Session R23: Free Surface Flows: General

Viscous-inviscid interaction of a just detached planar liquid film near the Taylor-Culick speed: waves, blow-up, reversed-flow breakdown

We consider a stationary developed thin liquid film having just passed a trailing edge of a horizontal plate under the action of gravity and surface tension. In the associated limit of large Reynolds and Froude numbers and long waves, the classical, double-deck type of viscous-inviscid interaction accounts for the rigorous treatment of the flow around the edge. The resultant asymptotic flow description is then solely parametrised by the reciprocal Weber number, T, suitably formed with the momentum flow of the just detaching film, and a rescaled, reciprocal Froude number of O(1), G. Correspondingly, our focus lies on the numerical and analytical treatment of the fully nonlinear interaction problem. Most interestingly, the capillary influence on the jet-type pressure-displacement (interaction) law reveals an unprecedented kind of choking of a capillary wave if T equals 1/2; a value disclosing that the momentum-based averaged speed of the detaching flow is given by the associated Taylor-Culick speed. Specifically, the asymptotic analysis in the least-degenerate, self-similar limit of vanishingly small values of ε given by T–1/2 and small ones of G discovers a surprising richness of phenomena. This condenses the various flow manifestations for all values of T smaller than 1 (a second critical threshold representing another type of choking) and all non-negative ones of G. For negative values of ε, the theory predicts nonlinear Squire modes of the flapping kind; for positive ones, no waves are found but the interacting-flow description terminates in strikingly different manner in dependence of the deviation of G from a unique critical threshold: for smaller values of G, the free jet undegoes massive flow reversal far downstream (compressive interaction); for larger ones, however, a blow-up singularity is encountered (expansive interaction, regularized on an Euler stage). Both types of breakdowns resemble those found in originally wall-bounded interaction. In all other cases, a WKBJ analysis predicts a Goldstein wake far downstream. This also quantifies how the wavelength diverges when T approaches 1/2 from below and how gravity increases the amplitude of the waves, pointing to inviscid cnoidal waves for sufficiently large values G. Viscous dissipation concentrated around their troughs attenuates them periodically.   

Rogue Nanowaves Driven by Thermal Fluctuations: A New Mechanism for Thin Film Rupture

Liquid films routinely coat solids, forming a protective layer on the eye or a pool of spilt coffee on a table. The most important practical question is whether these films can spontaneously rupture to create a dry spot.  Conventionally, rupture is assumed to be driven purely by attractive intermolecular forces between the solid and the liquid, a linear instability, but experiments discovered regimes in which thermal (Brownian) motion is instead responsible, especially for ultra-thin films.  

In this talk, a theoretical framework will be developed for the experimentally observed thermal regime, in which fluctuation-induced nanowaves rupture linearly stable films. Molecular simulations identify these regimes and are accurately reproduced by stochastic simulations based on fluctuating hydrodynamics. It is then shown how rare-event theory can be applied to, and developed for, this field for the first time to provide exceptional computational efficiency and accuracy that allows us to extend calculations deep within thermal regime.  Furthermore, analysis of the rare-event theory reveals a novel picture of how and when `rogue nanowaves’ are able to provide a route to film rupture.  Finally, future applications of the new theoretical framework and experimental verification will be discussed.   

Regime transitions in a laminar film flowing over a cylinder

Film flows over a cylinder have industrial applications in coating processes and heat exchangers. Previous studies on such flows have assumed that the flow from both sides of the cylinder reconnects at the bottom. This may not be true in realistic scenarios and the flow may separate before reaching the bottom, as seen on the underside of a spout due to teapot effect. We perform experiments on a cylinder of diameter 3.05cm with three sugar solutions of varying concentrations and observe four regimes. Regime I has two strands with dripping flow, Regime II has two independent falling strands of continuous streams, Regime III has re-connection of the falling film to a single strand, and Regime IV has re-connection of the two streams with an air gap aft of the cylinder. For a low viscosity solution with high inertia, the capillary forces dominate the reconnection regime. Regime IV occurs when when inertia overcomes the capillary forces to cause separation at an angle such that the two streams meet in air. As the viscosity increases, the reconnected Regime III narrows and transition to Regime IV happens at lower velocities. The regimes can be demarcated on a Reynolds Number vs product of Bond number and Weber number plot on a line of slope 2. On expanding the product of Bond number and Weber number, we get wetted perimeter as a constituent of Baudoin number and the regimes are demarcated by a modified form of Weber number.   

Thin film flow with an undulating surface: The inertial effects

Free surface flows driven by boundary undulations are observed in many biological phenomena, including the feeding and locomotion of water snails. The undulating boundaries deform the air-liquid interface and generate a net fluid flux inside the thin film. We experimentally observe that the fluid flux varies non-monotonically with the increasing wave speed in the low Reynolds number, while the non-monotonicity is eliminated in the finite Reynolds number regime. To rationalize these observations, we develop a two-dimensional thin-film mathematical model to explore the physical mechanism of free surface flows driven by periodic undulations. The model combines the effects of inertia, viscosity, gravity, and surface tension in a tractable way. In this talk, we will present the new model results along with the experimental findings and focus on how inertia influences the flow system.   

Retraction dynamics of highly viscous liquid sheets

Upon rupture, a planar liquid sheet undergoes retraction driven by unbalanced capillary forces. This process is governed by two dimensionless groups: the sheet's aspect ratio and the Reynolds number. In this talk, we will theoretically investigate the retraction dynamics of a two-dimensional viscous sheet in the regime of small aspect ratios and low Reynolds numbers, assuming that the sheet is free at one extremity and fixed at the other. An asymptotic model of the dynamics, derived using matched asymptotic expansions in an appropriate distinguished limit, will be presented. Our analysis reveals that the dynamics is dictated by a remote region where inertial and viscous effects balance. In that region, the flow has a conserved quantity, thereby enabling the reduction of the problem to a one-dimensional diffusion equation for the sheet thickness profile subject to effective boundary conditions at the free end. This reduced description facilitates the identification and analysis of distinct retraction regimes, which are characterized by the time elapsed since rupture and by the relative smallness of the aspect ratio to the Reynolds number.   

Surface dilatational viscosity: Fact or fiction?

The free surface of liquids, water in particular, can exhibit significant excess shear viscosity in the presence of surfactants, including monomolecular films of DPPC, the main constituent of lung surfactant. Detailed velocity measurements have been reconciled with numerical simulations for many different monolayers exhibiting either Newtonian or non-Newtonian responses to shear, where the interfacial velocity field is solenoidal. The situation is altogether different for surface dilatational viscosity, which is the more relevant quantity for non-solenoidal interfacial velocity fields. To avoid complications with interfacial accelerations, this work uses an oscillating floor and fluid inertia to produce a non-solenoidal velocity field at a flat interface with a fixed area, studying the competing effects of Marangoni stress associated with surface tension gradients and any intrinsic combination of surface dilatational and shear viscosity. The flow geometry is rectangular, with a width twice the depth and a span more than six times the depth. Interfacial velocity measurements verify the expected half-period-flip symmetry. These measurements are compared with 2D simulations utilizing the measured equation-of-state and diffusivity of DPPC.   

CO2-induced baroclinicity in an interfacially-driven flow

     Interfacial gas transport defines the behavior of many physical systems.  The knife-edge viscometer (KEV) is a flow device for the study of interfacial phenomena consisting of a thin circular knife edge contacting the air-liquid interface of a liquid-filled cylinder.  Knife-edge rotation conveys shear and mixing to the bulk via surface shear viscosity and fluid inertia.  This work examined hydrodynamic changes in a KEV caused by CO₂ gas transport from the air-liquid interface, including the impact of density changes due to dissolved CO₂.  Experiments consisted of planar laser-induced fluorescence flow visualization, a constant surface shear viscosity associated with a viscous monolayer, and variable rotation rate.  Numerical simulations used an axisymmetric second-order finite difference code with a Boussinesq-Scriven surface model and variable bulk density.  Results showed baroclinic production of vorticity by gradients of CO₂ that generated vigorous mixing and transport events, transients that decayed slowly in time due to CO₂’s large Schmidt number in water.   

Plunging liquid jets and the role of subharmonic gravity-capillary waves in air entrainment

As a jet of liquid falls into a pool, it can cause air bubbles to form at the plunge point and be drawn into the pool, a process known as air entrainment. This process is a useful analog for many two-phase flows including the ocean surface, metal casting, and aerated wastewater treatment and aquaculture ponds. Despite the ubiquity of air entrainment by plunging jets, it remains poorly understood. Prior works have tended to correlate air entrainment with factors that control it only indirectly, unconcerned with potentially important phenomena local to the plunge site. We attempt to remedy this by employing a plunging jet that does not entrain unless it is deliberately made to do so by the introduction of well-characterized harmonic disturbances. At lower jet forcing frequencies, the disturbance-driven air entrainment behaves similarly to past works with disturbed jets, but at higher frequencies, the jet disturbances cause subharmonic surface waves to arise and entrain air by interacting with the submerged jet flow. By capturing entrained bubbles to measure their flow rate, we explore this dramatic and novel interplay between turbulence and free-surface flow.