Schedule for: 19w5148 - Modelling of Thin Liquid Films-Asymptotic Approach vs. Gradient Dynamics

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

An overview is given about formulations of thin-film models in terms of a gradient dynamics on an underlying free energy. First, the approch is reviewed for films of nonvolatile and volatile simple liquids as well as for films of mixtures and suspensions [1]. Next, we discuss how the developed gradient dynamics form may be employed to develop models for liquids with more complex phase behaviour and intricate cross-couplings [2], e.g., in the case of concentration-dependent wettability [3]. Finally, we discuss cases where kinetic equations obtained via an asymptotic approach and via a gradient dynamics approach do not fully agree. [1] U. Thiele, Colloids Surf. A 553, 487-495 (2018). [2] U. Thiele, A. Archer, L. Pismen, Phys. Rev. Fluids 1, 083903 (2016). [3] U. Thiele, D. Todorova, H. Lopez, Phys. Rev. Lett. 111, 117801 (2013).

In this talk I will discuss non-equilibrium molecular dynamics simulations of droplets on inclined superhydrophobic and slippery liquid-infused infused porous surfaces. I will investigate steady state slipping and rolling motion of the droplets in viscous, surface, and contact line friction dominated limits, and consider the role of surface anisotropy on the steady state dynamics.

Helicoidal plywoods are an ubiquitous biological fibrous composites structure found in collagen and cellulosic materials. These cholesteric liquid crystal analogues display the Bouligand architecture which is associated with bulk and surface multifunctionalities such as sensor/actuator and structural color , as well as optimized mechanical and tribological properties. In this presentation, motivated and guided by biological surface topographies found in insects, plants, and fish scales, we present a model of surface pattern formation for chiral surfaces and reveal the elastic mechanisms that generate simple and complex wrinkling. Introducing the liquid crystal capillary vector we are able to efficiently map the relations between surface curvature, liquid crystal anchoring, chirality  and surface tension. Scaling laws of wrinkling amplitude and wave-length as a function of anchoring and chirality are derived. In the simplest case, a fiber orientation surface gradients generate a single scale harmonic whose amplitude is proportional to anchoring and whose wave-length is set by the orientation. We show how by manipulating the material property and fiber gradient space generates targeted patterns with desirable novel properties, such as low friction surfaces.Finally differential geometry-Lame surface stress curves relations is established.

The spreading of bacterial colonies at solid–air interfaces is determined by the physico-chemical properties of the involved interfaces and bioactive growth processes. The production of surfactant molecules by bacteria is a widespread strategy that allows the colony to efficiently expand over the substrate. On the one hand, surfactant molecules lower the surface tension of the colony, effectively increasing the wettability of the substrate, which facilitates spreading. On the other hand, gradients in the surface concentration of surfactant molecules result in Marangoni flows that drive spreading. These flows may cause an instability of the circular colony shape and the subsequent formation of fingers. In this work, we study the effect of bacterial surfactant production and substrate wettability on colony growth and shape within the framework of a hydrodynamic thin film model. We show that variations in the wettability and surfactant production are sufficient to reproduce four different types of colony growth, which have been described in the literature, namely, arrested and continuous spreading of circular colonies, slightly modulated front lines and the formation of pronounced fingers.

We discuss the use of feedback control in suppressing the inertial instabilities of a falling liquid film. In this setup, an input to the system, e.g. local fluid injection, is chosen in response to real time observations of the interface shape. If the control scheme is designed with perfect knowledge of the governing equations and implemented with access to instantaneous observations of the entire system state, feedback control offers almost unlimited scope to change the system dynamics. Both of these requirements fail in any practical implementation, but we would hope to be able to achieve effective control strategies based on reasonably accurate models and observations of only a few key variables. Falling liquid films have a hierarchy of long-wave models of increasing complexity, which offers an ideal environment to explore robustness to model choice. We find that the success of control schemes based on low order models is dependent on the method of actuation chosen; injection of fluid has a direct effect on interface dynamics and simple control strategies work well across many models, while selective substrate heating has a much more subtle effect on dynamics and control schemes are correspondingly more sensitive to details of the flow.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

Colloidal forces are known to influence the pattern deposition of nanoparticles off a volatile carrier liquid. Several experimental studies suggest a direct connection between colloidal forces and the morphology of the particulate deposit [1,2,3]. Specifically, variations in the zeta potential of the suspended particles and substrate; and variations in the ionic strength in the suspension were found to alter the geometry of the deposit. We use theory and experiment to investigate the connection between colloidal forces and the pattern deposition of colloidal particles off a volatile carrier liquid. We connect between colloidal forces and pattern deposition by considering the adhesion of particles to the solid substrate and the coagulation of particles in the suspension [4,5]. Our initial theoretical approach is based on an asymptotic – long wave type – model for the deposition process. In this model we employ the interaction–force boundary layer theorem to account for the rate of adhesion of particles to the solid substrate. The theorem allows for manifesting the dynamic process of adhesion in a form that is reminiscent of a first order chemical reaction in a dynamic advection-diffusion equation for the transport of particle mass in the liquid. Similarly, we employ the augmented Smoluchowski theorem to account for particle aggregation. Hence, the rate of aggregation is manifested in a form that is reminiscent of multiple second order chemical reactions. The coefficients of the reaction terms are associated with the energy barriers for adhesion or coagulation. Comparing the predictions of our asymptotic model to experiment, we find that the rate of adhesion, coagulation, and diffusion of particles in the volatile liquid as well as the rate of liquid evaporation govern the deposition of particles. Moreover, fast adhesion of particles to the solid substrate and fast diffusion of particles in the liquid disperse the spatial distribution of the particulate deposit. Fast coagulation and fast evaporation support the deposition of denser patterns of particles of sharper spatial boundaries. A different theoretical approach for modelling the deposition problem, which we currently pursue, is to employ phase change type energy functional to account for particle coagulation and adsorption in the volatile liquid film. Such an approach should naturally incorporate the physics associated with concentrated particulate systems and particle volume effects that are not accounted for in the current asymptotic model. References 1. R. Bhardwaj, X. Fang, et al. Self-Assembly of Colloidal Particles from Evaporating Droplets: Role of DLVO Interactions and Proposition of a Phase Diagram, Langmuir 26 (7833–7842) 2010 2. M. Anyfantakis, Z. Geng, et al. Modulation of the Coffee-Ring Effect in Particle/Surfactant Mixtures: the Importance of Particle−Interface Interactions
. Langmuir 31 ( 4113–4120) 2015 3. E. Homede, A. Zigelman, et al. Signatures of van der Waals and electrostatic forces in the deposition of nanoparticle assemblies; J. Phys. Chem. Lett. 9 (5226–5232) 2018 4. A. Zigelman and O Manor. Simulations of the dynamic deposition of colloidal particles from a volatile sessile drop, J. Colloids Interface Sci. 525 (282-290) 2018 5. A. Zigelman and O. Manor. The deposition of colloidal particles from a sessile drop of a volatile suspension subject to particle adsorption and coagulation, J. Colloids Interface Sci. 509 (195) 2018

We will describe the derivation of a nonlinear evolution equation that describes space-time self-organization at the free surface of a solid target undergoing irradiation by an energetic ion beam. Under this type of driving, for many materials the outermost surface layer of the target responds as a highly viscous fluid, displaying formation of nanoscale ripples in macroscopic time scales. In spite of the irrelevance of gravity at these small distances, the weakly nonlinear limit of the equation resembles the well known description of a macroscopic incompressible viscous thin film flowing down an incline, which is a paradigmatic instance of free surface flow systems for which the morphological instability responsible for pattern formation is controlled by inertial effects. The predictive power of the evolution equation for ion-beam surface nanopatterning underscores nonlinear effects that might have been expected to be of a secondary importance in such a nanoscopic-scale, Stokes-flow system. The content of this talk is joint work with Mario Castro (Universidad Pontificia Comillas) and Javier Muñoz-García (UC3M).

This talk focuses on the flow of a solidifying liquid film on a solid surface subject to a complex kinematics, a process relevant to pancake making or surface coating. The key question this study aims to address is: what is the optimal surface kinematics in order to spread the liquid layer uniformly on the surface? We present an optimal control method based on an adjoint formulation of governing partial differential equation. Key benefits of this method are that no assumption is required on the functional form of the controls and that significant improvement in thickness uniformity can be achieved at a low computational cost.

This talk will focus on recently developed models and computational techniques for thin films, with focus on nematic liquid crystal films. Models and computations are developed within the framework of long wave approach, augmented by inclusion of liquid-solid interaction forces via disjoining pressure model.  A particular aspect that will be discussed is the inclusion of the liquid-crystalline nature of the film into the model in a tractable manner. The simulation techniques include algorithms for GPU computing that allow for simulations of large domains and analysis of various instability mechanisms.

Asymptotic methods yield an efficient leading-order model of flow along a thin curved duct. The flow consists of a primary axial component along the duct and, because of the curvature, a secondary “Dean flow” in the cross section. Such flows are used in microfluidics to sort suspended particles by size; an important application is “liquid biopsy”, the isolation of a particular cell type within a (dilute) cell suspension. A description of the background flow, in the absence of particles, is pre-requisite to modelling the perturbation caused by a particle and the migration of the particle within the duct cross-section. I will discuss the suitability of a thin-film model for this purpose. Time permitting, I will also discuss the use of asymptotic methods for modelling of particle migration.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

In this talk, I will discuss various phase field models in the context of multi-scale diffusion dynamics. In particular, I will present our recent work on a family of dynamic boundary conditions which is to describe the force balance on the boundaries.

Modeling of partial differential equations is an art that leaves the scientists with a choice of what is deemed important, e.g., mathematical well-posendess, conservation laws, thermodynamic consistency, … ,or just simplicity - in multiphysics settings this calls for appropriate modeling strategies. In this talk, a thermodynamics inspired strategy for the modeling of free boundary flows with moving contact lines will be revisited. It will be argued that this mathematical structure is not just useful for modeling but also for the numerical approximation of solutions. Furthermore, possible strategies for their extension to non-smooth settings will be shortly motivated, e.g., hysteresis and particulate flows.

The long-wave dynamics of an elastic sheet lubricated by a thin layer of viscous fluid on a wetting substrate is examined using both numerical simulations and a quasistatic model. As a continuum model for a lipid bilayer, the elastic sheet has a electric capacitance that  stores changes across the membrane. Within the lubrication framework we investigate the electrohydrodynamics of the thin film coupled to the capacitive elastic membrane. Novel nonlinear dynamics is found and results are useful to understand electroformation, a standard experimental procedure to fabricate unilamellar vesicles from planar lipid bilayer membrane. We also extend this theoretical framework to investigate the rupture of vesicles when settled onto a glass substrate in an ionic solution.

The flow of an electrified liquid film down an inclined plane wall is investigated with the focus on coherent structures in the form of travelling waves on the film surface, in particular, single-hump solitary pulses and their interactions. The flow structures are analysed first using a long-wave model, which is valid in the presence of weak inertia, and second using the Stokes equations. Solitary-pulse solutions exist under certain conditions. The electric field increases the amplitude of the pulses, can generate recirculation zones in the humps and alters the far-field decay of the pulse tails from exponential to algebraic with a significant impact on pulse interactions. A weak-interaction theory predicts an infinite sequence of bound-state solutions for non-electrified flow, and a finite set for electrified flow. The existence of single-hump pulse solutions and two-pulse bound states is confirmed for the Stokes equations via boundary-element computations. In addition, the electric field is shown to trigger a switch from absolute to convective instability, thereby regularising the dynamics, and this is confirmed by time-dependent simulations of the long-wave model.

Catalysts are usually made of a dense but porous material such as activated carbon, zeolites, etc. that provide a large surface area. Liquids that are produced as a by-product of a gas reaction at the catalyst site transport to the surface of the porous material, slowing down transport of the gaseous reactants to the catalyst active site. Understanding the dynamics of the liquid films in the gas channel is critical to maintain performance and durability of the catalyst assembly. We develop a mathematical model for liquid films dynamics and study different regimes.

Self-organization or dynamic self-assembly is a mechanism responsible for the formation of complex structures through multiple interactions among the microscopic components of the system. We are interested in the formation of regular stripe patterns during the transfer of surfactant monolayers from water surfaces onto moving solid substrates by means of a generalized Cahn-Hilliard equation. A combination of numerical simulations and continuation methods is employed to investigate stationary and time-periodic solutions of the model. Further, the influence of the spatio-temporal forcing on the patterning process is discussed. We show that the occurring locking effects enable a control mechanism via properties of the forcing and facilitate the production of patterns with a broader range of features. In two dimensions, the production of a variety of complex patterns can be achieved through the competition of intrinsic properties of the pattern forming system and the external forcing.

We present a continuum-level description of dynamics of a thin electrolyte film on substrate which is characterized by spatially periodic variation of surface properties.The model couples together the electrostatic effects and viscous flow in the liquid. Linear stability analysis is carried out using a combination of numerical techniques for finding the eigenvalues of the discretized stability problem, asymptotic methods valid for small charge density variation, and Floquet theory. Substrate  non-uniformity can have either stabilizing or destabilizing effect. For the important practical case of a liquid film with oppositely charged boundaries and thickness comparable to the Debye length, transition from stabilizing to destabilizing influence is observed as the patterning wavelength is decreased. Numerical simulations of the strongly nonlinear evolution of the film are conducted, with emphasis on competition between patterns induced by substrate nonuniformity and by the intrinsic nonlinearity present even for uniform substrate. The topic of motion of contact line over patterned surface will also be discussed briefly. (Joint work with M. Jutley.)

When evaporated on a substrate, a droplet containing solutes usually deposits the solutes onto the substrate in a ``coffee-ring'' pattern. Recent experiments have shown that substrate permeability can suppress the coffee-ring pattern and promote more uniform solute deposition.  Motivated by these observations, we have developed a lubrication-theory-based model to describe imbibition and evaporation of droplets of colloidal suspensions on permeable substrates. The model consists of a system of one-dimensional partial differential equations accounting for the changing droplet shape and depth-averaged concentration of colloidal particles. We also incorporate a precursor film, disjoining pressure, and substrate topography to control contact-line motion of the droplet. Solvent evaporation is described using the well-known one-sided model, and imbibition of solvent by the substrate is assumed to only depend on the excess pressure on the liquid side. The governing equations are solved with finite-difference methods.  Our results reveal that solvent evaporation and solvent imbibition have the same qualitative effect on the final particle deposition pattern.  For the case where the substrate is smooth, we find that increasing imbibition or evaporation leads to a transition from a cone-shaped deposition pattern to a ring-shaped deposition pattern. For the case where the substrate is rough, the droplet contact line is pinned at a defect on the substrate, and the pinning-depinning transition leads to the ``bullseye'' deposition pattern often observed in experiments.  Finally, we also find that particle adsorption onto the substrate can promote more uniform particle deposition patterns for both smooth and rough substrates, and solvent imbibition can indirectly suppress the coffee-ring pattern by inducing more particle adsorption.

The drying of liquid droplets is a common daily life phenomenon that has long held a special interest in scientific research. We propose an Onsager variational principle theory that describes the droplet shape evolution and predicts the deposit distribution of nonvolatile components on the substrate. It is shown that for the drying of a single droplet, the deposition pattern changes continuously from a coffee ring to volcano-like and to mountain-like depending on the mobility of the contact line and the evaporation rate. When drying of two neighbouring droplets, asymmetrical ring-like deposition patterns are formed, including fan-like and eclipse-like deposition patterns. The same theoretical model is also used to explain the multi-ring patterns of the deposit that are formed when colloidal suspensions are dried on a substrate. Using a standard model for the stick-slip motion of the contact line, the theory predicts (a) the multi-ring patterns are not observed at high evaporation rate, (b) the spacing between rings decreases with the decrease of the ring radius, and (c) the multi-ring pattern is taken over by a disk pattern near the center. These results are in qualitative agreement with existing experiments, and the predictions of the theory about how the evaporation rate, droplet radius and receding contact angle affects the pattern can be tested experimentally.

Often macroscopic properties (e.g., wettability and slip of liquids or (micro)phase-separation in binary polymer systems) are dictated by the molecular architecture and microscopic pairwise interactions. In order to connect microscopic particle-based models with continuum descriptions (e.g., thin-film approximation or phase-field models) one has to identify key parameters that encode the microscopic structure into continuum models and devise computational techniques to compute these parameters. In turn, the continuum model may be coupled to particle-based description to speed-up the latter. In my talk I will discuss some aspects of this coupling using simple polymer liquids, polymer blends and copolymer systems.

Given suitable constitutive laws, thin-film equations can be derived for non-Newtonian fluids. In this talk, I will review the procedure for fluids with a yield stress, and then confront the resulting theory with experiments and computations that avoid the thin-layer approximation. The comparison with the computations illustrates that the theory can be effective, but the comparison with experiments highlights how additional physical considerations are often needed in realistic settings.

In this talk, the characteristic properties of the lubrication approximation are discussed and its weak ellipticity is established, in contradistinction to the commonly accepted parabolic character of the lubrication equations resulting from the underlying unidirectional flow assumption. The weak ellipticity property allows the lubrication analysis to capture flow topologies around stagnation points, contact lines, and flows over edges, all of which normally require elliptic operators to be accounted for.  This is used to explain the empirically observed over-performance of the lubrication approximation from the perspective of characteristic analysis.

The rupture of a thin film leads to a drop pattern that is somewhat coarser than that predicted by linear theory. The unstable evolution of a film implies the formation of a mesh of connected bridges as a consequence of the growth of the initial holes in it. This late stage of hole expansion is an important intermediate stage in the further evolution of the system to the final drop pattern. Simulations show that the mesh is broken into disconnected short filaments that evolve to drops. However, the analysis of this late stage requires some additional considerations not taken into account in the linear theory of thin films or infinite filaments. Therefore, a simple experiment consisting of a controlled regular grid of filaments is considered to give insight into the physics involved in this final stage. The experiments were performed both at millimetric (with PDMS oil) and nanometric (with liquid nickel) scales. After separation from the nodes each resulting short filament has a contraction in length and afterwards the bulges formed at their extremes break up leading to new drops. The iteration of this process leads to the final pattern. The retraction is modeled using a matching between Cox-Voinov and Blake models for the dewetting. On the other side, the breakup analysis considers a perturbed quasi equilibrium that leads to a Stokes flow at the necks. As a consequence, there are two characteristic distances from the breakup point to the unperturbed regions in the filament. They can be combined to determine whether or not there is coalescence of the retracting tips. The theory yields the number of drops as a function of the height and aspect ratio of the filaments. The observed data are compared with the predictions and show a good agreement.

We present a lattice-gas (generalised Ising) model for liquid droplets on solid surfaces. The time evolution in the model involves two processes: (i) Single-particle moves which are determined by a kinetic Monte Carlo algorithm. These incorporate into the model particle diffusion over the surface and within the droplets and also evaporation and condensation, i.e. the exchange of particles between droplets and the surrounding vapour. (ii) Larger-scale collective moves, modelling advective hydrodynamic fluid motion, determined by considering the dynamics predicted by a thin film equation. The model enables us to relate how macroscopic quantities such as the contact angle and the surface tension depend on the microscopic interaction parameters between the particles and with the solid surface. We present results for droplets joining, spreading, sliding under gravity, dewetting, the effects of evaporation, the interplay of diffusive and advective dynamics, and how all this behaviour depends on the temperature and other parameters.

The formation of iterated structures, such as satellite and subsatellite drops, filaments, and bubbles, is a common feature in interfacial hydrodynamics. Here we undertake a computational and theoretical study of their origin in the case of thin films of viscous fluids that are destabilized by long-range molecular or other forces. We demonstrate that iterated structures appear as a consequence of discrete self-similarity, where certain patterns repeat themselves, subject to rescaling, periodically in a logarithmic time scale. The result is an infinite sequence of ridges and filaments with similarity properties. The character of these discretely self-similar solutions as the result of a Hopf bifurcation from ordinarily self-similar solutions is also described. Joint work with M. Dallaston, D. Tseluiko and S. Kalliadasis.

The presence of a free-surface between air and liquid can lead to nonlinear behaviour, even in the absence of fluid inertia. Consequently, multiple distinct flow states may occur under the same driving conditions. A natural question to ask is whether simplified models of the system can capture the correct nonlinear behaviour, compared to the full field equations. For the (canonical) flow of a liquid film on the inner, or outer, surface of a rotating cylinder, we find multiple solutions and other complex behaviours can be accurately described over a wider range of parameters using a simplified model system derived via gradient dynamics rather than strict asymptotic arguments. (Joint work with Andre Von Borries Lopes and Uwe Thiele.)

with Mohar Dey and Xiaoyang Xu The tear film has a heterogeneous structure. Next to the ocular surface is a region hundreds of nanometers thick that is rich in mucin polymers. Atop this so-called mucus layer lies the aqueous layer, which is up to 5 microns thick. The outer surface of the aqueous layer is covered by a thin lipid layer that is exposed to the ambient air. The mucin protects the ocular epithelium against pathogens, and clinical evidence points to progressive destruction of the membrane-associated mucins (MAM) in eye infection. From a fluid mechanical viewpoint, the MAM modifies the wetting condition on the solid substrate, and modulates the van der Waals forces between the interfaces. We hypothesize that a change in the configuration of the MAM, say from complete coverage to partial coverage, induces a stronger van der Waals attraction and enhances slip on the ocular surface. Both factors should accelerate the breakup of the tear film. This will explain clinical observations of faster tear-film breakup in various eye diseases. We numerically simulate the tear-film breakup process to study the effect of the MAM through an elevated Hamaker constant and a modified slip length. Results show that the loss of MAM indeed precipitates tear-film breakup, and suggest that the tear-film breakup time can be a potential diagnostic of eye diseases.

Crystal growth is a model system for non-equilibrium physics. As a consequence, it has been widely studied, and in particular the nonlinear dynamics giving rise to complex shapes such as those of snow flakes have attracted much interest. However until now, most of these studies have been devoted to free growth, i.e. gowth in infinite systems. Confinement, which breaks translation invariance, gives rise to a novel class of nonlinear dynamics and morphologies, which are relevant in natural sciences such as geology –where crystal growth is often confined in small pores or faults of rocks, and for biomineralization, the process by which living organisms produce minerals (e.g. for their skeleton) –where growth is confined into a soft and complex environment. We propose a model for growth and dissolution of a crystal in confinement describing the non-equilibrium dynamics within the contact region using a continuum thin film equation [1]. Our model accounts self-consistently (in the lubrication regime) for surface tension effects, for the disjoining pressure, and for non-equilibrium transport processes such as diffusion and liquid convection. Based on this model, we study dissolution under a macroscopic load (pressure solution) [1] and growth under an applied supersaturation (crystallisation force) [2,3]. In pressure solution [1], the functional form of the crystal-substrate interaction potential appears to influence strongly the dynamics. For example, a divergent repulsion leads to flat contacts, and to a dissolution rate which increases indefinitely with the applied load. In contrast, a finite repulsion implies a sharp pointy contact shape, and a dissolution rate independent from the applied load. In confined growth it is well known that crystals can produce forces on their environment, and can exhibit a rim in the contact region. Our model shows the generic formation and growth of a precursor cavity which ultimately leads to the formation of the rim. The results are supported by experiments on NaClO3 in the University of Oslo [2].  We show that the formation of the cavity can be supercritical or subcritical, depending on the functional form of disjoining pressure [3]. Finally, the production of forces close to and far from equilibrium is also discussed [4]. References: [1] Thin film modeling of crystal dissolution and growth in confinement L Gagliardi, O Pierre-Louis, Physical Review E 97 (1), 012802 (2018) [2] Cavity formation in confined growing crystals F Kohler, L Gagliardi, O Pierre-Louis, DK Dysthe PHYSICAL REVIEW LETTERS 121, 096101 (2018) [3] Crystal growth in nano-confinement: Subcritical cavity formation and viscosity effects L Gagliardi, O Pierre-Louis, to be published in Journal of Crystal Growth (2019) [4] The non-equilibrium crystallization force L Gagliardi, O Pierre-Louis, preprint (2019).

Interfacial instabilities and pattern formation can occur in films on partially wetting surfaces. Lubrication models asymptotically reduce the governing equations for the free-surface flow to a fourth-order nonlinear parabolic partial differential equation for the evolution of the film height. Extensive studies have examined dynamics over a range of time-scales including finite-time rupture singularities and long-time droplet coarsening cascades when the fluid mass is conserved. We show that for volatile fluids, where the mass changes due to evaporation or condensation, the behaviors observed and the analysis needed yield new challenges and important differences from the non-volatile case. Some discussion will be given on dynamics that can occur when the form of the evaporative flux violates an energy dissipating structure. This is joint work with Hangjie Ji (UCLA).

I will discuss the problem of a thin-film flow down an inclined plane in the presence of gravity and surface tension, with the added complication that the flow is solidifying at its base.   This problem is motivated by the infiltration of liquid water into a porous snow-pack, which is governed by essentially similar equations and has significant climatic importance. Of interest is the stability of a uniformly propagating front, which is potentially subject to both capillary fingering (thicker regions advancing faster) and thermodynamic fingering (thicker regions solidifying less).

Recent experiments on thin films flowing down a vertical fibre with varying nozzle diameters present a wealth of new dynamics that illustrate the need for more advanced theory. We present a study of a full lubrication model that includes slip boundary conditions, nonlinear curvature terms and a film stabilization term. This study brings to focus the presence of a stable liquid layer playing an important role in the full dynamics. We propose a combination of these physical effects to explain the observed velocity and stability of travelling droplets in the experiments and their transition to isolated droplets.

The subject of this talk is the electrostatic stabilization of a viscous thin liquid film wetting the underside of a horizontal surface in the presence of an electric field applied parallel to the undisturbed interface. The formulated asymptotic model includes the effect of bounding solid dielectric regions typically found in experiments above and below the fluids. The competition between gravitational forces, surface tension, and the non-local effect of the applied electric field is captured analytically in the form of a nonlinear evolution equation. State-of-the-art computational tools based on the volume-of-fluid method are also implemented to both assess the range of validity of the derived model and guide these arguments towards practical (and highly nonlinear) contexts involving mixing, pumping and directed polymer assembly. Joint work with T.G. Anderson (Caltech), D.T. Papageorgiou (Imperial College London) and P.G. Petropoulos (NJIT).

The invasion of one fluid into a fluid occupying the narrow gap between parallel plates is a model for the displacement of a resident fluid in a porous medium by injecting a second fluid. In this paper, I describe equations formulated by Ruben Juanes and Luis Cueto-Felguereso based on a phase-field formulation. Numerical solutions of plane waves exhibit unusual behavior that is explored mathematically. Connections are made to finite speed propagation of a small amplitude wave solution of the porous medium equation, and to traveling waves that in a singular limit would be anomalous shock wave solutions of the equations. This research is documented in the thesis of Melissa Strait.

The Hele-Shaw problem is a classical example for studying interface dynamics or systems driven out of equilibrium. In this talk, I will first discuss numerical issues related to long-time computation of moving interfaces. Then I will present a time adaptive scheme for efficiently computing the dynamics of a moving interface. The idea is to design a time-space mapping such that in the new time scale, the interfaces can evolve at arbitrarily fast or slow speed. We then show numerical examples and present largest (expanding interface) and smallest (shrinking interface) Hele-Shaw simulation up to date.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.