The Hele-Shaw–Cahn–Hilliard model, coupled with phase separation, is numerically simulated to demonstrate the formation of anomalous fingering patterns in a radial displacement of a partially miscible binary-fluid system. The composition of injected fluid is set to be less viscous than the displaced fluid and within the spinodal or metastable phase-separated region, in which the second derivative of the free energy is negative or positive, respectively. Because of phase separation, concentration evolves non-monotonically between the injected and displaced fluids. The simulations reveal four areas of the concentration distribution between the fluids: the inner core; the low-concentration grooves/high-concentration ridges; the isolated fluid fragments or droplets; the mixing zone. The grooves/ridges and the fragments/droplets, which are the unique features of phase separation, form in the spinodal and metastable regions. Four typical types of patterns are categorized: core separation (CS); fingering separation (FS); separation fingering (SF); lollipop fingering, in the order of the dominance of phase separation, respectively. For the patterns of CS and FS, isolated fluid fragments or droplets around the inner core are the main features. Fingering formation is better maintained with droplets in the SF pattern if the phase separation is relatively weaker than viscous fingering (VF). Even continuous fingers are well preserved in the case of dominant VF; phase separation results in lollipop-shaped fingers. The evolving trend of the patterns is in line with the experiments. These patterns are summarized in a pattern diagram, mainly by the magnitude of the second derivative of the free energy profile.

The continental plates of Earth are known to drift over a geophysical time scale, and their interactions have led to some of the most spectacular geoformations of our planet while also causing natural disasters such as earthquakes and volcanic activity. Understanding the dynamics of interacting continental plates is thus significant. In this work, we present a fluid mechanical investigation of the plate motion, interaction and dynamics. Through numerical experiments, we examine the coupling between a convective fluid and plates floating on top of it. With physical modelling, we show the coupling is both mechanical and thermal, leading to the thermal blanket effect: the floating plate is not only transported by the fluid flow beneath, it also prevents the heat from leaving the fluid, leading to a convective flow that further affects the plate motion. By adding several plates to such a coupled fluid–structure interaction, we also investigate how floating plates interact with each other, and show that under proper conditions, small plates can converge into a supercontinent.

Large-eddy simulations (LES) of a hypersonic boundary layer on a $7^\circ$-half-angle cone are performed to investigate the effects of highly cooled walls (wall-to-recovery temperature ratio of $T_w / T_r \sim 0.1$) on fully developed turbulence and to validate a newly developed rescaling method based on volumetric flow extraction. Two Reynolds numbers are considered, $Re_m = 4.1 \times 10^6\ \text {m}^{-1}$ and $6.4 \times 10^6\ \text {m}^{-1}$, at free-stream Mach numbers of $M_\infty = 7.4$. A comparison with a reference laminar-to-turbulent simulation, capturing the full history of the transitional flow dynamics, reveals that the volumetric rescaling method can generate a synthetic turbulent inflow that preserves the structure of the fluctuations. Equilibrium conditions are recovered after approximately 40 inlet boundary layer thicknesses. Numerical trials show that a longer streamwise extent of the rescaling box increases numerical stability. Analyses of turbulent statistics and flow visualizations reveal strong pressure oscillations, up to $50\,\%$ of local mean pressure near the wall, and two-dimensional longitudinal wave structures resembling second-mode waves, with wavelengths up to 50 % of the boundary layer thickness, and convective Mach numbers of $M_c \simeq 4.5$. It is shown that their quasi-periodic recurrence in the flow is not an artefact of the rescaling method. Strong and localized temperature fluctuations and spikes in the wall-heat flux are associated with such waves. Very high values of temperature variance near the wall result in oscillations of the wall-heat flux exceeding its average. Instances of near-wall temperature falling below the imposed wall temperature of $T_w=300$ K result in pockets of instantaneous heat flux oriented against the statistical mean direction.

The dynamics of evolving fluid films in the viscous Stokes limit is relevant to various applications, such as the modelling of lipid bilayers in cells. While the governing equations were formulated by Scriven (1960), solving for the flow of a deformable viscous surface with arbitrary shape and topology has remained a challenge. In this study, we present a straightforward discrete model based on variational principles to address this long-standing problem. We replace the classical equations, which are expressed with tensor calculus in local coordinates, with a simple coordinate-free, differential-geometric formulation. The formulation provides a fundamental understanding of the underlying mechanics and translates directly to discretization. We construct a discrete analogue of the system using Onsager's variational principle, which, in a smooth context, governs the flow of a viscous medium. In the discrete setting, instead of term-wise discretizing the coordinate-based Stokes equations, we construct a discrete Rayleighian for the system and derive the discrete Stokes equations via the variational principle. This approach results in a stable, structure-preserving variational integrator that solves the system on general manifolds.

In this contribution, we develop a versatile formalism to derive unified two-phase models describing both the separated and disperse regimes as introduced by Loison et al. (Intl J. Multiphase Flow, vol. 177, 2024, 104857). It relies on the stationary action principle and interface geometric variables. This contribution provides a novel method to derive small-scale models for the dynamics of the interface geometry. They are introduced here on a simplified case where all the scales and phases have the same velocity and that does not take into account large-scale capillary forces. The derivation tools yield a proper mathematical framework through hyperbolicity and signed entropy evolution. The formalism encompasses a hierarchy of small-scale reduced-order models based on a statistical description at a mesoscopic kinetic level and is naturally able to include the description of a disperse phase with polydispersity in size. This hierarchy includes both a cloud of spherical droplets and non-spherical droplets experiencing a dynamical behaviour through incompressible oscillations. The associated small-scale variables are moments of a number density function resulting from the geometric method of moments (GeoMOM). This method selects moments as small-scale geometric variables compatible with the structure and dynamics of the interface; they are defined independently of the flow topology and, therefore, this model allows the coupling of the two-scale flow with an inter-scale transfer. It is shown, in particular, that the resulting dynamics provides partial closures for the interface area density equation obtained from the averaging approach.

Ciliated microorganisms near the base of the aquatic food chain either swim to encounter prey or attach at a substrate and generate feeding currents to capture passing particles. Here, we represent attached and swimming ciliates using a popular spherical model in viscous fluid with slip surface velocity that affords analytical expressions of ciliary flows. We solve an advection–diffusion equation for the concentration of dissolved nutrients, where the Péclet number ($Pe$) reflects the ratio of diffusive to advective time scales. For a fixed hydrodynamic power expenditure, we ask what ciliary surface velocities maximize nutrient flux at the microorganism's surface. We find that surface motions that optimize feeding depend on $Pe$. For freely swimming microorganisms at finite $Pe$, it is optimal to swim by employing a ‘treadmill’ surface motion, but in the limit of large $Pe$, there is no difference between this treadmill solution and a symmetric dipolar surface velocity that keeps the organism stationary. For attached microorganisms, the treadmill solution is optimal for feeding at $Pe$ below a critical value, but at larger $Pe$ values, the dipolar surface motion is optimal. We verified these results in open-loop numerical simulations and asymptotic analysis, and using an adjoint-based optimization method. Our findings challenge existing claims that optimal feeding is optimal swimming across all Péclet numbers, and provide new insights into the prevalence of both attached and swimming solutions in oceanic microorganisms.