This note presents a new supplement for Euripides, Heracles 767.

The quasi-steady shock refraction at a diffusive air–SF$_6$ interface (fast–slow type) is investigated numerically and theoretically. A new refraction pattern where both shock and expansion waves are simultaneously present in the reflected waves (named RRR-E) is first observed at the diffusive interface. The new refraction pattern is a regular pattern that is not expected to occur in classical shock refraction at a sharp fast–slow interface. Through the shock polar method, continuous refraction processes occur within the diffusion layer to satisfy the kinematic relationship between the reflected wave and the transmitted shock, which results in the RRR-E formation. Subsequently, the conditions for the RRR-E occurrence are obtained theoretically and verified numerically. In the phase diagram of the refraction patterns, the presence of RRR-E results in the transition boundaries of different refraction patterns at the sharp fast–slow interface no longer being valid. Specifically, the appearance of RRR-E delays the Mach reflection refraction (MRR) process, which is of great significance for the design of scramjet engines.

Brain Health Services are second-generation memory clinics that aim to reduce the risk of progression to dementia in at-risk individuals. We describe the rationale for such a service, and comment on its novel implementation by Venkataraman and colleagues that integrates digital technologies and biomarker testing. We describe the advantages and possible limitations of such an approach, then investigate areas for further work – namely, the need to account for multiple pathologies in biomarker testing and to formulate standards for genetic counselling.

Reflecting on the civil claim filed in France under the French Duty of Vigilance law (LdV) by members of the Union Hidalgo community in Mexico against the energy company Electricité de France (EDF), this article explores interactions between human rights due diligence in renewable energy projects. The lawsuit is one of the first cases brought under the LdV, and the first case claiming violations of Indigenous rights. The rights violations experienced by the community—the lack of free, prior and informed consent and violence against human rights defenders—epitomize the reality of harmful corporate tactics in the energy and extractive industries. Whereas the LdV enshrines a process through which communities affected by harmful corporate practices can access transnational legal avenues for redress, inconsistencies and ambiguities within the law call into question its ability to effectively regulate the human rights activities of French corporations involved in renewable energy projects.

Geophysical and astrophysical fluid flows are typically driven by buoyancy and strongly constrained at large scales by planetary rotation. Rapidly rotating Rayleigh–Bénard convection (RRRBC) provides a paradigm for experiments and direct numerical simulations (DNS) of such flows, but the accessible parameter space remains restricted to moderately fast rotation rates (Ekman numbers ${ {Ek}} \gtrsim 10^{-8}$), while realistic ${Ek}$ for geo- and astrophysical applications are orders of magnitude smaller. On the other hand, previously derived reduced equations of motion describing the leading-order behaviour in the limit of very rapid rotation ($ {Ek}\to 0$) cannot capture finite rotation effects, and the physically most relevant part of parameter space with small but finite ${Ek}$ has remained elusive. Here, we employ the rescaled rapidly rotating incompressible Navier–Stokes equations (RRRiNSE) – a reformulation of the Navier–Stokes–Boussinesq equations informed by the scalings valid for ${Ek}\to 0$, recently introduced by Julien et al. (2024) – to provide full DNS of RRRBC at unprecedented rotation strengths down to $ {Ek}=10^{-15}$ and below, revealing the disappearance of cyclone–anticyclone asymmetry at previously unattainable Ekman numbers (${Ek}\approx 10^{-9}$). We also identify an overshoot in the heat transport as ${Ek}$ is varied at fixed $\widetilde { {Ra}} \equiv {Ra}{Ek}^{4/3}$, where $Ra$ is the Rayleigh number, associated with dissipation due to ageostrophic motions in the boundary layers. The simulations validate theoretical predictions based on thermal boundary layer theory for RRRBC and show that the solutions of RRRiNSE agree with the reduced equations at very small ${Ek}$. These results represent a first foray into the vast, largely unexplored parameter space of very rapidly rotating convection rendered accessible by RRRiNSE.

This paper addresses the links between alchemy, mining and religion in the early modern alchemical iconography of the German mining theorist Martin Sturtz. He created a special version of the prayer Vaterunser (Our Father) for miners and metallurgists and used the images from Christian and alchemical iconography to demonstrate the history of the creation of metals and how they grow inside the Earth. In one of his images, the Saviour is crucified against the background of coloured strips or stripes, symbolizing the seven metals inside a sedimentary complex. In another miniature, the blood spilling from his wound gives rise to ore veins in which metals ripen. These and the other iconographical and textual examples from Sturtz’s treatises show the attempt to connect the mining process with alchemical theory and Christian religion. This paper will shed light on the origins of this unusual alchemical imagery and its connection with other alchemical and theological works, especially those by Paul Lautensack.

Previous studies on the scaling of pressure fluctuations in wall-bounded turbulent flows have typically employed the same frameworks as those used for mean flow, with inner scaling based on frictional velocity and viscous length scales, and outer scaling relying on boundary layer thickness or displacement thickness. These traditional scales primarily reflect the characteristics of the mean streamwise velocity profile and momentum balance. In this work, we propose novel scaling frameworks for pressure fluctuations in turbulent channel and pipe flows, derived from the Poisson equation for pressure fluctuations. Applying the scaling patch approach, we analyse the rapid and slow terms in the Poisson equation, and introduce new scaling for pressure fluctuation variance in both the inner and outer regions. These new scales are designed to better capture the influence of Reynolds stresses by incorporating their peak values. Additionally, we establish a strong correlation between the root mean square (r.m.s.) of pressure fluctuations and the Reynolds shear stress, resulting in an empirical equation that accurately predicts their ratio. This equation provides a practical method for estimating the r.m.s. of pressure fluctuations in the flow, which remains challenging to measure in experimental investigations.

The scramble to extract critical energy transition minerals creates risk of widespread negative human rights impacts. A just transition in the extraction of critical minerals must involve deep examination of the mine-community interface to gain a better understanding of the drivers of successful engagement between mining companies and communities. Drawing on fieldwork in South America’s lithium triangle, this paper finds that the nature of the corporate-community relationship is increasingly key to enabling a just transition whereby communities participate in the benefits of extraction with negative impacts mitigated. It establishes that key success factors are related to empowerment of Indigenous communities and have the potential to maximise positive outcomes for communities in the context of lithium extraction. Governments and companies must embed a more bottom-up process with an end goal of communities themselves defining the parameters of what a just transition means in the critical minerals context.

In this paper, we discuss the transport of sediment and the formation of bedforms in turbulent river flows, under flow conditions typical of flooding events. Through the implementation of an immersed boundary method, a wall model and a morphological model, we were able to simulate complex and mobile geometries under high Reynolds numbers at an affordable computational cost. In particular, we examined the evolution of bedforms on a loose sediment bed under turbulent flow conditions, using input parameters obtained from laboratory measurements. Over time, the bedforms become more three-dimensional and irregular in shape, leading to changes in the shear layer, crest angle and separation patterns. The bedforms continue to evolve until a quasi-steady equilibrium is reached. Our simulations highlight the crucial role played by the small-scale bedforms, which significantly affect the flow dynamics: an increase in the total drag is observed, related to the form drag generated by the local recirculation and the increased size of the large-scale recirculation bubble. Furthermore, a stronger turbulent activity ensues from the shear layers forming on the crests of the small-scale bedforms. Finally, a wider shedding angle of the shear layer is caused by the irregular crest line.

Motivated by microfluidic applications, we investigate drag reduction in laminar pressure-driven flows in channels with streamwise-periodic superhydrophobic surfaces (SHSs) contaminated with soluble surfactant. We develop a model in the long-wave and weak-diffusion limit, where the streamwise SHS period is large compared with the channel height and the Péclet number is large. Using asymptotic and numerical techniques, we determine the influence of surfactant on drag reduction in terms of the relative strength of advection, diffusion, Marangoni effects and bulk–surface exchange. In scenarios with strong exchange, the drag reduction exhibits a complex dependence on the thickness of the bulk-concentration boundary layer and surfactant strength. Strong Marangoni effects immobilise the interface through a linear surfactant distribution, whereas weak Marangoni effects yield a quasi-stagnant cap. The quasi-stagnant cap has an intricate structure with an upstream slip region, followed by intermediate inner regions and a quasi-stagnant region that is mediated by weak bulk diffusion. The quasi-stagnant region differs from the immobile region of a classical stagnant cap, observed for instance in surfactant-laden air bubbles in water, by displaying weak slip. As exchange weakens, the bulk and interface decouple: the surfactant distribution is linear when the surfactant is strong, whilst it forms a classical stagnant cap when the surfactant is weak. The asymptotic solutions offer closed-form predictions of drag reduction across much of the parameter space, providing practical utility and enhancing understanding of surfactant dynamics in flows over SHSs.

Carbon storage in saline aquifers is a prominent geological method for reducing CO2 emissions. However, salt precipitation within these aquifers can significantly impede CO2 injection efficiency. This study examines the mechanisms of salt precipitation during CO2 injection into fractured matrices using pore-scale numerical simulations informed by microfluidic experiments. The analysis of varying initial salt concentrations and injection rates revealed three distinct precipitation patterns, namely displacement, breakthrough and sealing, which were systematically mapped onto regime diagrams. These patterns arise from the interplay between dewetting and precipitation rates. An increase in reservoir porosity caused a shift in the precipitation pattern from sealing to displacement. By incorporating pore structure geometry parameters, the regime diagrams were adapted to account for varying reservoir porosities. In hydrophobic reservoirs, the precipitation pattern tended to favour displacement, as salt accumulation occurred more in larger pores than in pore throats, thereby reducing the risk of clogging. The numerical results demonstrated that increasing the gas injection rate or reducing the initial salt concentration significantly enhanced CO2 injection performance. Furthermore, identifying reservoirs with high hydrophobicity or large porosity is essential for optimising CO2 injection processes.

This article examines the concept of just energy transition in the context of Africa. It explores two key imperatives: (1) social inclusion and (2) an environmental rights-based approach to promote just energy transitions within African countries. The article looks at social inclusion from the perspective of local communities that host energy infrastructures, highlighting potential injustices and negative impacts that may arise from the energy transition. It further argues that social inclusion and environmental rights-based approaches can be useful tools for achieving just energy transitions in Africa. The article also analyses strategies that underpin social inclusion and environmental rights-based approaches within the governance and legal frameworks for energy transition projects in Africa, including empowering local communities to ensure the transition aligns with their socio-economic standing. The article suggests that adopting socially inclusive and environmental rights-based imperatives are significant steps towards overcoming and addressing injustices in energy transition projects in Africa.

Periodic gravity-capillary waves on a fluid of finite depth with constant vorticity are studied theoretically and numerically. The classical Stokes expansion method is applied to obtain the wave profile and the interior flow up to the fourth order of approximation, which thereby extends the works of Barakat & Houston (1968) J. Geophys. Res. 73 (20), 6545–6554 and Hsu et al. (2016) Proc. R. Soc. Lond. A 472, 20160363. The classical perturbation scheme possesses singularities for certain wavenumbers, whose variations with depth are shown to be affected by the vorticity. This analysis also reveals that for any given value of the physical depth, there exists a threshold value of the vorticity above which there are no singularities in the theoretical solution. The validity of the third- and fourth-order solutions is examined by comparison with exact numerical results, which are obtained with a method based on conformal mapping and Fourier series expansions of the wave surface. The outcomes of this comparison are surprising as they report important differences in the internal flow structure, when compared with the third-order predictions, even though both approximations predict almost perfectly the phase velocity and the surface profiles. Usually, this occurs when the wavenumber is far enough from a critical value and the steepness is not too large. In these non-resonant cases, it is found that the fourth-order theory is more consistent with the exact numerical results. With negative vorticity the improvement is noticeable both beneath the crest and the trough, whereas with positive vorticity the fourth-order theory does a better job either beneath the crest or beneath the trough, depending of the type of the wave.

African countries have increasingly emphasized adopting lower carbon, more efficient and environmentally responsible energy systems. Despite these efforts, little progress has been made in addressing the adverse human rights impacts of energy transition programs and projects, and the responsibilities of extractive sector corporations and operators. Existing legal and institutional frameworks supporting human rights face hindrances in adapting to local contexts to pursue clean energy transition and energy justice. Through the lens of community engagement, gender equality and other rights-based approaches, this article argues that socially excluding vulnerable groups in accessing energy markets is primarily a function of consolidating energy delivery in a way that navigates current discrimination and responds to the central roles played by different actors. The article explores how energy is produced, extracted, distributed and shared to help outline a future agenda for shaping discussions on just transitions in Africa, emphasizing the prioritization of fairness in these efforts.

The special issue brings together diverse academic and practitioner perspectives to explore the legal and governance aspects of implementing a just transition in practice. Recent studies have highlighted how efforts to advance clean energy transition programs in energy and extractive sectors have been increasingly linked to social exclusions, greenwashing, rising energy poverty levels and constraints to access to land and other resources in already vulnerable communities. While the need for a just transition is clear, an interdisciplinary and multijurisdictional examination of the practical challenges and gaps in the design and implementation of just transition programs has remained sparse. This special issue seeks to fill this gap in the existing literature. Through thematic and geographical case studies, the contributions herein critically examine the social, environmental and human rights implications of the clean energy transition, illuminating what a just transition should entail and how it can be realized in diverse contexts.