We derive a mathematical model for steady, unidirectional, thermoelectric magnetohydrodynamic (TEMHD) flow of liquid lithium along a solid metal trench, subject to an imposed heat flux. We use a finite-element method implemented in COMSOL Multiphysics to solve the problem numerically, demonstrating how the fluid velocity, induced magnetic field and temperature change depending on the key physical and geometrical parameters. The observed flow structures are elucidated by using the method of matched asymptotic expansions to obtain approximate solutions in the limit where the Hartmann number is large and the trench walls are thin.

Particle-laden flow through conduits is ubiquitous in both natural and industrial systems. In such flows, particles often migrate across the main fluid stream, resulting in non-uniform spatial distribution owing to particle–fluid and particle–particle interactions. The most relevant lateral particle migration mechanism by particle–fluid interaction is the Segré–Silberberg effect, which is induced by the inertial forces exerted on a particle, as the flow rate increases. However, methods to suppress it have not been suggested yet. Here, we demonstrate that adding a small amount of polymer to the particle-suspending solvent effectively suppresses the Segré–Silberberg effect in a square channel. To accurately determine the position of the particles within the channel cross-sections, we devised a dual-view imaging system applicable to microfluidic systems. Our analyses show that the Segré–Silberberg effect is effectively suppressed in a square microchannel due to the balance between the inertial and elastic forces at an optimal polymer concentration while maintaining nearly constant shear viscosity.

The controllability of passive microparticles that are advected with the fluid flow generated by an actively controlled one is studied. The particles are assumed to be suspended in a viscous fluid and well separated so that the far-field Stokes flow solutions may be used to describe their interactions. Explicit elementary moves parametrized by an amplitude $\varepsilon >0$ are devised for the active particle. Applying concepts from geometric control theory, the leading-order resulting displacements of the passive particles in the limit $\varepsilon \to 0$ are used to propose strategies for moving one active particle and one or two passive particles, proving controllability in such systems. The leading-order (in $\varepsilon$) theoretical predictions of the particle displacements are compared with those obtained numerically and it is found that the discrepancy is small even when $\varepsilon \approx 1$. These results demonstrate the potential for a single actuated particle to perform complex micromanipulations of passive particles in a suspension.

Despite the extensive research on bubble collapse near rigid walls, the bubble collapse dynamics in the presence of shear flow near a rigid wall is poorly understood. We conduct direct simulations of the Navier–Stokes equations to explore the bubble dynamics and pressures during bubble collapse near a rigid, flat wall under linear shear flow conditions. We examine the dependence of the bubble collapse morphology and wall pressures on the initial bubble location and shear rate. We find that shear distorts the bubble, generating two re-entrant jets – one developing from the side opposite to the mean flow and the other from the far end toward the wall. Upon impact of the jet on the opposite side of the bubble, water-hammer shocks are produced, which propagate outward and interact with the convoluted bubble shape. The shock stretches the bubble towards the wall, resulting in a closer impact location for the jet originating from the far end compared with the case with no shear flow. The water-hammer pressure location can be approximated as the theoretical distance travelled by a particle initialised at the bubble centre with the corresponding constant shear flow velocity. The maximum wall pressures can thus be predicted by considering the distance between the far jet impingement location and the wall along the wall-normal direction. As the shear rate is increased, the maximum wall pressure increases, although only marginally. We determine the critical initial stand-off distance from the wall at which the bubble morphology is shear dominated, i.e. characterised by converging re-entrant jets.

We introduce a new model equation for Stokes gravity waves based on conformal transformations of Euler's equations. The local version of the model equation is relevant for the dynamics of shallow water waves. It allows us to characterize the travelling periodic waves both in the case of smooth and peaked waves and to solve the existence problem exactly, albeit not in elementary functions. Spectral stability of smooth waves with respect to co-periodic perturbations is proven analytically based on the exact count of eigenvalues in a constrained spectral problem.

Impact dynamics have long fascinated due to their ubiquity in everyday phenomena, from rain droplets splashing on windscreens to stone-skimming on the surface of the ocean. Impacts are characterized by rapid changes over disparate length scales, which make them expensive or sensitive to capture experimentally and computationally. Here, reduced mathematical models come to the fore, offering a way to get significant physical insight at reduced cost. In this volume, Phillips & Milewski (J. Fluid Mech., 2024) develop a mathematical model allowing for air–water interactions in the low-impact speed regime, in which an impactor bounces or rebounds rather than splashes. Their model offers a reliable way to capture air effects in bouncing, with a range of potential applications including hydrodynamic-quantum analogues and biomimetic water walkers.

During a rainfall event, water infiltrates into the ground where it accumulates in porous rocks. This accumulation pushes the underlying groundwater towards neighbouring streams, where it runs to the sea. After the rain has stopped, the aquifer gradually releases its excess water, as the water table relaxes, until the next rain. In the absence of recharge, the water table would eventually reach its horizontal equilibrium position. The volume of groundwater stored above this level, which we call the active volume, sustains the river between two rainfall events. In this article, we use an experimental aquifer recharged by artificial rain to investigate how this active volume depends on the rainfall rate. Restricting our analysis to the steady-state regime, wherein the discharge into the stream balances rainfall, we explore a broad range of rainfall rates, for which the water table deforms significantly. We find that the active volume of water stored in the aquifer decreases with its depth. Using conformal mapping, we derive the flow equations and develop a numerical procedure that accounts for the active volume of groundwater in our experiments. In the case of an infinitely deep aquifer, the problem admits a closed-form solution, which provides a satisfying estimate of the active volume when the aquifer's depth is at least half its width. In the general case, a rougher estimate results from the energy balance of the dissipative groundwater flow.