The vertical, tip-to-tip arrangement of neighbouring caudal fins, common in densely packed fish schools, has received much less attention than staggered or side-by-side pairings. We explore this configuration using a canonical system of two trapezoidal panels (aspect ratio ${\textit{AR}}=1.2$) that pitch about their leading edges while heaving harmonically at a Strouhal number $St=0.45$ and a reduced frequency $k=2.09$. Direct numerical simulations based on an immersed-boundary method are conducted over a Reynolds-number range of $600\leq {\textit{Re}}\leq 1\times 10^{4}$, and complementary water-channel experiments extend this range to $1\times 10^{4} \leq {\textit{Re}}\leq 3\times 10^{4}$. Results indicate that when the panels oscillate in phase at a non-dimensional vertical spacing $H/c\leq 1.0$ with $c$ denoting the panel chord length, the cycle-averaged thrust of each panel rises by up to 14.5 % relative to an isolated panel; the enhancement decreases monotonically as the spacing increases. Anti-phase motion instead lowers the power consumption by up to 6 %, with only a modest thrust penalty, providing an alternative interaction regime. Flow visualisation shows that in-phase kinematics accelerate the stream between the panels, intensifying the adjacent leading-edge vortices. Downstream, the initially separate vortex rings merge into a single, larger ring that is strongly compressed in the spanwise direction; this wake compression correlates with the measured thrust gain. The interaction mechanism and its quantitative benefits persist throughout the entire numerical and experimental Reynolds-number sweep, indicating weak ${\textit{Re}}$-sensitivity within $600\leq {\textit{Re}}\leq 3\times 10^{4}$, and across multi-panel systems. These results provide the first three-dimensional characterisation of tip-to-tip flapping-panel interactions, establish scaling trends with spacing and phase, and offer a reference data set for reduced-order models of vertically stacked propulsors.

This paper considers the problem of water wave scattering by a rectangular anisotropic elastic plate mounted on the ocean surface, with either free, clamped or simply supported edges. The problem is obtained as an expansion over the dry modes of the elastic plate, which are computed using a Rayleigh–Ritz method. In turn, the component diffraction and radiation problems are solved by formulating a boundary integral equation and solving numerically using a constant panel method. The results are presented to highlight the resonant responses of the plate under different forcing scenarios. In particular, we illustrate how the excitation of certain modes can be forbidden due to symmetry.

Many species of fish, as well as biorobotic underwater vehicles (BUVs), employ body–caudal fin (BCF) propulsion, in which a wave-like body motion culminates in high-amplitude caudal fin oscillations to generate thrust. This study uses high-fidelity simulations of a mackerel-inspired caudal fin swimmer across a wide range of Reynolds and Strouhal numbers to analyse the relationship between swimming kinematics and hydrodynamic forces. Central to this work is the derivation and use of a model for the leading-edge vortex (LEV) on the caudal fin. This vortex dominates the thrust production from the fin and the LEV model forms the basis for the derivation of scaling laws grounded in flow physics. Scaling laws are derived for thrust, power, efficiency, cost-of-transport and swimming speed, and are parametrised using data from high-fidelity simulations. These laws are validated against published simulation and experimental data, revealing several new kinematic and morphometric parameters that critically influence hydrodynamic performance. The results provide a mechanistic framework for understanding thrust generation, optimising swimming performance, and assessing the effects of scale and morphology in aquatic locomotion of both fish and BUVs.

Dense granular flows exhibit pronounced non-local behaviours, particularly in creeping regions and shear-localised zones, which challenges classical local inertial rheologies. In this work, we develop a continuum framework for dense granular flows by extending the $\mu (I)$ rheology through the inclusion of granular temperature as an explicit state variable, thereby establishing a direct link between grain-scale velocity fluctuations and macroscopic stresses, and enabling the representation of non-local effects. The model is implemented within a finite-volume computational framework, and systematically validated against three canonical configurations spanning steady and transient regimes: heap flows, split-bottom Couette flows, and granular column collapse. Across these benchmarks, the formulation captures key non-local features observed experimentally and numerically, including sustained creeping below yield, shear-band broadening and migration, and the transient evolution of free surfaces and runout dynamics. Overall, the granular-temperature-extended $\mu (I)$ rheology provides a unified continuum description that reconciles local and non-local behaviour in dense granular flows, retains the predictive capability of inertial rheology in rapid regimes, and extends its applicability to creeping and shear-localised flows. The proposed framework offers a physically interpretable and scalable basis for modelling granular processes in both geophysical and industrial contexts.

A classical and central problem in the theory of water waves is to classify parameter regimes for which non-trivial solitary waves exist. In the two-dimensional, irrotational, pure gravity case, the Froude number $ \textit{Fr}$ (a non-dimensional wave speed) plays the central role. So far, the best analytical result $ \textit{Fr} \lt \sqrt {2}$ was obtained by Starr (1947 J. Mar. Res., vol. 6, pp. 175–193), while the numerical evidence of Longuet-Higgins & Fenton (1974 Proc. A, vol. 340, pp. 471–493) states $ \textit{Fr} \leq 1.294$. On the other hand, as shown recently by Kozlov (2023 On the first bifurcation of Stokes waves), the hypothetical upper bound $ \textit{Fr} \lt 1.399$ is related to the existence of subharmonic bifurcations of Stokes waves. In this paper, we develop a new strategy and rigorously establish the improved upper bound $ \textit{Fr} \lt 1.3451$, which is the first rigorous improvement of Starr’s bound. In this process, we establish several new inequalities for the relative horizontal velocity, which are of separate interest and for which we delicately make use of the bound on the slope of the surface profile established by Amick (1987 Arch. Ration. Mech. Anal., vol. 99, pp. 91–114). As an application we show that the velocity at the bottom below the crest of any solitary wave does not exceed $47\,\%$ of the propagation speed.

While studying soap film bursting to validate their opening velocity, i.e. the Taylor–Culick velocity, Mysels and co-workers discovered fifty years ago a compression region propagating in front of the hole that they called the aureole. In the wake of such a discovery, a series of papers ‘Bursting of soap films’ focused on the study of such peculiar Marangoni flow resulting from the rapid surfactant compression. Their pioneering theory postulates that surfactants remain insoluble at the interface, leading to a self-similar process that has been verified on small films. In the present study, by using films large enough to allow the surfactant to relax, we reveal a previously unexplored regime of aureole development. The surfactants forming the aureole initially behave as if they were insoluble, with an aureole front propagating at a constant speed. After a few milliseconds, however, the front slows down until it matches the hole-opening velocity, and the aureole length then becomes constant. In this steady regime, a model taking into account surfactant advection/diffusion in the film is developed. Our theory accurately captures the thickness and velocity exponential profiles observed in experiments, demonstrating that the observed deviations arise from a balance between the surfactant rapid compression and a desorption flux. Furthermore, measurements of the characteristic aureole lengths provide estimates of physico-chemical properties of the monolayer, which are discussed in the light of predictions based on adsorption laws. The present study highlights the transition from the insoluble limit to the soluble limit, and paves the way for measurement of out-of-equilibrium dynamics of surfactants.

We develop a weakly nonlinear model of duct acoustics in two and three dimensions (without flow). The work extends the previous work of McTavish & Brambley (2019 J. Fluid Mech., vol. 875, pp. 411–447) to three dimensions and significantly improves the numerical efficiency. The model allows for general curvature and width variation in two-dimensional ducts, and general curvature and torsion with radial width variation in three-dimensional ducts. The equations of gas dynamics are perturbed and expanded to second order, allowing for wave steepening and the formation of weak shocks. The resulting equations are then expanded temporally in a Fourier series and spatially in terms of straight-duct modes, and a multi-modal method is applied, resulting in an infinite set of coupled ordinary differential equations for the modal coefficients. A linear matrix admittance and its weakly nonlinear generalisation to a tensor convolution are first solved throughout the duct, and then used to solve for the acoustic pressures and velocities. The admittance is useful in its own right, as it encodes the acoustic and weakly nonlinear properties of the duct independently from the specific wave source used. After validation, a number of numerical examples are presented that compare two- and three-dimensional results, the effects of torsion, curvature and width variation, acoustic leakage due to curvature and nonlinearity and the variation in effective duct length of a curved duct due to varying the acoustic amplitude. The model has potential future applications to sound in brass instruments. Matlab source code is provided in the supplementary material.

Free-surface cusps are a generic feature of externally driven, viscous flow bounded by a free surface, in that their form is stable under small perturbations. Here we present an alternative to the boundary integral description found recently (J. Eggers, Phys. Rev. Fluids, vol. 8, 2023, 124001), which is based directly on a local analysis of the Stokes equation. The new description has the advantage of greater simplicity and transparency, allowing us to understand the connections with bifurcation theory, as well as with other physical systems displaying similar singularities. To illustrate this, we construct cusp solutions corresponding to higher-order singularities, as well as time-dependent solutions.

We develop, simulate and extend an initial proposition by Chaves et al. (J. Stat. Phys., vol. 113, no. 5-6, 2003, pp. 643–692) concerning a random incompressible vector field able to reproduce key ingredients of three-dimensional turbulence in both space and time. In this paper we focus on the important underlying Gaussian framework. Presently, the statistical spatial structure of this velocity field is consistent with a divergence-free fractional Gaussian vector field that encodes all known properties of homogeneous and isotropic fluid turbulence at a given finite Reynolds number, up to second-order statistics. The temporal structure of the velocity field is introduced through a stochastic evolution of the respective Fourier modes. In the simplest picture, Fourier modes evolve according to an Ornstein–Uhlenbeck process, where the characteristic time scale depends on the wave-vector amplitude. For consistency with direct numerical simulations (DNS) of the Navier–Stokes equations, this time scale is inversely proportional to the wave-vector amplitude. As a consequence, the characteristic velocity that governs the eddies is independent of their size and is related to the velocity standard deviation, which is consistent with some features of the so-called sweeping effect. To ensure differentiability in time while respecting the Markovian nature of the evolution, we use the methodology developed by Viggiano et al. (J. Fluid Mech., vol. 900, 2020, A27) to propose a fully consistent stochastic picture. We finally derive analytically all statistical quantities in a continuous set-up and develop precise and efficient numerical schemes of the corresponding periodic framework. Both exact predictions and numerical estimations of the model are compared with DNS provided by the Johns Hopkins database.

A data-driven algorithm is proposed that employs sparse data from velocity and/or scalar sensors to forecast the future evolution of three-dimensional turbulent flows. The algorithm combines time-delayed embedding together with Koopman theory and linear optimal estimation theory. It consists of three steps: dimensionality reduction, currently with proper orthogonal decomposition (POD); construction of a linear dynamical system for current and future POD coefficients; and system closure using sparse sensor measurements. In essence, the algorithm establishes a mapping from current sparse data to the future state of the dominant structures of the flow over a specified time window. The method is scalable (i.e. applicable to very large systems), physically interpretable and provides sequential forecasting on a sliding time window of prespecified length. It is applied to the turbulent recirculating flow over a surface-mounted cube (with more than $10^8$ degrees of freedom) and is able to forecast accurately the future evolution of the most dominant structures over a time window at least two orders of magnitude larger that the (estimated) Lyapunov time scale of the flow. Most importantly, increasing the size of the forecasting window only slightly reduces the accuracy of the estimated future states.

This study utilises chromocapillary stresses induced by light-actuated photosurfactants to demonstrate theoretically that a stable uniform liquid layer wetting a substrate can be sculpted and stirred on the microscale. A mathematical model is presented for two photosurfactant species that can switch from trans to cis states. Switching takes place in the bulk and on the interface, and convection–diffusion–reaction equations describe the local concentrations there. Under uniform light illumination (e.g. blue light) the equilibrium concentrations of trans and cis are non-uniform with layer depth, and a quiescent state with a flat interface exists. A non-uniform light intensity along the layer is superimposed to drive the system out of equilibrium, and induce interfacial deformations and flow in the bulk. This is carried out asymptotically for small-intensity non-uniformities and the first-order non-uniform solutions are found in semi-analytic form. The solutions show that a local increase in intensity increases the surface tension locally by sweeping surfactant off the interface to generate an inward trapping flow (known as a ‘Marangoni tweezer’ in experiments). Light intensities with a sinusoidal variation along the interface are also considered to show that vortical mixing motions are set up. Additionally, the liquid sculpting problem is analysed and a class of inverse problems are solved to predict the distribution of the light intensity required to produce a desired target interfacial shape. Finally, a parametric study is carried out to evaluate the effect of Biot, Damköhler and Marangoni numbers on the maximum light-induced interfacial velocity.

Slender fibres, including textile-derived microplastics, are abundant in aquatic environments and often extend beyond the Kolmogorov length scale. While breakup at dissipative scales has been characterised by velocity-gradient statistics, no closure existed for inertial-range spans where eddy turnover sets the clock. Here we develop a turbulence-informed kinetic theory of fibre fragmentation bridging turbulence forcing and slender-beam mechanics. First, we derive a load-to-curvature mapping showing that spanwise forcing generates peak bending moments scaling as $\sim U_L L^2$, with $U_L$ the velocity increment across fibre length $L$. Second, we construct a breakup hazard $h(L)$ from curvature-threshold exceedances over eddy-time blocks, which identifies a turbulence-defined critical span $\ell _c$. For $L\gt \ell _c$, breakup is eddy-time-limited, $h(L)=O(\bar \varepsilon ^{1/3}L^{-2/3})$ with $\bar \varepsilon$ the mean turbulent energy dissipation rate, whereas for $L\lt \ell _c$, it is a rare-event process with $h(L)\propto L^{5/3+\alpha }$, $\alpha$ denoting the small correction from intermittency. Embedding this hazard in a self-similar binary kernel yields a closed population-balance equation for the fragment distribution $n(L,t)$ with sources and sinks. The framework produces explicit predictions: intermittency-corrected curvature scalings, critical spans set by material and flow parameters, start-up and halving times linked to surf-zone conditions and scaling profiles in the cascade. The steady-state bulk distribution on the subcritical branch, with vertical removal induced by horizontal convergence, follows $n(L)\propto L^{-8/3-\alpha }\simeq L^{-2.7}$, in striking agreement with the mean slope $\simeq -2.68$ observed for environmental microfibres in recent surveys. The reported variability of slopes is naturally explained in our framework by the coexistence of supercritical and subcritical branches together with $L$-dependent removal-driven sinks.

Turbulent flows, despite their apparent randomness, exhibit coherent structures that underpin their dynamics. Proper orthogonal decomposition (POD) has been widely used to extract these structures from experimental data. Periodic features such as vortex shedding can appear as POD mode pairs in strongly periodic flows, but detecting propagating structures in more complex flows is challenging. Hilbert proper orthogonal decomposition (HPOD) addresses this by applying POD to the analytic signal of the turbulent fluctuations, which yields complex modes with a $\pi /2$ phase shift between the real and imaginary components. These modes capture propagating structures effectively but introduce spectral leakage from the Hilbert transform used to derive the analytic signal. The current work investigates the relationship between the modes of the POD and those of the HPOD on the velocity fluctuations in the wake of a sphere. By comparing their outputs, POD mode pairs that correspond to the same propagating structures revealed by HPOD are identified. Furthermore, this study explores whether computing the analytic signal of the POD modes can replicate the HPOD modes, offering a more computationally efficient method for determining the pairs of POD modes that represent propagating structures. The results show that the pairs of POD modes identified by the HPOD can be determined more efficiently using the Hilbert transform directly on the POD modes. This method enhances the interpretive power of POD, enabling more detailed analysis of the turbulent dynamics without the need to compute the analytic signal of the entire turbulent fluctuation data.