This work investigates experimentally and numerically the dynamics of rigid particles with two orthogonal symmetry planes settling under gravity in a highly viscous fluid at a Reynolds number much smaller than one. Joshi & Govindarajan (2025 Phys. Rev. Lett. 134(1), 014002), showed theoretically that for such shapes, the dynamics are qualitatively different for different signs of the product of two rotational–translational mobility coefficients, evaluated with respect to the particle centre of mass in a symmetric reference frame. However, upon examining a particle’s shape, it is not immediately evident if this product is negative, positive or zero. In this paper, we demonstrate how to estimate these coefficients and the sign of their product from experiments, using special initial orientations, and also numerically, based on the Stokes equations. Especially interesting are the ‘settlers’ – such particles that reorient and approach a stationary stable orientation, and we focus our study on this class of shapes. We show experimentally that cones, crescent moons, arrowheads and open flat rings are the settlers, and we evaluate from the experiments their rotational–translational mobility coefficients. Then, we reconstruct each experimental shape as a rigid conglomerate of many touching beads, and use the precise Hydromultipole code to calculate the mobility coefficients for the conglomerate. The numerical and experimental values are close enough to determine that the particles are the settlers, and to estimate the characteristic reorientation time scales. Our findings apply to non-Brownian micro-objects in water-based solutions – experimentally by the similarity principle and theoretically based on the Stokes equations. The reorientation of sedimenting rigid particles to a stationary stable configuration in a relatively short time might be used for environmental, biological, medical or industrial applications.

The nonlinear evolution of free-stream vortical disturbances entrained in the entrance region of a channel is investigated using asymptotic and numerical methods, building on the linear framework developed by Ricco & Alvarenga (2021 J. Fluid Mech., vol. 927, A18). The focus is on low-frequency disturbances that induce streamwise-elongated structures at Reynolds numbers for which the entrance flow is locally stable according to classical linear stability theory. The perturbation flow along the channel entrance is generated by free-stream vortical disturbances located at the channel inlet. These disturbances are symmetric or antisymmetric with respect to the centreplane and their amplitude is sufficiently intense to provoke nonlinear interactions within the channel. The formation and evolution of the perturbation flow are described by the nonlinear unsteady boundary-region equations. Combined with physically realistic initial conditions, the resulting initial-boundary-value problems are solved numerically using a streamwise integration method. A parametric study is conducted to elucidate how the nonlinear channel flow is influenced by the Reynolds number and the inlet-disturbance properties, i.e. the amplitude and the streamwise, wall-normal and spanwise wavelengths. Nonlinearity is found to stabilise the intense algebraic growth and to drive the formation of elongated channel-entrance structures that span the entire cross-section. These structures, characterised by low- and high-speed regions and streamwise vortices, meander along the streamwise direction and persist even when the base flow is fully developed. They exhibit a half-turn rotational symmetry with respect to the vortex centres. These properties emerge downstream regardless of the symmetry of the initial perturbation flow, provided nonlinear interactions are sufficiently intense. The occurrence of travelling waves is detected sufficiently downstream, and their similarity to those found in the fully developed region by other researchers is discussed. Our results show good agreement with theoretical predictions, numerical results and experimental measurements for both the mean flow and the perturbation flow.

Rapid granular free-surface flows on inclined planes can develop secondary vortices aligned with the dominant flow direction. The reason for their formation remains a subject of research, but plausible mechanisms include instabilities driven by (i) dilatation/compressibility, (ii) normal stress differences and (iii) a self-induced Raleigh–Taylor instability caused by segregation of large–dense and small–light particles. In this paper, a set of novel experiments are performed with large and small particles (of the same bulk density), which form longitudinal stripes due to a combination of secondary recirculation and particle-size segregation. A conceptual model is formulated, in which large particles concentrate in the downwelling sections, small particles concentrate in the upwelling sections and a breaking-size-segregation wave separates the two pure phases from one another. In each secondary vortex, the breaking waves allow the large and small particles to continuously recirculate. Assuming that a series of counter-rotating vortices exist, it is shown that this internal cross-slope structure emerges naturally from solving the gravity-shear-driven segregation-advection equations. When viewed from above, this generates a series of alternating bands of large and small particles, that are sharply separated from one another and are aligned with the downslope direction. Each complete stripe (measured from centre to centre of each large band) is formed by two counter-rotating secondary vortices. Despite the apparent order of the steady-state stripes, it is shown that the individual large and small particle paths form complex interpenetrating co-rotating sub-vortices as they avalanche downslope.

We study a reaction–advection–diffusion model of a target–offender–guardian system designed to capture interactions between urban crime and policing. Using Crandall–Rabinowitz bifurcation theory and spectral analysis, we establish rigorous conditions for both steady-state and Hopf bifurcations. These results identify critical thresholds of policing intensity at which spatially uniform equilibria lose stability, leading either to persistent heterogeneous hotspots or oscillatory crime–policing cycles. From a criminological perspective, such thresholds represent tipping points in guardian mobility: once crossed, they can lock neighbourhoods into stable clusters of criminal activity or trigger recurrent waves of hotspot formation. Numerical simulations complement the theory, exhibiting stationary patterns, periodic oscillations and chaotic dynamics. By explicitly incorporating law enforcement as a third interacting component, our framework extends classical two-equation models. It offers new tools for analysing non-linear interactions, bifurcations and pattern formation in multi-agent social systems.

Although stably stratified shear flows, where the base velocity shear is quasi-continuously forced externally, arise in many geophysically and environmentally relevant circumstances, the emergent dynamics of their ensuing statistically steady stratified turbulence is still an open question. We address this phenomenon in a series of three-dimensional direct numerical simulations using spectral element methods. We consider a forced, stably stratified shear flow with an initial bulk Reynolds number $\textit{Re}_{0} = 50$, an initial bulk Richardson number $\textit{Ri}_{0} = 1/80$ (also corresponding to the initial minimum gradient Richardson number $\textit{Ri}_{{g}}$) and a fluid of Prandtl number ${\textit{Pr}} = 1$ in horizontally extended domains. Although the initial configuration is unstable to a primary Kelvin–Helmholtz instability, the ensuing turbulence is sustained by continuously relaxing the resulting flow back towards the initial profiles of streamwise velocity and buoyancy. We study statistical as well as structural aspects of the final statistically steady flows, including the flux coefficient $\varGamma _{\chi }$ and dynamically emergent length scales $\varLambda$ associated with the large-scale dynamics, respectively. Despite the ongoing stirring and mixing, we find that the shear layer half-depth converges to a finite value of $d \approx 8$ (i.e. $\varLambda _{z} \approx 16$) once the horizontal extent of the domain $L_{{h}} \gtrsim 96$. While this implies a final ${{Re}} \approx 400$ and ${Ri} \approx 0.1$, we hypothesise that such forced flows ‘tune’ themselves eventually to a state of a gradient Richardson number $\textit{Ri}_{{g}} \lesssim 0.2$, consistently with several previous studies. Moreover, provided sufficiently extended domains, we observe the emergence of large-scale flow structures with spanwise $\varLambda _{\!y} \approx 50$ and streamwise $\varLambda _{x} \lesssim 115$. Clearly, these observations demonstrate the marked anisotropy of characteristic emergent length scales, even for such ‘weakly stratified’ forced shear flows. We conjecture that the actual emergent streamwise structures are a vestigial ‘imprint’ in the sheared turbulent flow of the primary linear instability of the converged deepened turbulent shear layer.

We study the behaviour of a thin fluid filament (a rivulet) flowing in an air-filled Hele-Shaw cell. Transverse and longitudinal deformations can propagate on this rivulet, although both are linearly attenuated in the parameter range we use. On this seemingly simple system, we impose an external acoustic forcing, homogeneous in space and harmonic in time. When the forcing amplitude exceeds a given threshold, the rivulet responds nonlinearly, adopting a peculiar pattern. We investigate the dance’ of the rivulet both experimentally using spatiotemporal measurements, and theoretically using a model based on depth-averaged Navier–Stokes equations. The instability is due to a three-wave resonant interaction between waves along the rivulet, the resonance condition fixing the pattern wavelength. Although the forcing is additive, the amplification of transverse and longitudinal waves is effectively parametric, being mediated by the linear response of the system to the homogeneous forcing. Our model successfully explains the mode selection and phase-locking between the waves, it notably allows us to predict the frequency dependence of the instability threshold. The dominant spatiotemporal features of the generated pattern are understood through a multiple-scale analysis.

Low Reynolds number hydrodynamic interactions are generally considered both deterministic and reversible due to their linearity. However, the role of soft interactions in deformable suspensions drives nonlinear effects with ambiguous consequences. On the one hand, nonlinearities can be responsible for soft chaos, i.e. long-time apparent randomisation resulting from sensitivity to initial conditions. On the other hand, they can also drive steady streaming and/or drifting effects leading to alignment and ordering. Here, we conduct a comprehensive study on the binary interaction of elastic capsules positioned in different shear planes using high-fidelity particle-resolved simulations. The effects of alignment angle, inter-surface distance, capillary number and size ratio are systematically explored. Based on interaction stability, three regimes are identified: leapfrog, minuet and a novel capturing regime. Unlike leapfrog and minuet motions, where the satellite capsule ultimately escapes from the reference capsule, the capturing motion forms a stable doublet aligned along the vorticity direction. We reveal that capturing is a gentle interaction, which induces only minimal deformation and stress. The mechanism underlying the capturing regime is attributed to the interplay between periodic oscillations induced by the central capsule and steady drift along the vorticity direction. Harmonic analysis of interaction frequencies further underscores the nonlinearity inherent to this dynamics. Extending beyond binary systems, we show that this mechanism relays into ternary alignment, suggesting a generic route to chain formation, demonstrating that nonlinear hydrodynamic interactions alone can drive spontaneous ordering of deformable particles.

Doubly diffusive convection describes the fluid motion driven by the competing buoyancy forces generated by temperature and salinity gradients. While the resulting convective motions usually occupy the entire domain, parameter regions exist where the convection is spatially localised. Although well studied in planar geometries, spatially localised doubly diffusive convection has never been investigated in a spherical shell, a geometry of relevance to astrophysics. In this paper, numerical simulation is used to compute spatially localised solutions of doubly diffusive convection in an axisymmetric spherical shell. Several families of spatially localised solutions, named using variants of the word convecton, are found and their bifurcation diagram computed. The various convectons are distinguished by their symmetry and by whether they are localised at the poles or at the equator. We find that, because the convection rolls that develop in the spherical shell are not straight but curve around the inner sphere, their strength varies with latitude, making the system prone to spatial modulation. As a consequence, spatially periodic states do not form from primary bifurcations and localised states are forced to arise via imperfect bifurcations. While the direct relevance of this work is to doubly diffusive convection, parallels drawn with the Swift–Hohenberg equation suggest a wide applicability to other pattern-forming systems in similar geometries.

The emergence of large-scale spatial modulations of turbulent channel flow, as the Reynolds number is decreased, is addressed numerically using the framework of linear stability analysis. Such modulations are known as the precursors of laminar–turbulent patterns found near the onset of relaminarisation. A synthetic two-dimensional base flow is constructed by adding finite-amplitude streaks to the turbulent mean flow. The streak mode is chosen as the leading resolvent mode from linear response theory. In addition, turbulent fluctuations can be taken into account or not by using a simple Cess eddy viscosity model. The linear stability of the base flow is considered by searching for unstable eigenmodes with wavelengths larger than the base flow streaks. As the streak amplitude is increased in the presence of the turbulent closure, the base flow loses its stability to a large-scale modulation below a critical Reynolds-number value. The structure of the corresponding eigenmode, its critical Reynolds number, its critical angle and its wavelengths are all fully consistent with the onset of turbulent modulations from the literature. The existence of a threshold value of the Reynolds number is directly related to the presence of an eddy viscosity, and is justified using an energy budget. The values of the critical streak amplitudes are discussed in relation with those relevant to turbulent flows.

The Cahn–Hilliard model with reaction terms can lead to situations in which no coarsening is taking place and, in contrast, growth and division of droplets occur which all do not grow larger than a certain size. This phenomenon has been suggested as a model for protocells, and a model based on the modified Cahn–Hilliard equation has been formulated. We introduce this equation and show the existence and uniqueness of solutions. Then, formally matched asymptotic expansions are used to identify a sharp interface limit using a scaling of the reaction term, which becomes singular when the interfacial thickness tends to zero. We compute planar solutions and study their stability under non-planar perturbations. Numerical computations for the suggested model are used to validate the sharp interface asymptotics. In addition, the numerical simulations show that the reaction terms lead to diverse phenomena such as growth and division of droplets in the obtained solutions, as well as the formation of shell-like structures.

Vertical thermal convection exhibits weak turbulence and spatio-temporally chaotic behaviour. For this configuration, we report seven new equilibria and 26 new periodic orbits. These orbits, together with four previously studied in Zheng et al. (J. Fluid Mech., 2024b, vol. 1000, p. A29) bring the number of periodic-orbit branches computed so far to 30, all solutions to the fully nonlinear three-dimensional Navier–Stokes equations. These new and unstable invariant solutions capture intricate spatio-temporal flow patterns including straight, oblique, wavy, skewed and distorted convection rolls, as well as bursts and defects. These interesting and important fluid mechanical processes in a small flow unit are shown to also appear locally and instantaneously in a chaotic simulation in a large domain. Most of the solution branches show rich spatial and/or spatio-temporal symmetries. The bifurcation-theoretic organisation of these solutions is discussed; the bifurcation scenarios include Hopf, pitchfork, saddle-node, period-doubling, period-halving, global homoclinic and heteroclinic bifurcations, as well as isolas. Furthermore, these orbits are shown to be able to reconstruct statistically the core part of the attractor, so that these results may contribute to a quantitative description of transitional fluid turbulence using periodic orbit theory.

Interfaces subjected to strong time-periodic horizontal accelerations exhibit striking patterns known as frozen waves. In this study, we experimentally and numerically investigate the formation of such structures in immiscible fluids under high-frequency forcing. In the inertial regime – characterised by large Reynolds and Weber numbers, where viscous and surface tension effects become negligible – we demonstrate that the amplitude of frozen waves scales proportionally with the square of the forcing velocity. These results are consistent with vibro-equilibria theory and extend the theoretical framework proposed by Gréa & Briard (2019 Phys. Rev. Fluids 4, 064608) to immiscible fluids with large density contrasts. Furthermore, we examine the influence of both Reynolds and Weber numbers, not only in the onset of secondary Faraday instabilities – which drive the transition of frozen wave patterns toward a homogenised turbulent state – but also in selecting the dominant wavelength in the final saturated regime.

Understanding the interplay between thermal, elastic and hydrodynamic effects is crucial for a variety of applications, including the design of soft materials and microfluidic systems. Motivated by these applications, we investigate the emergence of natural convection in a fluid layer that is supported from below by a rigid surface, and covered from above by a thin elastic sheet. The sheet is laterally compressed and is maintained at a constant temperature lower than that of the rigid surface. We show that for very stiff sheets, and below a certain magnitude of the lateral compression, the system behaves as if the fluid were confined between two rigid walls, where the emergent flow exhibits a periodic structure of vortices with a typical length scale proportional to the depth of the fluid, similar to patterns observed in Rayleigh–Bénard convection. However, for more compliant sheets, and above a certain threshold of the lateral compression, a new local minimum appears in the stability diagram, with a corresponding wavenumber that depends solely on the bending modulus of the sheet and the specific weight of the fluid, as in wrinkling instability of thin sheets. The emergent flow field in this region synchronises with the wrinkle pattern. We investigate the exchange of stabilities between these two solutions, and construct a stability diagram of the system.

The constant temperature and constant heat flux thermal boundary conditions, both developing distinct flow patterns, represent limiting cases of ideally conducting and insulating plates in Rayleigh–Bénard convection flows, respectively. This study bridges the gap in between, using a conjugate heat transfer (CHT) set-up and studying finite thermal diffusivity ratios $\kappa _s \! / \! \kappa _f$ to better represent real-life conditions in experiments. A three-dimensional Rayleigh–Bénard convection configuration including two fluid-confining plates is studied via direct numerical simulations given a Prandtl number ${Pr}=1$. The fluid layer of height $H$ and horizontal extension $L$ obeys no-slip boundary conditions at the two solid–fluid interfaces and an aspect ratio of ${\Gamma }=L/H=30$ while the relative thickness of each plate is ${\Gamma _s}=H_s/H=15$. The entire domain is laterally periodic. Here, different $\kappa _s \! / \! \kappa _f$ are investigated for moderate Rayleigh numbers $Ra=\left \{ 10^4, 10^5 \right \}$. We observe a gradual shift of the size of the characteristic flow patterns and their induced heat and mass transfer as $\kappa _s \! / \! \kappa _f$ is varied, suggesting a relation between the recently studied turbulent superstructures and supergranules for constant temperature and constant heat flux boundary conditions, respectively. Performing a linear stability analysis for this CHT configuration confirms these observations theoretically while extending previous studies by investigating the impact of a varying solid plate thickness $\Gamma _s$. Moreover, we study the impact of $\kappa _s \! / \! \kappa _f$ on both the thermal and viscous boundary layers. Given the prevalence of finite $\kappa _s \! / \! \kappa _f$ in nature, this work is a starting point to extend our understanding of pattern formation in geo- and astrophysical convection flows.

Volcanic fissure eruptions typically start with the opening of a linear fissure that erupts along its entire length, following which, activity localises to one or more isolated vents within a few hours or days. Localisation is important because it influences the spatiotemporal evolution of the hazard posed by the eruption. Previous work has proposed that localisation can arise through a thermoviscous fingering instability driven by the strongly temperature dependent viscosity of the rising magma. Here, we explore how thermoviscous localisation is influenced by the irregular geometry of natural volcanic fissures. We model the pressure-driven flow of a viscous fluid with temperature-dependent viscosity through a narrow fissure with either sinusoidal or randomised deviations from a uniform width. We identify steady states, determine their stability and quantify the degree of flow enhancement associated with localised flow. We find that, even for relatively modest variations of the fissure width (${\lt } 10$ %), the non-planar geometry supports strongly localised steady states, in which the wider parts of the fissure host faster, hotter flow, and the narrower parts of the fissure host slower, cooler flow. This geometrically driven localisation differs from the spontaneous thermoviscous fingering observed in planar geometries and can strongly impact the localisation process. We delineate the regions of parameter space under which geometrically driven localisation is significant, showing that it is a viable mechanism for the observed localisation under conditions typical of basaltic eruptions, and that it has the potential to dominate the effects of spontaneous thermoviscous fingering in these cases.

A linear stability model based on a phase-field method is established to study the formation of ripples on the ice surface. The pattern on horizontal ice surfaces, e.g. glaciers and frozen lakes, is found to be originating from a gravity-driven instability by studying ice–water–air flows with a range of water and ice thicknesses. Contrary to gravity, surface tension and viscosity act to suppress the instability. The results demonstrate that a larger value of either water thickness or ice thickness corresponds to a longer dominant wavelength of the pattern, and a favourable wavelength of 90 mm is predicted, in agreement with observations from nature. Furthermore, the profiles of the most unstable perturbations are found to be with two peaks at the ice–water and water–air interfaces whose ratio decreases exponentially with the water thickness and wavenumber.