Session T13: Flow Instability: Interfacial and Thin Film (8:00am - 8:45am CST)

Nonuniformities in Miscible Two-Layer Two-Component Thin Films

Obtaining uniform liquid films is a problem integral to many industries and requires an understanding of capillary leveling, Marangoni flow, evaporation, and many other phenomena. Multilayer films arise in various contexts where films must have multiple layers with distinct properties for optimal performance. Simultaneously coating multiple liquid layers presents a multitude of technological challenges that add to the complexity of obtaining uniformity. It has been experimentally demonstrated that two-layer films with miscible layers can undergo dewetting, but theoretical understanding of this phenomenon is lacking. Through a lubrication-theory-based model, we study the mechanisms initiating dewetting in miscible two-layer two-component films. The model film consists of nonvolatile solvent and solute with constant density and viscosity. Two coupled fourth-order nonlinear PDEs that describe the time-evolution of the film height and solute concentration are derived and solved with a spectral method. A disparity in the initial solute concentration between the film layers drives flows that lead to height nonuniformities and eventually dewetting. A parametric study is conducted to examine the influence of system parameters on this behavior and develop several scaling relations.   

Dynamics of Coating Flow on Rotating Circular and Elliptical Cylinders

Coating the exterior of an object in a layer of fluid is a fundamental problem in fluid mechanics, and perhaps the most well-known example of this problem is coating a uniformly rotating horizontal circular cylinder with a thin film of fluid, which was studied in the pioneering papers by Moffatt (1977) and Pukhnachev (1977). While this problem has been well studied in recent years (and has been extended to incorporate a variety of other physical effects), there has been almost no work on non-circular cylinders and, in many practical applications (such as the coating of chocolate bars and orthopaedic implants), the substrate may not be circular. Two-dimensional flow on the surface of a rotating elliptical cylinder was first studied by Hunt (2008) and more recently by Li et al.\ (2017), both of which used Direct Numerical Simulation (DNS). We use lubrication theory to derive and analyse a reduced model for thin-film flow on a uniformly rotating elliptical cylinder. This approach retains the essential physics inherent in the full two-dimensional Navier-Stokes problem, but is much less computationally expensive than DNS. Our calculations show that even a small eccentricity can cause a significant difference in the behaviour compared to the perfectly circular case.   

Effect of Marangoni stresses on the three-dimensional wave dynamics of surfactant-covered falling liquid films

The ubiquity of falling films in industrial processes and daily applications has inspired the scientific community for decades, producing a number of crucial reviews in recent years. In the context of falling film flows, surfactants reduce surface tension, additionally introducing variations of this quantity that give rise to Marangoni stresses, which have a flow stabilizing effect. In recent years, the effect of insoluble surfactants on falling films has been studied using stability theory. This work aims to study the effect of insoluble surfactants on the nonlinear dynamics of falling liquid films using fully three-dimensional numerical simulations. This study will present the effect of varying the surfactant Marangoni parameter on the emergent wave dynamics and critical vortical structures.   

stability of a uniaxial marangoni flow

Marangoni flows result from surface-tension gradients, and while these flows may occur over finite distances on the surface, subsequent secondary flows can be observed on much larger lengthscales. These flows play major roles in various phenomena, from foam dynamics to microswimmer propulsion. We show here that if a Marangoni flow of soluble surfactants is confined laterally, the flow forms an inertial surface jet. A full picture of the flows on the surface is exhibited, and the velocity profile of the jet is predicted analytically, and is successfully compared with the experimental measurements. Moreover, this straight jet eventually destabilizes into meanders. Quantitative comparison between the theory and our experimental observations yields a very good agreement in terms of critical wavelengths. The characterization and understanding of the 2D flows generated by confined Marangoni spreading is a first step to understand the role of inertial effects in the Marangoni flows with and without confinement.   

Nonlinear Dynamics and Plug Formation in a Long-wave Model of Film Flows Inside a Tube in the Presence of Surfactant

A long-wave model based on lubrication theory is developed for the flow of a viscous liquid film lining the interior of a vertical tube in the presence of an insoluble surfactant on the interface; no thin-film assumption is made. Linear stability analysis identifies two modes; in the absence of base flow, the `interface' mode is the only unstable mode. The growth rates of this mode serve as an accurate predictor of how surfactant concentration increases plug formation time, and the effects of film thickness on this increase are quantified. For a falling film, both the interface mode and `surfactant' mode may be unstable, resulting in a richer variety of free-surface dynamics. Previous work has shown that turning points in families of traveling wave solutions for gravity-driven film flow with a clean interface can be a good indicator of $h_c$, the critical thickness past which plugs may form. In the presence of surfactant, it is found that turning points in branches of traveling waves that arise from an unstable surfactant mode give an estimate of $h_c$, provided the interface mode is linearly stable. When both modes are unstable, interpretation of these turning points as they relate to plug formation is more complicated.   

The influence of viscoelasticity on the dewetting of ultrathin polymer films

The influence of viscoelasticity on the dewetting of ultrathin polymer films is unraveled by means of theoretical analysis and numerical simulations. The Oldroyd-B, Giesekus, and FENE-P models are employed to analyze the dynamics of film rupture in the limit of negligible inertia. The onset of temporal instability is analyzed for the first time using linear theory. For times close to the rupture singularity, the self-similar regime recently obtained by Moreno Boza et al. (Phys. Rev. Fluids 5, 2020), is asymptotically established independently of the rheological model. The spatial structure of the flow is characterized by a Newtonian core and a thin viscoelastic boundary layer at the free surface, where polymeric stresses become singular as rupture is approached. The Deborah number and the solvent-to-total viscosity ratio emerge as the relevant parameters controlling the rupture time and the length scale of the resulting dewetting pattern. The asymptotic flow structure close to rupture is however unaffected by the choice of rheological model, which is thus shown to be universal.   

Thin-Liquid-Film Flow on Three-Dimensional Topographically Patterned Rotating Cylinders

The coating of rotating discrete objects with surface topography is a problem commonly encountered in manufacturing processes. To study this problem, we model the flow of thin liquid coatings in three dimensions on topographically patterned cylinders that rotate about their horizontal axes. An evolution equation describing variations in the coating thickness is solved numerically. In the limit of a rapidly rotating cylinder, we find that liquid accumulates at either pattern crests or pattern troughs. Using a long-wave analysis, we derive an expression for the critical Weber number separating these regimes. When gravity is reincorporated, the accumulation of liquid at crests or troughs may cause the coating to sag, leading to the formation of droplets or rings whose average spacing at large rotation rates is controlled by the balance between centrifugal and surface-tension forces. Flow visualization experiments yield results that agree quantitatively with predictions of the simulations and long-wave analysis. We observe the most uniform coatings in experiments at moderate rotation rates, where disturbances in the coating thickness develop slowly. This indicates that to obtain nearly uniform coatings in practice, the coating must be solidified faster than disturbances can develop.   

Stability of binary evaporating thin films

The dynamics and stability of thin liquid films over curved substrates affect us on a daily basis, from tear films to lubricant foaming. Even compositionally simple systems can exhibit complex behavior under the appropriate circumstances. In our work, we compare and contrast the behavior of an evaporating, binary silicone oil film over a glass dome and over an air bubble. We demonstrate that the interplay between Marangoni flows (related to liquid composition), evaporation, gravity, diffusion and capillarity dictates the behavior of these films. Our interferometric experiments show that when the Marangoni driving force is large enough, the initially axisymmetric film will eventually break symmetry. In computational solutions of the appropriate 2D lubrication model, we observe the same type of symmetry breaking bifurcation above a critical condition, defined by the liquid composition and diffusivity (i.e. Peclet number). A linearized disturbance analysis further reveals that the system is linearly unstable above a given Peclet number, but nonlinear effects can either be stabilizing or destabilizing in time.   

On the origin of draperies structures in limestone caves: two-dimensional analysis of the impulse response

We investigate the role of hydrodynamic instabilities in the morphogenesis of typical draperies structures encountered among other speleothems in limestone caves. The problem is tackled using the lubrication approximation for the fluid film flowing under an inclined plane, in the presence of substrate perturbations that grow according to a classical deposition law. We generalize to the two-dimensional case the spatio-temporal analysis of the linear impulse response resulting from linear simulations. We exploit the concepts of Riesz transform and monogenic signal, the multi-dimensional complex continuation of a real signal, to retrieve the asymptotic properties of the wavepacket. The isotropy of the pure hydrodynamic solution is broken and the deposition process selects predominant streamwise structures on the substrate as the response is advected away. Furthermore, the presence of an initial localized perturbation on the substrate produces a quasi-steady region characterized by streamwise structures both in the fluid film and on the substrate. We suggest that these linear selection mechanisms contribute to the formation of draperies under inclined cave ceilings.   

Elastic amplification of the Rayleigh-Taylor instability in solidifying melts

The concomitant deformation and solidification of melts is relevant to a broad range of phenomena. Examples include the preparation of cotton candy, the atomization of metal, the manufacture of glass fibers... The shape of the solids formed in these processes is typically determined through the competition between the deformation of the liquid phase and its solidification, such that solid-like deformations halfway through solidification are rarely envisioned. Here we show that very soft solids in the midst of solidification (G $\sim$ 100Pa) can be permanently deformed to form previously unknown periodic structures. Namely, we generate an array of droplets on a thin layer of liquid elastomer melt coated on the outside of a rotating cylinder through the Rayleigh-Taylor instability. Then, as the melt goes through its gelation point, we stretch these drops into elastic hairs. The ongoing solidification eventually hardens the material, erasing the memory of the deformations. Using experiment, simulation and theory, we demonstrate that this coupled liquid-elastic hair pattern can be rationalized by combining tools from fluid mechanics and elasticity.   

Self Templating Assembly of Drop Lattices

We study the recursive Rayleigh-Plateau instability of neighboring viscous threads. We have recently found that the successive breakup of viscous threads deposited in an immiscible bath from a moving nozzle generates periodic drop patterns. In addition to the low-energy hexagonal lattice we report a variety of other non-hexagonal lattices obtained by adjusting nozzle translation speed and exploring diverse extrusion toolpaths, e.g. spirals. In order to elucidate this self-assembly mechanism, we study the instability of a single thread close to a periodic template. We find that the presence of a boundary drives the dynamics of the instability and affects the breakup pattern in a certain regime of parameters we will specify. We quantify the ``memory'' of the system, predicting when the patterns bear an imprint of the initial conditions or instead evolve towards universal solutions. We leverage this understanding to engineer the lattice morphology and characteristics.   

Modelling and Simulation of Spin Coating on a Spherical Substrate

What do solar cells, printed circuit boards, microprocessors, and LED displays have in common? They are essential to 21st-century life, and they are all limited to flat geometries by the use of spin coating during manufacturing. Here, we present a lubrication-based model for the flow of a thin film on a rotating sphere. This was used to model spin coating a polymer film, which cures over time, on a spherical substrate. We see that centrifugal force causes the accumulation of fluid in a distinct peak near the equator of the sphere. We investigate the effect of varied substrate kinematics and film thickness, as well as the impact of different fluid properties. Finally, we briefly consider the effect of rotation on the spreading of a fluid film from a non-uniform initial condition. Overall, the uniformity and smoothness of the coated film consistently worsened as a result of spin coating, compared to the effect of gravity only.   

Dynamics of a drying viscoelastic polymer solution

Drying polymeric solutions with a volatile solvent are encountered frequently in technological applications. So far, a number of studies have focused on the effect of solutocapillary and thermal Marangoni effects, while the effect of the rheological characteristics of the polymeric film have been largely ignored. Polymeric solutions, though, are known to exhibit viscoelastic behavior with properties (i.e. viscosity, relaxation time) that may depend on the local concentration of the solvent. Here, we develop a theoretical model that fully takes into account the viscoelastic nature of these solutions. In the limit of slow evaporation, we perform a linear stability analysis under the quasi-steady assumption and perform an extensive parametric study. In order to examine the dynamics in the non-linear regime, we also perform time-dependent simulations, based on a finite element formulation. Our numerical results indicate that the increasingly important effect of viscoelasticity (due to the continuous increase of the polymer solute concentration) destabilizes the flow and also leads to patterns with smaller wavelengths. Finally, we discuss the mechanisms which give rise to these instabilities.   

A hierarchical modelling approach for the active control of thin liquid film flows

The ability to robustly and efficiently control liquid film dynamics is a challenging topic which lies at the heart of applications such as coating (where the liquid-gas interface should ideally be close to flat) and heat or mass transfer (where an increase in interfacial area is desirable). Mathematically this is a framework in which progress can be made based on reduced-order modelling and asymptotic analysis. This leads to an extended range of long-wave models incorporating levels of simplification that balance our theoretical interrogation abilities with access into a wider practical parameter space. Our goal here is to develop powerful feedback strategies at lower (and more cost-effective) levels of the modelling hierarchy and investigate/extend their ability to translate into real-life solutions by using direct numerical simulations (DNS) of the multi-phase Navier-Stokes equations as an in silico experimental platform. We discuss both distributed and point-actuated mechanisms in a unified analytically-informed high performance computing context in which the DNS can accurately capture the relevant nonlinearities and put the rigorously developed control strategies to the test, even beyond their traditional ranges of applicability.   

Helical Instability of an Eccentric Coated Fiber

We study the destabilization of a gravity-driven viscous flow coating a vertical fiber. Numerous studies have focused on the transition of a liquid thread into a downward traveling train of beads along a fiber, a phenomenon known as Rayleigh-Plateau instability, in the limit of small Bond numbers, $Bo$, where the surface tension dominates over gravity. We here explore the limit of large $Bo$, i.e. centimetric radial sized liquid column. The experiments are carried out using highly viscous silicone oils to focus on inertialess flows (large Ohnesorge number, $Oh$). We observe the formation of a helical interface, coiling around the thin fiber, in this limit of high $Oh$ and $Bo$. We address theoretically the physical mechanism underlying the observed interface coiling and its associated geometric and hydrodynamic thresholds, by means of the linear stability analysis of a unidirectional flow along a rigid eccentric fiber. The asymmetry of the drainage velocity (shear distribution) above a certain threshold induces coiling. Overall, small fiber radius and large eccentricity tend to promote the coiling of the interface ($m=1$ modes), while reducing $Bo$ tends to preserve an axisymmetric interface ($m=0$ modes). We will compare the predictions of our model with experimental results.   

Grow or perish: Dynamics of a pendant drop sliding on a thin film

Anyone who has ever painted a ceiling knows that thin coatings can destabilize into drops spontaneously under the action of gravity via the Rayleigh-Taylor instability. Once formed, these pendant drops interact with the thin film on which they lie on, producing interesting non-linear dynamics; they spontaneously move even on perfectly horizontal surfaces. Using experiments and numerical simulations we study the dynamics of such pendant drops on slightly inclined pre-wet substrates. We show that for a given film thickness, both the drop size evolution and its velocity on the substrate are highly sensitive to the inclination angle. For sufficiently large angles ($\sim 1-2$ deg), the drop shrinks and leaves a thicker film in its wake, i.e. a rivulet, while at lower angles the drop grows and depletes the thin film as it moves. Steady motion only occurs at the transition, when the drop neither grows nor shrinks.   

Optimal electrostatic control of fluid films

Controlling film flows has long been a central target for fluid dynamicists due to its ubiquitous applications, in fields from heat exchangers, to biochemical recovery, to semiconductor manufacture. However, despite its significance in the literature, most analyses have focussed on the “forward” problem: what effect a given control has on the flow. Often these problems are already complex, incorporating the - generally multiphysical - interplay of hydrodynamic phenomena with the mechanism of control. Indeed many systems still defy meaningful agreement between models and experiments. The inverse problem - determining a suitable control scheme for producing a specified flow - is considerably harder, and much more computationally expensive. Performing such calculations for the full Navier-Stokes problem is generally prohibitive. Using an electric field as a control mechanism, we examine the inverse problem. We derive a low-order model that is accurate even deep into the shortwave regime. A rapid solver allows repeated solution of both the forward and adjoint problems on sub-second timescales, allowing both terminal and regulation optimal control studies to be implemented. We exploit this in a variety of novel ways in combination with direct numerical simulations.   

Interfacial instabilities in electrified liquid films

The electrohydrodynamic instabilities of fluid-fluid interfaces can be exploited in various microfluidic applications in order to enhance mixing, replicate well-controlled patterns or generate drops of particular size. In this work, we study the dynamics and stability of a system of three superimposed layers of two immiscible fluids subject to a normal electric field. Following the Taylor-Melcher leaky dielectric model, the bulk remains electroneutral while a net charge accumulates on the interfaces. The interfacial charge dynamics is captured by a conservation equation accounting for Ohmic conduction, advection by the flow and finite charge relaxation. Using this model, we perform a linear stability analysis and uncover different modes of instability in terms of the relevant dimensionless groups of the problem. Further, we perform numerical simulations using the boundary element method in order to study the full nonlinear problem. We demonstrate how the coupling of flow and interfacial charge dynamics in different modes of instability gives rise to non-linear phenomena such as tip streaming or pinching into droplets.   

Overcoming Rayleigh-Plateau instabilities in liquid metal streams via electrochemical oxidation

Liquid streams emerging from a nozzle break up rapidly into droplets due to Rayleigh–Plateau instabilities driven by surface tension. We find that a room-temperature liquid metal, eutectic gallium indium, can be formed into stable cylindrical streams by applying an oxidizing potential to a slowly-injected, high-surface-tension metal. We observe a range of morphologies, including droplets, fine (100-$\mu$m diameter) wires, and irregular shapes. The wire-like streams are stable enough to flow and bend around obstacles, suggesting their utility as means of producing and controlling metallic structures at room temperature.   

DNS of superimposed fluids under the action of DC electric field

This study aims to explore the electrohydrodynamic-driven instability of the interface separating two horizontal immiscible liquid layers. The fluids are confined between two electrodes, where the light and less conducting liquid is overlaid on top of the heavy and more conducting one. Direct Numerical Simulations are performed using a front tracking/finite difference scheme in conjunction with Taylor-Melcher leaky dielectric model. The interface remains stable below a critical electric field strength and became unstable beyond it. For a sinusoidal initial perturbation, the instability leads to formation of a liquid column that penetrates from the lower fluid into the upper one. It is shown that the relative importance of the ratio of the electric conductivity and permittivity of the two fluids play a key role in determination of the shape of the columns and depending on this parameter, the liquid column may settle to an equilibrium height or continue to grow until its growth is limited by the upper wall. Examination of the structure of the flow field provides insight about the two observed behaviors.   

Bifurcation Leading to Periodic Structuring of Nanofilms By External Modulation of a Thermocapillary Instability

While temporal modulation of a driving field provides an effective means to control fluid systems, far less attention has been focused on spatial modulation to enforce a high degree of uniformity in pattern and growth rate, especially in phenomena triggered by noise. Here we explore the nonlinear dynamics accompanying spatial modulation of a thin slender film prone to an intrinsic long-wavelength thermocapillary instability arising from noise. The frequency of the spatially periodic modulation can either be close or far from the frequency characterizing the instrinsic instability. The modulation can be enforced either by substrate thermal control or periodic features patterned into the substrates used to heat and cool the film. Here we present results of a combined analytic and numerical study to probe early, intermediate and late time behavior of an evolving film in the linear, weakly nonlinear and fully nonlinear regime. Frequency analysis coupled with simulations of the governing nonlinear interface equation elucidate the mechanism leading to bifurcation, whose behavior changes with modulation amplitude and frequency. Based on our findings, we provide estimates of various experimental quantities useful to the thermocapillary design of micro-optical arrays.