This study offers the first comprehensive analysis of convective onset in a uniformly internally heated, rotating porous sphere. By incorporating the Coriolis force into Darcy’s law, we formulate the linear stability problem using a vector potential expansion combined with spherical harmonic decomposition. The resulting equations are solved numerically via a spectral method based on Worland polynomials. In the absence of rotation, the most unstable mode corresponds to a spherical harmonic degree $l = 2$, with a critical Rayleigh number of 91.95. When rotation is introduced, the Coriolis force further stabilises the flow, causing the critical Rayleigh number to increase monotonically with the Taylor number. Notably, the system consistently favours an azimuthal wavenumber $m = 2$ across nearly the entire parameter range. This behaviour contrasts sharply with rotating fluid convection, where the preferred wavenumber increases indefinitely with the rotation rate. This fundamental difference arises from the disruption of the quasi-geostrophic balance by the strong Darcy resistance, leading to the failure of the Proudman–Taylor theorem. Under strong rotation, the system exhibits power-law scaling and develops an Ekman-like boundary layer localised near the polar axis, which channels global heat and mass transport. These findings provide novel theoretical insights into the thermal structure and evolution of early evolutionary bodies, such as rapidly rotating planetesimals or undifferentiated asteroids with high permeability, and lay the groundwork for future nonlinear studies of rotating porous convection.

The present study investigates the hydroelastic response of a floating ice plate clamped to a vertical wall and subjected to an oscillatory point pressure, with particular emphasis on the effects of second-order nonlinearity on bending stresses in ice. A nonlinear potential flow model is coupled with a nonlinear thin-plate theory to systematically investigate the role of second-order nonlinearities in wave–ice interactions. It is shown that nonlinear contributions from the plate equation are negligible at second order, allowing for a linearised structural model without loss of accuracy. In contrast, nonlinear fluid–structure interactions significantly influence the strain response, particularly for loading frequencies close to resonant frequency. At these frequencies, the nonlinear self-interaction of the primary wave mode excites freely propagating second harmonics, resulting in secular growth of the second-order solution. Through a regular perturbation analysis, we derive second-order corrections to the ice deflection and show that the nonlinearity leads to localised amplification of curvature and hence strain, especially near the forcing location. Numerical results further show that for loading frequencies close to the resonant frequency, the strain distribution attains its maximum amplitude at the location of the applied load, substantially exceeding the corresponding strain levels near the clamped edge of the ice. At moderate frequencies, wave reflections from the wall cause the strain to localise near the boundary in an oscillatory pattern, with linear theory remaining accurate. However, second-order corrections may still amplify or shift strain peaks away from the wall, influencing potential fracture zones. The results demonstrate that classical linear models may severely underestimate local stress concentrations. The study underscores the importance of incorporating second-order nonlinearities and boundary effects to accurately predict strain localisation, energy transfer and potential failure zones in ice-covered waters subjected to dynamic loading.

The concept of inverse energy cascades has played a central role in the development of turbulence theory, with applications in two-dimensional and quasi-two-dimensional flows. We examine the presence or absence of inverse energy cascades in rotating stably stratified flows constrained to anisotropic yet fully three-dimensional domains, in a range of parameters that are relevant for planetary atmospheres. Our results show that inverse energy cascades can indeed emerge when rotation overcomes a certain threshold that depends on the stratification. Implications for the self-organisation processes of planetary atmospheres are discussed.

We explore the bifurcation structure of mode-1 solitary waves in a three-layer fluid confined between two rigid boundaries. A recent study Lamb (2023 J. Fluid Mech., vol. 962, A17) proposed a method to predict the coexistence of solitary waves with opposite polarity in a continuously stratified fluid with a double pycnocline by examining the conjugate states for the Euler equations. We extend this line of inquiry to a piecewise-constant three-layer stratification, taking advantage of the fact that the conjugate states for the Euler equations are exactly preserved by the strongly nonlinear model that we will refer to as the three-layer Miyata-Maltseva-Choi-Camassa (MMCC3) equations. In this reduced setting, solitary waves are governed by a Hamiltonian system with two degrees of freedom, whose critical points are used to explain the bifurcation structure. Through this analysis, we also discover families of solutions that have not been previously reported for a three-fluid system. Using the shared conjugate state structure between the MMCC3 model and the full Euler equations, we propose criteria for distinguishing the full range of solution behaviours. This alignment between the reduced and full models provides strong evidence that partitioning the parameter space into regions associated with distinct solution types is valid within both theories. This classification is further substantiated by numerical solutions to both models, which show excellent agreement.

We present a method to simulate non-coalescing impacts and rebounds of droplets onto the free surface of a liquid bath, together with new experimental data, focused on the low-speed impact of droplets. The method is derived from first principles and imposes only natural geometric and kinematic constraints on the motion of the impacting interfaces, yielding predictions for the evolution of the contact area, pressure distribution and wave field generated on both impacting masses. This work generalises an existing kinematic-match method whose prior applications dealt with deformation of the surface of the bath only; i.e. neglecting that of the droplet. The method’s extension to include droplet deformation gives predictions that compare favourably with existing experimental results and our new experiments conducted in the low-Weber-number regime.

Active flow control often exploits disturbance amplification mechanisms to achieve desired flow properties. Recently, theoretical predictions of optimal control based on stability analysis have gained traction. However, these methods are limited in their ability to predict nonlinear control strategies, such as burst-mode actuation for separated flows, which involve intermittent and high-amplitude forcing. To address this limitation, we developed a nonlinear optimal forcing analysis based on optimal perturbation theory. This method is specifically designed to capture non-harmonic forcing patterns and the nonlinear temporal evolution of the disturbance field. We applied this method to the two-dimensional high-subsonic, low-Reynolds number flow around a NACA0012 airfoil to reattach the separated flow and investigate the onset mechanism of low-frequency oscillation. The analysis identified an optimal temporal forcing pattern characterized by damped oscillation. This forcing exploits flow amplification mechanisms over the separation bubbles, promoting the formation of spanwise vortices in the shear layer. When implemented as a periodic forcing concentrated at the separated point, these vortices were stably generated, resulting in a significant lift increase via momentum exchange. A key finding is that the application of this optimal forcing induced long-term changes in the flow field, driven by the transient emergence of low-frequency oscillations. Furthermore, we explored the intermittent application of this forcing and found that an appropriate duty cycle can enhance the lift coefficient while reducing energy consumption.

Surfactants at the air–sea interface are known to alter surface wave dynamics by modifying surface tension and Marangoni stresses. In this study, we perform two-dimensional direct numerical simulations of gravity-capillary waves with insoluble surfactants using a coupled phase field and volume-of-fluid method. We consider a nonlinear equation of state for surface tension and resolve Marangoni stresses induced by surfactant concentration gradients. We explore a broad parameter space characterised by initial wave steepness $ak$, Bond number $\textit{Bo}$ (comparing gravity and surface tension), Reynolds number $\textit{Re}$ (comparing inertia and viscosity), and the importance of surfactant concentration and strength of the gradient, characterised by a surfactant parameter $\beta$. We analyse the impact of surfactants on wave patterns, surface roughness, wave breaking, energy dissipation and surface vorticity. Our results reveal a non-monotonic dependence of wave shape, roughness, vorticity and energy dissipation on $\beta$, which is found to be governed by Marangoni effects that peak at intermediate surfactant concentrations. Wave regime transition at high $\textit{Bo}$ is governed by an effective $\textit{Bo}$, which accounts for the reduction in surface tension induced by surfactants. We further introduce a rescaled parameter $\textit{Bo}\,\textit{Re}^{-1/2}\,(ak)^{-1}$ based on force balance, which collapses the transition boundaries across different $\textit{Re}$. These findings provide a systematic understanding of surfactant-modulated wave dynamics for both laboratory and geophysical applications.

This study investigates the reflection of a moving shock on a stationary oblique shock – a prototype for supersonic vehicle encounters. Combining computational fluid dynamics (CFD) with a simplified model with key assumptions checked against CFD, we reveal how triple-point trajectories and pressure peaks evolve with wedge angle, and identify mechanisms governing transitions between interference types. It is shown that: (i) for Type V interference, the triple points move at distinct velocities, so the equations must be set in each triple point’s moving frame rather than in a single nominal intersection point’s frame of the incident and oblique shocks. Reducing the wedge angle weakens confinement, lowering overpressure and slowing triple-point motion. (ii) At the Type VI–V transition, a sudden Mach stem emergence creates a sharp pressure spike. (iii) For Type II and IV interferences, a major difficulty arises in determining the postreflection pressure behind the shock – a key to closing the model. This obstacle is overcome by treating the flow as a normal shock impinging on a wall, an analogy that yields the missing parameter and is checked by CFD. We also find that transitions between interference types are governed by the emergence and disappearance of triple points in their moving frames, accounting for deviations from classical critical conditions. These results uncover fine-scale flow physics previously overlooked in global studies.