We derive a generalised asymptotic model for the flow of a thin fluid film over an arbitrarily parameterised non-axisymmetric curved substrate surface based on the lubrication approximation. In addition to surface tension, gravity and centrifugal force, our model incorporates the effects of the Coriolis force and disjoining pressure, together with a non-uniform initial condition, which have not been widely considered in existing literature. We use this model to investigate the impact of the Coriolis force and fingering instability on the spreading of a non-axisymmetric spin-coated film at a range of substrate angular velocities, first on a flat substrate, and then on parabolic cylinder- and saddle-shaped curved substrates. We show that, on flat substrates, the Coriolis force has a negligible impact at low angular velocities, and at high angular velocities results in a small deflection of fingers formed at the contact line against the direction of substrate rotation. On curved substrates, we demonstrate that, as the angular velocity is increased, spin-coated films transition from being dominated by gravitational drainage with no fingering to spreading and fingering in the direction with the greatest component of centrifugal force tangent to the substrate surface. For both curved substrates and all angular velocities considered, we show that the film thickness and total wetted substrate area remain similar over time to those on a flat substrate, with the key difference being the shape of the spreading droplet.

Recent studies of viscous dissipation mechanisms in impacting droplets have revealed distinct behaviours between the macroscale and nanoscale. However, the transition of these mechanisms from the macroscale to the nanoscale remains unexplored due to limited research at the microscale. This work addresses the gap using the many-body dissipative particle dynamics (MDPD) method. While the MDPD method omits specific atomic details, it retains crucial mesoscopic effects, making it suitable for investigating the impact dynamics at the microscale. Through the analysis of velocity contours within impacting droplets, the research identifies three primary contributors to viscous dissipation during spreading: boundary-layer viscous dissipation from shear flow; rim geometric head loss; and bulk viscous dissipation caused by droplet deformation. This prompts a re-evaluation of viscous dissipation mechanisms at both the macroscale and nanoscale. It reveals that the same three kinds of dissipation are present across all scales, differing only in their relative intensities at each scale. A model of the maximum spreading factor (βmax) incorporating all forms of viscous dissipation without adjustable parameters is developed to substantiate this insight. This model is validated against three distinct datasets representing the macroscale, microscale and nanoscale, encompassing a broad spectrum of Weber numbers, Ohnesorge numbers and contact angles. The satisfactory agreement between the model predictions and the data signifies a breakthrough in establishing a universal βmax model applicable across all scales. This model demonstrates the consistent nature of viscous dissipation mechanisms across different scales and underscores the importance of integrating microscale behaviours to understand macroscale and nanoscale phenomena.

Craters form as the lander's exhaust interacts with the planetary surfaces. Understanding this phenomenon is imperative to ensuring safe landings. We investigate the crater morphology, where a turbulent air jet impinges on granular surfaces. To reveal the fundamental aspect of this phenomenon, systematic experiments are performed with various air-jet velocities, nozzle positions and grain properties. The resultant crater morphology is characterized by an aspect ratio. We find a universal scaling law in which the aspect ratio is scaled by a dimensionless variable consisting of the air velocity at the nozzle, the speed of sound in air, the nozzle diameter, the nozzle-tip distance from the surface, the grain diameter, the density of the grains and the density of air. The obtained scaling reveals the cross-over of the length scales governing the crater aspect ratio, providing a useful guideline for ensuring safe landings. Moreover, we report a novel drop-shaped sub-surface cratering phenomenon.

The nonlinear evolution of free-stream vortical disturbances entrained in the entrance region of a circular pipe is investigated using asymptotic and numerical methods. Attention is focused on the low-frequency disturbances that induce streamwise elongated structures. A pair of vortical modes with opposite azimuthal wavenumbers is used to model the free-stream disturbances. Their amplitude is assumed to be intense enough for nonlinear interactions to occur inside the pipe. The formation and evolution of the perturbation flow are described by the nonlinear unsteady boundary-region equations in the cylindrical coordinate system, derived and solved herein for the first time. Matched asymptotic expansions are employed to construct appropriate initial conditions and the initial–boundary value problem is solved numerically by a marching procedure in the streamwise direction. Numerical results show the stabilising effect of nonlinearity on the intense algebraic growth of the disturbances and an increase of the wall-shear stress due to the nonlinear interactions. A parametric study is carried out to evince the effect of the Reynolds number, the streamwise and azimuthal wavelengths, and the radial length scale of the inlet disturbance on the nonlinear flow evolution. Elongated pipe-entrance nonlinear structures (EPENS) occupying the whole pipe cross-section are discovered. EPENS with $h$-fold rotational symmetry comprise $h$ high-speed streaks positioned near the wall, and $h$ low-speed streaks centred around the pipe core. These distinct structures display a striking resemblance to nonlinear travelling waves found numerically and observed experimentally in fully developed pipe flow. Good agreement of our mean-flow and root mean square data with experimental measurements is obtained.

Droplet clustering in sprays refers to the dynamic evolution of highly concentrated regions due to the preferential accumulation of the polydisperse droplets in the turbulent airflow entrained by the spray. In the current study, we aim to experimentally investigate the collective vaporization of the droplets in droplet clusters in an air-assisted acetone spray characterized by the Group number, $G$. The magnitude of $G$ depends on the cluster length scale and interdroplet spacing, and it indicates the vaporization mode that may vary from the isolated mode ($G \ll 1$) to external group mode ($G \gg 1$). The droplet measurements were obtained under atmospheric conditions at different axial and radial locations within the spray. Application of the Voronoi analysis to particle image velocimetry images of the spray droplets facilitated the identification and characterization of the droplet clusters, which allowed the measurement of $G$ for each cluster. The results highlighted that multiscale clustering of the evaporating droplets leads to multimode group evaporation of the clusters (characterized by a wide range of $G$: 0.001–10). The trend of interdroplet spacing versus cluster area allowed the classification of the droplet clusters into small-scale clusters (which are of the order of the Kolmogorov length scale) and large-scale clusters (that scale with the large-scale turbulent eddies), that are found to exhibit distinct group evaporation behaviour. A theoretical model is invoked to correlate $G$ with the droplet evaporation rate for individual clusters, and some interesting observations are identified, which are explained in the paper.

We report on the melting dynamics of frozen olive oil in quiescent water for Rayleigh numbers up to $10^9$. The density difference results in an upward buoyancy-driven flow of liquid oil forming a thin film around the frozen oil. We experimentally investigate flat, cylindrical and spherical shapes and we derive theoretical expressions for the local film thickness, velocity and the local melt rate for these three canonical geometries. Our theoretical models predict the correct order of magnitude and the correct scaling as compared with our experimental findings.

Toroidal bubbles (TBs) represent cases of vortex rings with a gas–liquid interface where a gas vortex ring is encased within a liquid vortex ring, and can serve as effective media for mass conveyance, process mixing, noise reduction and reaction regulation. In this study, we carry out a systematic study on the interaction between a TB and a free surface. According to the high-speed photographic images from the experiments, we identify strong and weak interactions in terms of the normalized maximum free surface deformation $h_{max}^*$. Then, we perform numerical simulations based on the volume of fluid (VOF) method in the OpenFOAM platform. Based on both the experimental and the numerical results, we conclude that the Froude number, $Fr$, determines the main characteristics during the interaction process. The TB–free surface interaction is essentially the interaction between the liquid vortex ring enveloping the TB and the free surface, supplemented by the TB's complex behaviour. Next, we establish the scaling law of $h_{max}^*$ based on the energy balance condition. Based on this, we provide the critical $Fr$ and the slenderness of the TB, $\eta$, for identifying the strong and weak interactions, and a parametric plot of the interactions in terms of $Fr$ and $\eta$.

We study the effects of Prandtl number $\mathit {Pr}$ and Rayleigh number $\mathit {Ra}$ in two-dimensional Rayleigh–Bénard convection without boundaries, i.e. with periodic boundary conditions. For Prandtl numbers in the range $10^{-3} \leqslant \mathit {Pr} \leqslant 10^2$, the viscous dissipation scales as $\epsilon _\nu \propto \mathit {Pr}^{1/2}\mathit {Ra}^{-1/4}$, which is based on the observation that enstrophy $\langle {\omega ^2}\rangle \propto \mathit {Pr}^0 \mathit {Ra}^{1/4}$, and the Nusselt number tends to follow the ‘ultimate’ scaling $\mathit {Nu} \propto \mathit {Pr}^{1/2}\mathit {Ra}^{1/2}$ for all values of $\mathit {Pr}$ considered. The inverse cascade of kinetic energy forms the power-law spectrum $\hat {E}_u(k) \propto k^{-2.3}$, which is close to $k^{-11/5}$ proposed by the Bolgiano–Obukhov (BO) scaling. The potential energy flux is not constant, in contrast to one of the main assumptions underlying the BO phenomenology. So, the direct cascade of potential energy forms the power-law spectrum $\hat {E}_\theta (k) \propto k^{-1.2}$, which deviates from the expected $k^{-7/5}$. Finally, at $\mathit {Pr} \to 0$ and $\infty$, we find that the dynamics is dominated by vertically oriented elevator modes that grow without bound, even at high Rayleigh numbers and with large-scale dissipation present.