument. Specifically, we obtain closed-form estimates, up to an amplitude constant, for the coefficients of the Leith and Smagorinsky eddy-viscosity closures, a biharmonic eddy-viscosity closure and the Jansen–Held backscattering closure with a prescribed backscattering fraction. The amplitude constant appears in the turbulent kinetic energy direct-cascade spectrum and can be diagnosed from a few direct numerical simulation (DNS) or eddy-resolving snapshots. For isotropic cases ($\beta =0$), the diagnosed amplitude constant is consistent with previous theoretical estimates based on closure, renormalisation-group and mode-coupling methods. The resulting semi-analytical parameters closely match the online-learned values obtained using ensemble Kalman inversion across several 2-D geophysical turbulence set-ups (Guan et al. Phys. Rev. Res. vol. 8, 2026, p. 023215). LES using these parameters reproduces key DNS statistics, including the tails of the vorticity distribution, and robustly outperforms dynamic Leith and Smagorinsky baselines.

Minimising bacterial surface adhesion and subsequent biofilm formation in industrial and medical settings requires understanding how bacteria are transported and adhere to complex surface geometries in the presence of non-uniform flow. In this paper, we consider the transport of a dilute suspension of motile bacteria through a corrugated two-dimensional channel with perfectly adhesive walls. Considering flows much faster than the bacterial swimming speed, we asymptotically analyse the diffusive boundary layer that forms using a curvilinear coordinate system based on the fluid streamfunction. We present a similarity solution to the diffusivity-varying diffusion-type equation that arises, from which we derive an analytical expression for the bacterial adhesion rate as a function of surface arclength and the spatially varying wall shear rate. Our model predicts that bacterial adhesion becomes localised on curved surfaces, with bacteria showing preferential adhesion to wall ‘peaks’ at lower shear rates and preferential adhesion to wall ‘valleys’ at higher shear rates. More broadly, our results highlight how spatially varying flows generated by complex geometries can lead to localised bacterial adhesion, with potential implications for both enhancing and minimising biofilm formation.

We study the onset of two-dimensional secondary shear instability (SSI) in the braid regions connecting primary Kelvin–Helmholtz billows in stratified shear flows. While strain induced by the billows stabilises the braids, it also compresses their tilted isopycnals, enhancing baroclinic shear that enables rapid perturbation growth. By modifying the classical analysis of Corcos & Sherman (1976 J. Fluid Mech. 73, 241–264) in braid-aligned coordinates and adding an instability criterion based on the ratio of strain rate to shear, we develop an inviscid, time-dependent model for the braid evolution and the onset of SSI. We show that the criterion for instability can be achieved significantly earlier than the saturation of the primary billow at sufficiently high initial Richardson number $ \textit{Ri}$, because increased stratification slows the rate of billow growth relative to the rate of baroclinic shear production. Two-dimensional direct numerical simulations up to Reynolds numbers $ \textit{Re}=10^7$ quantify the role of viscosity. At high $ \textit{Re}$, we find that SSI indeed develops early in the braid, as predicted by the inviscid model, while the primary billow is still growing and before viscosity slows braid thinning. These results provide a mechanistic explanation for field observations of braid-dominated mixing and suggest that, at geophysically relevant $ \textit{Ri}$ and $ \textit{Re}$, SSI can control the three-dimensional transition and ensuing mixing by preceding and pre-empting both vortex pairing instabilities and secondary convective instabilities in the billow core.

Wall-resolved large-eddy simulations are used to investigate the effects of elevated surface temperature on the leading edge of a swept wing. Earlier stability analyses demonstrate a strong destabilising effect resulting from surface heating, but do not reveal the ultimate effects on transition and relaminarisation. The current results support the findings from stability analysis with elevated surface temperatures, but show a more complex dependency between instability and turbulence than is observed for adiabatic conditions. For attachment-line Reynolds numbers above the adiabatic relaminarisation threshold, instability leads to fully turbulent flow. At Reynolds numbers below the adiabatic relaminarisation threshold, instability leads to finite-amplitude equilibrium states and intermittent turbulent bursting, but not fully turbulent flow.

Integral identities relating the skin-friction coefficients and the Reynolds shear stresses in annular cylindrical flows are derived. In the pressure-driven case, an identity for the radial location of the maximum streamwise velocity is obtained and, in the cylinder-driven case, an identity for the bulk velocity is found. The formulas are used to analyse existing numerical and experimental data and simplify to classical channel-flow and pipe-flow identities in the limiting cases of vanishingly small and infinitely large radius ratios.

When a less viscous fluid displaces a more viscous one in a porous medium, the interface becomes unstable, forming viscous fingers. While viscous fingering has been studied in depth, previous work has focused on the pre-breakthrough dynamics – that is, before the injected fluid reaches any outlet boundary. Here, we investigate post-breakthrough behaviour in a radial micromodel through experiments and pore-network simulations. Post-breakthrough displacement is key for practical applications, as it governs how much of the defending fluid is ultimately recovered. We show that the displacement pattern either ‘freezes’ or continues to grow, depending on the capillary number $ \textit{Ca}$ and viscosity ratio $M$. A pore-network model predicts a sharp drop in growth rate at breakthrough and a geometry-controlled cross-over $ \textit{Ca}_\times$ below which the displacement pattern is arrested. These results highlight the role of domain size, pore geometry, fluid properties and injection rate in regulating the post-breakthrough dynamics and provide a framework for optimising displacement efficiency.

We study by means of direct numerical simulations the influence of the dimensionality of convection on flow properties. We call attention to a few general principles from considering in totality the results from one dimension, two dimensions, three dimensions and four dimensions. In particular, we explore two practical aspects: (i) the transient time, or the amount of time it takes for the flow to reach the steady state; and (ii) possible implications for the so-called ultimate state.

We investigate theoretically the propagation of dendrites along an undercooled solid substrate. Our approach takes into account the heat diffusion in the substrate, and the solid and liquid domains, coupled by the Stefan condition at the solid–liquid interface. Using dimensional and asymptotic analysis, we deduce a simplified model in which diffusion in the liquid and horizontally in the substrate can be neglected. We obtain a self-similar solution consisting of a half-parabolic dendrite shape propagating at constant speed. We derive a scaling law for the maximum velocity, set by the Gibbs–Thomson and kinetic effects, which is in good agreement with experimental results from the literature.

Surface-swimming animals generate capillary-gravity waves whose patterns reflect the unsteady, sinuous nature of their trajectories. For uniform circular motion, cusp formation and wavefront folding have been observed at supercritical speeds. Here, we show that slaloming trajectories at supercritical speeds produce wavefront singularities of higher codimension than previously reported, namely a swallowtail catastrophe. Combining observations of surface-swimming snakes with numerical simulations of unsteady wakes, we identify a swallowtail catastrophe embedded in the wave envelope. This is explained by an integral formulation with a delay term accounting for the timing of wave emission along the trajectory, enabling a description of singularity formation and degeneracy within catastrophe theory. These results provide a general framework for predicting the emergence of caustics and cusps in wavefronts generated by unsteady motions.

Pendant-drop neck contraction under external stretching and moderate finite gravity $(0.1\lt Bo\lesssim 0.5)$ is investigated via theoretical modelling and high-speed photography, focusing on the dynamic surface-tension-balancing mechanism and its transition to the inertial–capillary self-similar regime. A slender-jet model is developed to account for the coupled effects of inertia, viscosity, surface tension and external stretching. From the stretching dynamics and mass conservation, a viscosity-independent scaling $h\sim \tau ^{1/2}$ for the neck radius is derived for $ \textit{We}\sim 1$. Experiments reveal a three-stage contraction: an early balancing regime with $h\sim \tau ^{1/2}$, followed by an inertial–capillary equilibrium stage and finally an inertial–capillary self-similar regime with $h\sim \tau ^{2/3}$. A smaller $ \textit{We}$ prolongs the $\tau ^{1/2}$ regime. Within the present Bond-number range, $ \textit{Bo}$ mainly provides a weak prefactor-level correction to the contraction factor $A_g$, without measurably altering the observed scaling transition. An increasing Ohnesorge number $ \textit{Oh}$ narrows the balancing region and reduces $A_g$ from $0.56$–$0.59$ to $0.43$–$0.46$. Multivariate nonlinear regression yields the dimensionless group $\varPi =We^{0.06}Bo^{-0.01}Oh^{-0.08}$ and the empirical correlation $A_g=0.49\varPi$, with $R^2=0.77$. The contraction factor $A_g$ is weakly enhanced by inertia, reduced by viscosity and only weakly affected by gravity within the moderate finite-gravity range considered here.

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.