This study reveals the presence of frequency components lower than the wake vortex-shedding frequency within the classical Mode A. Primaries are one-third of the vortex-shedding frequency and frequencies corresponding to recirculation bubble pumping, as previously studied. However, when the spanwise domain size $L_z$ in numerical simulations is sufficiently large, their interaction relationship becomes obscured. To clarify the interaction relationship, we introduce a process in which distinct frequency components gradually emerge by starting with a small spanwise domain of $3.3D$ and then increasing it to $4.7D$, where $D$ represents the diameter of the cylinder. At $L_z \leqslant 3.5D$, only the vortex-shedding frequency harmonics are present. One-third of the vortex-shedding frequency component appeared in $L_z \geqslant 3.6D$. Bispectral mode decomposition and energy transfer analysis reveal that the difference interaction between the one-third-shedding frequency and the vortex-shedding frequency component transfers energy to another low-frequency component. The recirculation bubble pumping is evident in the flow fields $L_z \geqslant 3.8D$. The frequency components after this emergence are not only the harmonics of the lowest-frequency component, and the periodic nature is disrupted, which is marked as a quasi-periodic state. Nonlinear interactions between the lowest-frequency component corresponding to recirculation bubble pumping, primary frequency components such as wake vortex shedding, and approximately one-third of the vortex-shedding frequency complicate the temporal behaviour of the flow field. Utilising the constraint of the spanwise domain size, our approach effectively reveals the interaction relationship among frequency components inherent in a flow field with several coherent spectral components.

We present a theoretical framework for modelling a plane-strain hydraulic fracture propagating in a poroelastic rock in the toughness-dominated regime. The formulation explicitly incorporates two-dimensional (2-D) pore-pressure diffusion, thereby generalising the classical Carter leak-off model, which can be interpreted as the limiting case of one-dimensional (1-D) diffusion. The poroelastic response is captured by superposing pore pressure and backstress contributions from a spatial and temporal distribution of instantaneous point sources along the extending fracture. A scaling analysis reveals the existence of a class of large-time, self-similar solutions for which the fracture length grows as $\ell \sim t^{1/2}$, with a prefactor function of a dimensionless injection rate $\mathcal{I}$ and a poroelastic stress coefficient $\eta$. The injection rate $\mathcal{I}$ emerges as the dominant controlling parameter. Asymptotic analysis provides large-time closed-form solutions in the limits of both large and small $\mathcal{I}$, which show excellent agreement with full numerical simulations. For large $\mathcal{I}$, diffusion reduces to 1-D and the solution converges to the classical toughness- and leak-off-dominated solution governed by Carter’s law. For small $\mathcal{I}$, fracture growth is instead controlled by pseudo-steady (2-D) diffusion. The transition from 2-D to 1-D diffusion is characterised by an increase in the fracture length prefactor and a reduction in leak-off. The poroelastic coefficient $\eta$ acts to shorten and narrow the fracture while increasing both leak-off and driving pressure. This framework delineates the transition between 2-D and 1-D diffusion and establishes quantitative conditions under which Carter’s law remains valid in the large-time limit.

We propose a novel stability criterion for incompressible shear flows by combining input–output analysis and the small gain theorem. The criterion yields an explicit threshold on the magnitude of velocity perturbations about a given base flow that guarantees stability. If this threshold is crossed – either due to non-modal growth, exponential growth or a bypass transition scenario – our analysis predicts a loss of stability that may lead to transition to turbulence. We consider three approximated models for nonlinearity: unstructured, structured with non-repeated blocks and structured with repeated blocks. We show that the imposed threshold obtained by these three methods complies with a hierarchical relationship, where the unstructured case is the most conservative, imposing the lowest bound on disturbance magnitude. We apply this approach to three canonical and well-studied base flows: Couette, plane Poiseuille and Blasius. For these three base flows, we compare our results with experiments, direct numerical simulation results, non-modal nonlinear stability results and linear stability theory (LST). In the limit of infinitesimally small perturbation magnitude, our stability criterion for the unstructured case recovers the results of LST. For finite perturbations, the structured cases that account for nonlinear interactions provided stability thresholds that are consistent with experimental observations and simulation results of transition at both subcritical and post-critical Reynolds numbers for the considered base flows in our study. In particular, we utilise our stability criterion to demonstrate that Couette flow can become unstable and transition can be triggered at different Reynolds numbers, which is consistent with past experimental observations.

We present a three-dimensional numerical study of the splashing dynamics of non-spherical droplets impacting a quiescent liquid film, covering a wide range of aspect ratios ($A_r$) and Weber numbers ($ \textit{We}$). The simulations reveal distinct impact dynamics, such as spreading, splashing type-1, splashing type-2 and canopy formation, which are delineated in a regime map constructed in the $A_r$–$ \textit{We}$ parameter space. Our results demonstrate that droplet morphology during the impact significantly influences crown evolution and splash initiation, with oblate drops promoting finger growth and fragmentation due to enhanced rim deceleration, while prolate drops tend to form canopies. We observe that the hole instability, which becomes more prominent at higher Weber numbers, arises from lamella rupture in the thinnest region of the film, located just beneath the crown rim. A linear stability analysis, supplemented by the temporal evolution of the crown obtained from the numerical simulations, adequately predicts the number of fingers formed along the crown rim by accounting for both Rayleigh–Plateau (RP) and Rayleigh–Taylor (RT) instabilities. The theoretical analysis demonstrates the dominant role of the RP instability in determining the number and wavelength of early undulations, with the RT instability serving to amplify the growth rate of the disturbances. Our findings highlight the critical role of the droplet shape in splash dynamics, which is relevant to a range of applications involving droplet impact.

The current work analyses the onset characteristics of buoyancy and thermocapillary-driven instabilities in two-layer binary fluid systems near their upper critical solution temperature (UCST). To account for the non-trivial thickness of the fluids’ interface and the temperature-dependent solubility in such regimes, the present analysis utilises the phase-field approach with a modified free-energy expression. The spatial discretisation of the field variables is carried out here using the spectral collocation approach with a suitable grid mapping strategy to accurately evaluate the field gradients around the diffuse-interface region. The results reveal that in the case of pure buoyancy-driven (Rayleigh–Bénard) convection, the parametric range for oscillatory onset is found to shrink when the system approaches the UCST, as the increased solubility results in less favourable conditions for oscillatory onset. The marginal stability curves of different fluid combinations considered here exhibit unique drift patterns based on their thermo-physical and transport properties. For systems with added thermocapillarity effects (Rayleigh–Bénard–Marangoni convection), the changing solubilities and the interfacial thickness, like the interfacial tension, exhibit a dual role that results in system-specific expansion/shrinkage of the parametric space for oscillatory flow onset.

The description of riblets and other drag-reducing devices has long used the concept of longitudinal and transverse protrusion heights, both as a means to predict the drag reduction itself and as equivalent boundary conditions to simplify numerical simulations by transferring the effect of riblets onto a flat virtual boundary. The limitation of this idea is that it stems from a first-order approximation in the riblet-size parameter $s^+$, and as a consequence it cannot predict other than a linear dependence of drag reduction upon $s^+$; in other words, the initial slope of the drag-reduction curve. Here the concept is extended to a full asymptotic expansion using matched asymptotics, which consistently provides higher-order protrusion coefficients and higher-order equivalent boundary conditions on a virtual flat surface. While the majority of this expansion, though nonlinear in $s^+$, remains linear in velocity, and therefore we shall not directly address the shape of the drag-reduction curve, this procedure will also allow us to explore the way nonlinearities of the Navier–Stokes equations first enter the $s^+$ expansion, with somewhat surprising negative results.

The accuracy obtained with computational fluid dynamics and process simulations of flotation critically depends on the quality and robustness of the underlying models for the non-resolved subprocesses. An important issue in flotation is the collision between particles and air bubbles. Many models have been developed, but their accuracy for applications in flotation is limited. In particular, the significant size difference between particles and bubbles and their intricate coupling to the turbulent flow field pose severe challenges. The present paper first reviews presently employed collision models, highlighting their advantages and disadvantages when applied to flotation. On this basis, the `integrated multisize collision model’ (IMSC) is proposed. After a detailed evaluation, it combines existing approaches from various sources and introduces new developments designed to address present shortcomings. The model is validated by own direct numerical simulation data as well as data from the literature. It is shown that, overall, the IMSC provides better predictions for the collision rate in typical flotation conditions than presently employed collision models and covers the entire parameter range of the flotation process very well. Using the available data, some of the underlying modelling assumptions are validated. Finally, a comprehensive overview of the model is provided for further use in Euler–Euler frameworks or process simulations also beyond flotation.

Results of an experimental study investigating issues of Coriolis effects on the fluid dynamics associated with vortex rings propagating through a rotating fluid are presented. The vortex rings are generated at the top of a large, water-filled rotating tank and they propagate downwards along the axis of rotation. The motion and the decay of the rings in a rotating fluid are expected to be accompanied by an inertial-wave field being established in the fluid surrounding the rings. However, the existence of this inertial-wave field had previously never been verified experimentally. Particle image velocimetry measurements were performed with the goal of demonstrating the existence of the inertial-wave field. Datasets were processed to extract individual inertial-wave modes and, for the first time, experimentally construct the dispersion relation for the inertial waves associated with vortex rings. For rotation rates when Coriolis forces dominate the dynamics, the experimental data are found to be in very good agreement with the well-established theoretical dispersion relation for inertial waves. The generation of inertial waves implies that kinetic energy is radiated away from the vortex rings. First results relating to the redistribution process of the kinetic energy are briefly discussed.

Slow viscous flow around a fixed body generates a shape-dependent drag. We explore the drag-minimising shapes of bodies centred between two parallel plates in two-dimensional viscous flow. The channel width introduces a length scale so that the optimal profile is area-dependent. We solve the shape optimisation problem numerically over a wide range of areas. We also compute the optimal elliptical shapes and this identifies how these shapes should be slightly altered to reduce the drag with reductions of up to $3.8\,\%$ attained at high areas. More broadly, we derive two properties of general optimal shapes within the confined flow: the magnitude of the surface vorticity is approximately (but not exactly) constant and the noses have sharp angles that are independent of area. For relatively small bodies, the optimal shape becomes identical to that in an unconfined geometry, but the drag is qualitatively different owing to the influence of confinement; within a channel, it is proportional to the inverse of the logarithm of the body area. At relatively large areas, the optimal body becomes long and its surface is approximately parallel to the channel boundaries, except in the vicinity of the noses. Using a lubrication approximation, we recast the optimisation problem as an Euler–Lagrange equation that is solved to determine the drag-minimising shape, finding that the drag is proportional to the body area in this regime.

Sustainable Development Goal 6.1 seeks universal access to safe drinking water for all by 2030, yet persistent disparities remain even in high-income countries. Indigenous, remote and small communities are disproportionately affected by poor drinking water quality, but comparable evidence to evaluate performance across communities is very limited due to inconsistent monitoring and reporting. To this end, we constructed a community-level meta-panel dataset of 839 communities (4,137 observations) across 4 Australian jurisdictions (Northern Territory, South Australia, Victoria and Western Australia) and Ontario, Canada, over the period 2018–2022. Drinking water quality was assessed using the Australian Drinking Water Guidelines and Canadian Boil Water Advisories. Logistic regression was employed to estimate the probability of accessing good-quality drinking water, with Indigenous status, remoteness, population size and socio-economic condition as key explanatory variables. Results reveal systematic disparities: Indigenous and very remote communities are statistically significantly less likely to have good-quality drinking water than non-Indigenous and regional communities after controlling for other factors. Our findings indicate that structural inequities – rather than geographic or demographic variation alone – are critical determinants of poor drinking water outcomes in small, Indigenous communities in both Australia and Canada.