The superlinear scaling relationship between the hydrodynamic dispersion coefficient and the Péclet number in porous media has been widely acknowledged. Nevertheless, the mechanisms driving this behaviour remain inadequately understood. In this work, we investigate the mechanism responsible for this superlinear scaling using a Lagrangian framework that combines a statistical model, which links the global probability density function of tracer transition time to flow variability in porous media, with a continuous time random walk framework. Our analysis reveals that the intra-pore and inter-pore flow variabilities are the primary sources responsible for the superlinear scaling, with their relative significance characterised by a structure-specific parameter, $\chi$. Specifically, the inter-pore flow variability dominates when $\chi \gt 1$, while the intra-pore variability prevails for $0\lt \chi \lt 1$. The parameter $\chi$ is derived exclusively from the statistical distributions of pore-throat radius, length and orientation angle, which can be readily obtained from structural characterisation techniques such as X-ray computed tomography imaging. These theoretical predictions are validated through extensive numerical simulations on tube networks with substantial structural variation. This study resolves discrepancies in previous studies regarding the mechanisms of superlinear scaling in hydrodynamic dispersion and offers valuable insights into modulate dispersion and mixing in porous media.

The grazing of livestock on arid and hyper-arid rangelands is the dominant land use across much of the Arabian Peninsula, and is a human activity of large historical, cultural, and economical importance. In this perspective, we outline the historical trajectory of pastoralism in the Arabian Peninsula, examine its transformation over the past century, and propose a framework for sustaining this practice amid contemporary environmental and social pressures. Our context is the seven countries that make up the Peninsula, but with an emphasis on the Kingdom of Saudi Arabia, given its size and geopolitical influence. A substantial part of Arabian rangelands is severely degraded due to overgrazing by goats, sheep and camels. This degradation has been accompanied by declines in customary pastoral and social practices such as al hima, which allowed rangelands to be grazed and managed sustainability for more than a thousand years. Major transformation in current pastoral practices is needed to ensure that Arabian rangelands continue to sustain people and ecosystems into the next century. An important need is to build capacity and capability in organisations and individuals charged with managing rangelands. This requires the establishment of programmes that include, but are not limited to, a greater investment in the science and management of pastoralism, including the need to develop and promote systems to monitor environmental change under different grazing practices. Rangeland management and conservation programmes should also aim to incorporate elements of traditional pastoral systems such as al hima, managed by local herders, to promote a more sustainable utilisation of rangeland resources. Together, it is hoped that these actions will create a core group of local scientists working on local solutions, with a reduced reliance on international experts. Finally, we advocate for a greater strengthening of communal governance of rangeland resources. This should include greater support for pastoral groups to ensure that land management is socially grounded, and co-investment in conservation and pastoralism. By linking science, tradition, and local empowerment, the Arabian Peninsula has the potential to lead a new era of sustainable pastoralism in the world’s arid and hyper-arid rangelands.

Sinking marine snow particles, composed primarily of organic matter, control the global export of photosynthetically fixed carbon from the ocean surface to depth. The fate of sedimenting particles is partly regulated by their encounters with suspended objects, which leads to mass accretion and potentially alters their buoyancy, and with bacteria that can colonise the particles and degrade them. Their collision rates are typically calculated using two types of models focusing either on direct (ballistic) interception with a finite interaction range, or advective-diffusive capture with zero interaction range. Yet, since many relevant marine encounter scenarios span across both regimes, quantifying such encounters remains challenging because the two models yield asymptotically different predictions at high Péclet numbers. We reconcile the two approaches by quantifying encounters in the general case using theoretical analysis and simulations. By solving the advection-diffusion equation in Stokes flow around a sphere to model mass transfer to a sinking particle by finite-sized objects, we determine a new formula for the Sherwood number as a function of the Péclet number and the ratio of particle sizes. Contrary to the common assumption, we find that diffusion still plays a significant role in generating encounters even at high Péclet numbers. We predict that at Péclet numbers as high as 106 the direct interception model underestimates the encounter rate by up to two orders of magnitude. This overlooked contribution of diffusion to encounters suggests that processes affecting the fate of marine snow may proceed at a rate much higher than previously thought.

Bubble dynamics constitutes a fundamental scientific problem in fluid mechanics. Although the oscillation can be predicted through theories for bubble dynamics in previous studies, the viscous effects on the bubble migration remains difficult to predict accurately. In this study, we establish a theoretical model for bubble migration across the entire cycle. The theoretical model derives a drag coefficient expression under dynamic Reynolds numbers, and incorporates corrections to account for non-spherical bubble dynamics. A key advance is the capability to account for viscous drag without relying on constant empirical drag coefficients. Validation against experimental results demonstrates that the theoretical model effectively predicts the bubble migration. Furthermore, we discuss the correlation between drag coefficient and Reynolds number, and elucidate the effects of viscous domain range and bubble deformation on the drag coefficient of the present model.

Linear and weakly nonlinear stability analyses are carried out to understand the influence of anisotropic slip on the instability and transition characteristics of pressure-driven parallel flow in the fluid overlying a porous medium. The slip is induced on the upper plate dominating in the streamwise direction. The investigation is made by imposing Navier slip on the classical model considered by Aleria et al. (SIAM J. App. Math., vol. 84, 2024, pp. 433–463). For finite-amplitude disturbances, a weakly nonlinear stability analysis based on the cubic-Landau theory is exploited. The bifurcation phenomena are investigated as a function of slip length at the critical instability point (CIP) and as a function of Reynolds number away from the CIP. The linear stability analysis shows that Squire’s theorem does not hold for anisotropic slip, and the mode of instability along the neutral curve is sensitive to slip length. Along the instability boundary, slip stabilises (destabilises) the porous mode (odd-fluid mode), whereas in the even-fluid mode, slip can have either a stabilising or destabilising effect. When the porous mode or odd-fluid mode dominates the flow instability, only the supercritical bifurcation exists at and away from the CIP. For each value of the depth ratio, there exists a finite interval of slip parameter in which the three-dimensional disturbances are least stable and the critical mode of instability is the even-fluid mode. Both the subcritical and supercritical bifurcations are possible for the even-fluid mode of instability and the supercritical bifurcation at the CIP always shifts to a subcritical bifurcation away from the CIP. The nonlinear kinetic energy analysis reveals that modifications in energy due to gradient production and viscous dissipation are mainly responsible for inducing the subcritical instability. The role of spanwise slip, Darcy number, porosity and Beavers–Joseph coefficient is also investigated. The results demonstrates a stabilising (destabilising) impact of spanwise slip (porosity and Beavers–Joseph coefficient), and instability as well as bifurcation characteristics is a function of ${\sqrt {\textit{Da}}}/{\hat{d}}$, rather than individual Darcy number ($Da$) and depth ratio ($\hat{d}$). Overall, this study finds a significant relationship among the critical modes of instability, dimension of the least stable disturbances, bifurcation phenomena and skin-friction coefficient. The present results also witness good experimental support for the stability of flow in the fluid overlying a porous medium and slippery flow in a single-fluid layer configuration.

The settling dynamics of fractal aggregates in constant-density environments and through miscible density interfaces are investigated via particle-resolved direct numerical simulations, which provide the settling velocity as a function of the fractal dimension, Galileo number, and particle and fluid densities. In a fluid of uniform density the settling velocity increases with the fractal dimension and the Galileo number. This behaviour is captured by an empirical relationship that holds over a broad range of parameter values. In the presence of a miscible density interface, consistent with earlier observations we observe that lighter fluid is carried into the denser layer by the aggregate’s pore spaces, which we quantify based on the concept of $\alpha$-shapes. This causes the aggregate to slow down, until the lighter pore fluid is replaced by the denser fluid via a combination of diffusion and convection. The degree of the aggregate’s slowdown depends on the ratio of the density differences between the aggregate and the two fluids, and it can again be captured by an empirical relationship. The duration of the slowdown is determined by the pore fluid replacement time, which in turn depends on the relative importance of convection and diffusion, and hence on the aggregate’s geometry. A relationship is derived that captures the dependence of this replacement time on the shape of the aggregate, the ratio of the density differences, and the Galileo number.

Miniature vortex generators (MVGs) are a promising passive flow control technique for viscous drag reduction by producing large-scale vortical motions that manipulate turbulence structures in turbulent boundary layers (TBLs) without significant device drag. This study conducts hot-wire anemometry experiments to investigate the influence of the Reynolds number and the ratio of MVG height $h$ to TBL thickness $\delta _0$ (MVG height ratio $h/\delta _0$) on turbulence structures. Experiments encompass two MVG height ratios, $h/\delta _0=0.09,\;0.18$, friction Reynolds numbers ranging from ${\textit{Re}}_\tau =400$ to 2000 and measure the velocity information at various downstream stations. Spectral analysis confirms the MVG-induced vortices amplify large-scale structures in the outer region, sustaining up to 100 times the MVG height downstream. The MVGs are also found to attenuate turbulence energy across a wide range of turbulence structures below the amplification location in the logarithmic region, connected with the MVG-induced spanwise motion. Increasing the friction Reynolds number from ${\textit{Re}}_\tau =400$ to $900$ or doubling the MVG height ratio causes the amplified structures to develop into longer motions and move away from the wall, while increasing the turbulence energy attenuation proportion to the log region. Moreover, the energy attenuation amplitude of large-scale structures in the near-wall region increases with a larger MVG height ratio but decreases with increasing Reynolds numbers. The findings indicate that, at friction Reynolds numbers ${\textit{Re}}_\tau \geqslant 900$, MVGs induce spanwise motions that attenuate near-wall structures and modulate large-scale outer motions. The present configuration does not yield a global viscous drag reduction, but the turbulence modulation trends suggest the potential for viscous drag reduction when the MVG configurations are optimised to enhance favourable buffer-layer spanwise motions.

Entropically driven fluid–solid transitions in monodisperse, purely repulsive hard spheres (MPRHS) are well established in theory, simulation and experiment for atomic and colloidal systems. For MPRHS, however, coexistence is usually located via bulk free-energy calculations; the underlying microscopic balance between configurational and vibrational entropy is left implicit. Frenkel clarified this mechanism explicitly as an exchange of long-range configurational entropy for short-range vibrational entropy, but in the pristine MPRHS limit the nucleation barrier near coexistence is so high that phase separation is predicted only on astronomical time scales. Consistent with this, even unbiased simulations do not show spontaneous, equilibrium fluid–crystal coexistence; transient mixtures are mostly overtaken by a single phase; observed coexistence is still algorithmically driven. Nearly hard-sphere colloid experiments do observe fluid–crystal coexistence, but always in the presence of unavoidable triggers such as gravity and walls. We treat the hard-sphere phase diagram as settled and ask how the entropic exchange mechanism can be revealed in nearly hard-sphere colloidal simulations. We probe the mechanism on finite time scales by introducing minimal perturbations that trigger phase separation: small reductions in hardness that increase locally accessible free volume (and thus gently increase vibrational entropy), and 2 %–4 % distributed crystal seeds. These perturbations produce coexisting fluid and crystal domains with crystal fraction, phase envelope and osmotic pressure that, with systematically increasing particle hardness, approach the hard-sphere limit. These results demonstrate that slight enhancements to vibrational entropy provide a dynamically accessible route to realising the long-range/short-range entropy exchange required for phase separation.

Since the early 1990s, numerous theoretical methods have been proposed to predict Mach stem height in steady supersonic shock reflections by assembling sub-models for local flow structures, including incident/reflected shocks, the triple point, the slipline, and Mach stem curvature. We constructed an updated model and employed it as a benchmark to evaluate the performance of various sub-models corresponding to typical flow regions. The results show that the curved assumption for the free part of the slipline outperforms the straight-line approximation, considering the differences in regions after the reflected shock can improve the predictive accuracy, while using compatibility relations in the interactive part of the slipline is superior to the wave reflection model and better captures the linear slope of Mach stem height with wedge trailing edge height. Nevertheless, prediction errors in the slope and systematic biases in the overall Mach stem height prediction persist. To address these shortcomings, we developed a calibrated scaling law for the coefficient of a linear Mach stem model. Grounded in asymptotic reasoning and high-fidelity numerical simulations, this law yields a compact, easy-to-implement expression that achieves substantially higher accuracy than existing analytical composite models across the full parameter space. It retains well-established limiting cases, clarifies how inadequate sub-modelling degrades prediction accuracy, and provides uncertainty estimates for practical engineering applications.

The behaviour of near-inertial waves (NIWs) in baroclinic currents is investigated, with a focus on wave trapping and critical layer dynamics. We present theoretical analysis, supported by numerical simulations, of the Sawyer–Eliassen equation, which describes waves in the plane perpendicular to an arbitrary balanced background flow. Gradients of the background velocity and buoyancy field modulate the wave properties, in particular defining the range of frequencies for which the Sawyer–Eliassen equation is hyperbolic, i.e. wave-like. Variations in the lower limit of this range, the local minimum frequency, lead to the trapping of low-frequency, typically sub-inertial, waves through either total internal reflection or the formation of critical layers. Both mechanisms are studied. We introduce a local coordinate rotation that not only elucidates these dynamics by simplifying the governing equation, but also allows us, through direct analogy, to draw upon theory and intuition developed for barotropic problems. In the majority of physically relevant cases, the transformed coordinates are aligned with and perpendicular to isopycnals, and are thus easily utilised. Employing the coordinate transformation, we consider along-isopycnal modes, and study the behaviour of waves approaching a critical level without the need for a full ray-tracing approximation. Finally, we find qualitative differences in the critical layers that form in strongly and weakly baroclinic flows, most notably in their location. In the weakly baroclinic case, NIW energy accumulates about the minimum in the relative vorticity of the background flow, whereas in the strongly baroclinic case, we find slantwise critical layers concentrated in fronts.