Evolution of waves in liquid films on moving substrates

Using low-dimensional numerical simulations, we investigate the characteristics of complex and three-dimensional surface waves in a liquid film flowing over a rotating disk, focusing on large flow rates from a nozzle. Existing integral boundary layer (IBL) models, which are based on spatially averaged variables along the direction normal to the disk surface, have primarily focused on the formation of axisymmetric waves under relatively small flow rates. In this study, an extended IBL model that accounts for both laminar and turbulent regimes is developed by considering the non-uniformity of the local flow rate in the spreading film flow and incorporating closure models dependent on the local Reynolds number. Our numerical results successfully capture the generation of concentric waves by an impinging circular liquid jet and their transition into three-dimensional solitary waves. These findings are in good agreement with visualization images and time-series data of free-surface fluctuations from a displacement sensor. The backscattering of small-scale three-dimensional turbulence into large-scale horizontal turbulence inside the film plays a critical role in determining the transition of wave modes and the nonlinear dynamics of the waves in the turbulent regime. Furthermore, the behaviour of three-dimensional waves in the downstream region, including frequent wave coalescence in the transition region and the breakup of small-scale solitons, is distinct from that of gravity-driven falling film flows. The amplitude of the three-dimensional waves is inversely related to the generalized Reynolds number defined for rotating films.

Section: Introductionmentioning

confidence: 99%

“…Recently, the IBL models have been applied for the investigation of three-dimensional waves in more practical conditions such as film flows on moving (Ivanova et al. 2023) or cylindrical (Ding & Wong 2017) surfaces and film flows with surfactant (Batchvarov et al. 2021).…”

Section: Introductionmentioning

confidence: 99%

Low-order modelling of three-dimensional surface waves in liquid film flow on a rotating disk

Kim,

Kim

2024

On the coupling instability of a gas jet impinging on a liquid film

Barreiro-Villaverde,

Gosset,

Lema

et al. 2024

We investigate the dynamics of a gas jet impinging perpendicular to a thin liquid film dragged by a rising vertical substrate. This configuration is relevant to the jet-wiping process in hot-dip galvanization and it is unstable. Previous studies analysed the dynamics of the instability in the case of liquids with low Kapitza numbers (highly viscous liquids), more amenable to experimental and numerical investigations. This work extends the previous investigations by focusing on the wiping at much higher Kapitza numbers, which are more relevant to the galvanizing process. The simulations are carried out by combining volume of fluid and large-eddy simulations, and the dynamics of the gas–liquid interaction is analysed using extended multiscale proper orthogonal decomposition. The simulations allowed for analysing the jet-wiping instability in new flow conditions. Despite the largely different conditions, the results show that the interaction between the gas jet and the liquid film is qualitatively similar, featuring two-dimensional waves in the liquid correlated with oscillations and deflections of the gas jet in all cases. The wave characteristics (e.g. frequency and propagation speed) scale remarkably well using the Shkadov-like scaling based on the liquid, suggesting a dominant role of the liquid film in the coupling, and potentially enabling extrapolation of the results to a broader range of wiping conditions. Finally, we use the numerical results to discuss the limitations of liquid-film models, which constitute currently the only possible approach to study the jet-wiping process in industrial conditions.

Linear stability analysis of a vertical liquid film over a moving substrate

Pino,

Mendez,

Scheid

2024

The stability of liquid-film flows is essential in many industrial applications. In the dip-coating process, a liquid film forms over a substrate extracted at a constant speed from a bath. We studied the linear stability of this film considering different thicknesses $\hat {h}$ for four liquids, spanning an extensive range of Kapitza numbers ( $Ka$ ). By solving the Orr–Sommerfeld eigenvalue problem with the Chebyshev–Tau spectral method, we calculated the threshold between growing and decaying perturbations, investigated the instability mechanism, and computed the absolute/convective threshold. The instability mechanism was studied by analysing the perturbations’ vorticity distribution and the kinetic energy balance. It was found that liquids with low $Ka$ (e.g. corn oil, $Ka = 4$ ) are stable for a smaller range of wavenumbers compared with liquid with high $Ka$ (e.g. liquid zinc, $Ka = 11\,525$ ). Surface tension has a stabilising and a destabilising effect. For long waves, it curves the vorticity lines near the substrate, reducing the flow under the crests. For short waves, it fosters vorticity production at the interface and creates a region of intense vorticity near the substrate. In addition, we discovered that the surface tension contributes to both the production and dissipation of perturbation's energy depending on the $Ka$ number. Regarding the absolute/convective threshold, we identified a window in the parameter space where unstable waves propagate throughout the entire domain (indicating absolute instability). Perturbations affecting Derjaguin's solution ( $\hat {h}=1$ ) for $Ka<17$ and the Landau–Levich–Derjaguin solution ( $\hat {h}=0.945 Re^{1/9}Ka^{-1/6}$ ), are advected by the flow (indicating convective instability).