We study the settling of a heavy particle-laden phase layered above an unladen fluid. When the suspension has finite dilution, the dynamic of this set-up has remained largely unexplored. This system, which is fundamentally analogous to that found when studying the Rayleigh–Taylor instability, generates a mixing front whose width grows at an anomalous rate. In the present work, we explore the fluid and particle dynamics underpinning the mixing layer expansion. We perform particle-resolved direct numerical simulations to investigate the set-up: we use particle-to-fluid density ratios $\gamma =\{2,4,6,8\}$ and particle-laden phase volume fractions $\phi _{b}\approx 0.1$. The results show the emergence of flow structures analogous to Rayleigh–Taylor plumes. Downward-directed structures are preferentially sampled by the particles, and within their cores, particles are able to accelerate past their terminal velocity in still fluid, resulting in strongly enhanced settling rates. We quantify the intensity of turbulence in the mixing layer, showing possible evidence of an incipient energy cascade.

It is well-known that viscoelasticity plays a stabilising role in thin liquid film flow at moderate Reynolds numbers, which, however, cannot be resolved by current long-wave models. All these long-wave models can only predict the destabilising effect of viscoelasticity. This paper proposes a novel integral boundary layer model for a two-dimensional Oldroyd-B liquid film flow down a vertical plane. It consists of four dynamical variables – the film thickness $h$, local flow rate $q$, depth-integral normal viscoelastic stress $A$, and depth-integral shear viscoelastic stress $B$ – which successfully capture the destabilising mechanism of viscoelasticity at small Reynolds numbers and the stabilising mechanism of viscoelasticity at moderate Reynolds numbers by a linear stability analysis. Energy budget analysis reveals that the overall effect of viscoelasticity results from competition between destabilising surface shear stress and stabilising total polymeric stress work. Nonlinear travelling wave analysis reveals that wave speed is enhanced by weak viscoelasticity but suppressed by strong viscoelasticity, while the maximum film thickness grows monotonically with viscoelasticity at low Reynolds numbers, displaying a non-monotonic response characterised by an initial increase followed by a subsequent decrease under moderate Reynolds number conditions. Direct numerical simulations show that our integral boundary layer model is remarkably accurate when the Reynolds number is moderate, which predicts the nonlinear wave speed within $2\,\%$ errors against the full Navier–Stokes equations.

Absorbing sets are a central organising concept in the long-time analysis of dissipative partial differential equations, especially those arising in fluid mechanics. Roughly, an absorbing set is a bounded region in a suitable function space that eventually contains every trajectory that starts from any bounded set of initial data. Once the dynamics enter this region, they remain controlled in norm, and this ultimate boundedness becomes the gateway to more refined statements: existence of global attractors, finite-dimensional long-time behaviour. A variational framework, based on the background method, is developed to determine an ‘absorbing ball’ in the state space of incompressible shear flows described by the Navier–Stokes equations. This region is defined by the Reynolds–Orr energy identity and is guaranteed to contain all long-term dynamics, including chaotic and non-chaotic attractors. We employ a gradient-based optimisation to find a background flow corresponding to minimal absorbing radius for plane Poiseuille and Couette flows over a range of Reynolds numbers. The optimised background profiles are compared with the turbulent mean flow and they exhibit significantly steeper near-wall gradients than turbulent mean profiles obtained from direct numerical simulations and do not reproduce the universal law of the wall. Despite this quantitative discrepancy, the methodology provides rigorous, provable bounds on the region of state space accessible to the flow dynamics. This offers a novel and promising foundation for improving the global stability limit of the laminar state.

Extracting energy from a flow is a fundamental problem in fluid mechanics of significant practical engineering importance. To generate power from a flow, a resistance must be applied. Open questions remain on how to optimise this resistance, particularly for non-uniform flows. In this paper, we extend the multi-streamtube theory to address this gap. The extended theory allows for an arbitrary resistance distribution across an actuator strip (representing either a single turbine or an array of turbines) and is formulated as a power-coefficient maximisation problem to determine the optimal resistance distribution for both uniform and non-uniform flows. When the undisturbed kinetic energy flux projected onto the strip’s frontal area is used to normalise the extracted power, a uniform resistance maximises the resulting power coefficient for both uniform and non-uniform incoming flows. When the upstream kinetic energy flux of the flow through the strip is used for normalisation, the same optimisation result is obtained for uniform incoming flow, regardless of the assumed resistance distribution. However, for a non-uniform incoming flow, the optimal resistance distribution is non-uniform, with greater resistance applied in regions of higher velocity within the shear flow. This different optimisation result for non-uniform flow arises physically because the kinetic energy flux used in the second power-coefficient definition depends on the resistance applied across the strip, whereas the first does not. Two-dimensional direct numerical simulations are employed to examine the applicability and limitations of the multi-streamtube theory. The numerical and optimisation results together demonstrate that optimising the resistance distribution requires accounting not only for the non-uniformity of the incoming flow but also for the local flow variability around the strip.

Flash evaporation, a liquid-to-gas phase transition phenomenon in real fluids, is prevalent in aerospace propulsion systems. To elucidate the wave structures of flash evaporation in the equilibrium limit and provide theoretical benchmarks for computational fluid dynamics simulations, this paper formalises the flash evaporation Riemann problem (FeRP) characterised by the expansion branch crossing the saturation line, within the framework of homogeneous equilibrium and vapour–liquid equilibrium assumptions. An exact solution framework that analytically resolves all thermodynamic derivatives of equilibrium two-phase fluids is established for arbitrary two-parameter equations of state. By evaluating the Landau fundamental derivative, the non-classical wave structures arising in the FeRP are analysed, for which a stable iterative solution strategy incorporating the Chapman–Jouguet condition as an outer constraint is proposed. Furthermore, the FeRP framework is extended to the Riemann problem (RP) with a given thermodynamic non-equilibrium speed of sound, enabling a comprehensive evaluation of how this non-equilibrium closure affects the RP solution. Results indicate that such a non-equilibrium model alters the definition of the two-phase mixture entropy in the Euler equations, introducing a different isentropic path characterised by non-physical entropy decrease and density lag relative to the equilibrium path. Consequently, as the given speed of sound shifts from the complete equilibrium limit towards Wood’s mechanical equilibrium, the intermediate pressure, velocity and vaporisation extent in the RP solution decrease.

The classical problem of steady streaming induced by an oscillating object has been studied extensively, but prior work has focused almost exclusively on single-frequency oscillations, which result in symmetric, quadrupole-like flows. Here we demonstrate that dual-frequency oscillations induce asymmetric steady streaming with a non-zero net flux in a direction determined by the polarity of the oscillation – the oscillator serves as a pump. We use numerical simulations and asymptotic analysis at small amplitude to examine two-dimensional steady streaming around a cylinder, first focusing on frequency ratio 2. The computational experiments show asymmetrical streaming and pumping, i.e. net flux downstream. It is well known from asymptotic analysis that steady streaming is second order in amplitude, and we show that pumping occurs at third order. We then extend the analysis to general frequency ratios, where we give necessary conditions for pumping, and predict the order in amplitude at which pumping occurs. Finally, we corroborate the theoretical results with computational simulations for different frequency ratios, and we discuss the implications for using dual-mode vibrations to pump fluids in lab-on-a-chip and other applications.