Existing theoretical analyses on the Faraday instability in Hele-Shaw cells typically adopt gap-averaged governing equations and rely on Hamraoui’s model coming from molecular kinetics theory, thereby oversimplifying essential transverse information, such as contact line velocity and capillary hysteresis, and conflicting with the unsteady meniscus dynamics. In this paper, a gap-resolved approach is developed by directly modelling the transverse gap flow and the contact angle dynamics, which overcomes the aforementioned limitations, ultimately yielding a modified damping with respect to the static contact angle and hysteresis range. A novel amplitude equation for linear Faraday instability is derived that combines this damping and the gap-averaged counterpart based on the oscillatory Stokes boundary layer, with the viscous dissipation preserved. By means of Lyapunov’s first method, an explicit analytical expression for the critical stability boundary is established. Two series of laboratory experiments are performed that focus, respectively, on evolutions of the lateral meniscus and the longitudinal free surface near the Faraday onset, from which key parameters relevant to the theory are precisely measured. Based on the experimental data, the validity of the proposed mathematical model for addressing the Faraday instability problem in Hele-Shaw cells is confirmed, and the generation and development mechanisms of the onset are clarified. In the asymptotic analysis, the inclusion of contact angle dynamics increases the overall damping and thus partially compensates for the frequency detuning introduced by oscillatory Stokes flow approximation.

A finite wall-mounted cylinder (FWMC) refers to a cylinder mounted to a flat wall, characterised by flow over its free end and at the wall junction. In the present study, finite wall-mounted harbour seal vibrissal cylinders with different spanwise aspect ratios ($\textit{AR} = L/D$, where $L$ is the length of the cylinder and $D$ is its diameter) are examined at a Reynolds number of $\textit{Re} = 12\,000$ based on the cylinder diameter, with an incoming turbulent boundary layer thickness of $\delta /D = 0.16$. The far-field acoustic spectra of the harbour seal vibrissal FWMCs are broadband for all aspect ratios and effectively suppress the tonal peaks observed in the spectra of circular FWMCs. However, the effectiveness of the suppression mechanism diminishes with decreasing aspect ratio. For the shortest harbour seal vibrissal FWMC ($\textit{AR}= 3.2$), the overall sound pressure level exceeds that of its circular counterpart, and the tonal peak is replaced by broadband noise of comparable magnitude. Spanwise-intermittent recirculation zones and cellular vortex shedding are identified in the near wake of the harbour seal vibrissal FWMCs. For all harbour seal vibrissal FWMCs, as well as for circular FWMCs with $\textit{AR}$ = 6.5, hairpin vortex shedding is observed rather than classical von Kármán vortex shedding. This corresponds to the absence of tonal peaks in the far-field acoustic spectra. Wavy separation lines play a crucial role in the formation of hairpin vortices and in inducing phase differences in vortex shedding in the spanwise direction. Concentrated quadrupole streamwise vortical structures periodically appear along the span in the wake of the longest harbour seal vibrissal FWMC, which diminish with $\textit{AR}$ decreasing to $\textit{AR}$ = 6.5. This concentrated quadrupole arrangement of positive–negative streamwise vortices suppresses the vortex shedding observed along the span of circular FWMCs by disrupting the acoustic-coherent structures into smaller, spanwise-wavy organised elements. It also stabilises the flow and balances transverse ($y$) forces.

We study the collective behaviour of clusters of cylinders placed in the wake of a fixed cylinder and free to move in a direction perpendicular to that of the incoming flow, with no structural damping or stiffness. We keep the Reynolds number, defined based on the cylinder diameter, at 100 and consider five different configurations for the initial positions of the cluster cylinders: linear, rectangular, V-shaped, triangular and circular. In each configuration, we consider progressively increasing numbers of cylinders in the cluster. We show that overall, the cylinders tend to form final linear configurations, in which, after their transition, the cylinders form one or more lines. Some free-to-move cylinders might take the lead position in some of these linear formations depending on the initial configuration. These steady-state positions are achieved when the mean value of lift that acts on the cylinders becomes negligible. As a byproduct of these reconfigurations, the overall drag force that acts on the collection of cylinders reduces at their final steady-state locations in comparison with their original configurations. The complicated wakes that are observed in the fixed counterparts of these configurations are replaced by a series of vortex rows in the wake of separate lines of cylinders. Reducing the mass ratio allows the cylinders to oscillate about their mean displacement paths, but their transient paths and their final steady-state positions are not affected significantly by the decrease in the mass ratio.

Motivated by the understanding of fluid mechanics behind dry eye syndrome during the winter season, we perform the linear and nonlinear stability analysis of the tear film. To reflect the biological structure of the tear film, we model it as a viscoelastic film on a substrate with an insoluble surfactant at the air–film interface, destabilisation by van der Waals forces, variation of viscosity and elasticity along the film height, and impose a substrate-normal temperature gradient. The air–film interface tension is assumed to decrease linearly with temperature and surfactant concentration. To conduct general linear stability analysis (GLSA), we employ the pseudo-spectral method. The GLSA predicts the existence of a longwave thermocapillary-induced instability, which overcomes the stabilisation by the solutocapillary effect due to the surfactant. The elasticity of the film, van der Waals forces and slip at the film–substrate interface contribute to further destabilisation, while a decrease in viscosity along the film height stabilises the film. The longwave instability is the dominant mode of instability. Thus, we derive longwave evolution equations whose linear stability analysis confirms the predictions of GLSA. The nonlinear analysis of the derived evolution equations, carried out using COMSOL 6.2, demonstrates tear-film rupture due to pure thermocapillary instability for sufficiently high thermal Marangoni numbers. In contrast, diseased eyes suffering from lipid layer dysfunction can undergo tear-film rupture at a much lower temperature difference between the ambient air and the cornea. Consideration of a viscosity decrease with increasing film height delays tear film rupture, while van der Waals forces decrease tear film rupture time. The rupture time range predicted by our model is in good agreement with clinical observations, thereby confirming the impact of the winter season on tear film dynamics.

A new model of the mean temperature profiles within incompressible wall-bounded turbulence is developed based on a relationship between the momentum and thermal eddy diffusivities for a wide range of Prandtl numbers ($ \textit{Pr}$). Flow media with $ \textit{Pr}$ and Schmidt numbers ($ \textit{Sc}$) other than unity are of great engineering importance, and the proposed model for passive scalar mean profiles is able to suitably account for $ \textit{Pr}$ and $ \textit{Sc}$ variations and their effects. Existing models have a higher degree of error at the lower $ \textit{Pr}$ ranges due to significant inaccuracies of the modelled thermal eddy diffusivity in this region. Considering incompressible wall-bounded turbulence at $0.007 \leqslant \textit{Pr} \leqslant 10$, the proposed methodology reduces the average error to just around $4\,\%$ across all cases considered, lower than previously proposed models due to its ability to capture low $ \textit{Pr}$ behaviour, showcasing a new temperature–velocity relationship that can account for variations in $ \textit{Pr}$ for all the cases considered.

This study develops a novel theoretical model for predicting the nonlinear evolution of Richtmyer–Meshkov instability (RMI) in particle-laden flows at large Stokes numbers. We construct a coupled multiphase potential flow theory framework incorporating two key models: (i) a postshock interface velocity attenuation model based on exponential decay accounting for momentum dissipation and (ii) a unified bubble and spike growth model for multiphase conditions. The multiphase-unified model maintains compatibility with classical single-phase RMI theories in the dilute limit, meanwhile revealing stronger particle-induced nonlinear decay in the amplitude growth rate. Model validation demonstrates good quantitative agreement across key predictive metrics – including dilute to dense particle volume fractions, Atwood numbers, particle sizes and initial perturbation amplitudes. This wide-range predictive capability for nonlinear instability growth may improve theoretical understanding of phenomena relevant to engineering applications. The results reveal that increased particle loading significantly reduces the growth rate of interface disturbances due to enhanced damping effects, leading to the blunting of spikes and flattening of bubbles. Vorticity dynamics analysis further shows that particle-induced vorticity weakens baroclinic production, thereby stabilizing the flow and inhibiting RMI development.

We model filtration of a feed solution, containing both small and large foulant particles, by a membrane filter. The membrane interior is modelled as a network of pores, allowing for the simultaneous adsorption of small particles and sieving of large particles, two fouling mechanisms typically observed during the early stages of commercial filtration applications. In our model, first-principles continuum partial differential equations model transport of the small particles and adsorptive fouling in each pore, while sieving particles are assumed to follow a discrete Poisson arrival process with a biased random walk through the pore network. Our goals are to understand the relative influences of each fouling mode and highlight the effect of their coupling on the performance of filters with a pore-size gradient (specifically, we consider a banded filter with different pore sizes in each band). Our results suggest that, due to the discrete nature of pore blockage, sieving alters qualitatively the rate of the flux decline. Moreover, the difference between sieving-particle sizes and the initial pore size (radius) in each band plays a crucial role in indicating the onset and disappearance of sieving–adsorption competition. Lastly, we demonstrate a phase transition in the filter lifetime as the arrival frequency of sieving particles increases.