# SCIENTIFIC PROGRAMS AND ACTIVITIES

August  1, 2015

THE FIELDS INSTITUTE
FOR RESEARCH IN MATHEMATICAL SCIENCES

 February 22-24, 2012 Workshop on Surfactant Driven Thin Film Flows to be held at the Fields Institute, Toronto Hosted by the Fields Institute Organizers: Marina Chugunova, University of Toronto Rachel Levy, Harvey Mudd College Linda B. Smolka, Bucknell University
 Abstracts

Edgar J. Acosta,
University of Toronto
Predicting lung mechanics from dynamic surface tension evaluations of lung surfactants (slides)

This presentation introduces an algorithm that predicts dynamic changes in lung pressure (P=f1(t)) and lung area (A=f2(t)) for a given ventilation protocol (lung volume, V=f3(t)). The lung pressure (P=f1(t)) was calculated considering the capillary (Pc=f4(?)) and tissue (Pt=f5(V)) contributions. The dynamic surface tension (?=f6(t)) was predicted using a compression-relaxation model (CRM) for lung surfactants and the dynamic changes in lung area. The CRM was also used to mine in-vitro dynamic surface tension data to determine the elasticity and the adsorption and relaxation rate constants of lung surfactants. Finally, the algorithm also includes a thermodynamic analysis that was used to link the instantaneous values of P,V, A and ?. The dynamic values of P,V, A and ? predicted with this CRM-PV algorithm were compared to experimental values obtained with rabbits, mice, and acid-injured mice models of acute respiratory distress syndrome (ARDS). Potential uses and limitations of this algorithm will be discussed.

Eugene Benilov
, University of Limerick
On advancing contact lines with a $180^{\circ}$ contact angle (slides)

This work builds on the foundation laid by Benney & Timson (1980),who examined the flow near a contact line and showed that, if the contact angle is $180^{\circ}$, the usual contact-line singularity does not arise. Their local analysis, however, does not allow one to determine the velocity of the contact line and their expressionfor the shape of the free boundary involves undetermined constants - for which they have been severely criticized by Ngan & Dussan V. (1984). As a result, the ideas of Benny & Timson (1980) have been largely forgotten. The present work shows that the criticism of Ngan & Dussan V. (1984) was, in fact, unjust. We consider a two-dimensional steady Couette flow with a free boundary, for which the local analysis of Benney & Timson (1980) can be complemented by an analysis of the global flow (provided the slope of the free boundary is small, so the lubrication approximation can be used). We show that the undetermined constants in the solution of Benney & Timson (1980)
can all be fixed by matching their local solution to the global one. The latter also determines the contact line's velocity, which we compute among other characteristics of the global flow.

Josh Bostwick
, North Carolina State University
Spreading and bi-stability of droplets driven by thermocapillary and centrifugal forces (slides)

We consider the spreading of an axisymmetric drop on a radially-heated, partially-wetting solid substrate in a slowly rotating geometry. Non-uniform heating of the substrate can generate both axial and radial temperature gradients along the drop interface, which produce distinct thermocapillary forces and equivalently flows that affect the spreading process. We utilize the lubrication approximation to derive an evolution equation for the contact-line radius that implicitly describes the interface shape and flow field. We show that competition between surface chemistry (wetting) and thermocapillary flows give rise to bi-stability in certain regions of parameter space. We also identify parameter regimes in which the droplet spreads indefinitely and compute spreading laws to compare with experiments on spreading droplets.

Richard Braun,
University of Delaware
Models for Tear Film Dynamics (slides)

The tear film is a critical element for ocular function and health. Tears must rapidly form a smooth multi-layer structure after a blink; the lipid layer must float atop the aqueous layer with the aid of being spread by polar lipids that function as insoluble surfactants. The lipid layer slows evaporation of water from the aqueous layer and helps to preserve it. Evaporation also concentrates salts in the tear film, collectively referred to as osmolarity, and elevated osmolarity is thought to be an important factor in dry eye.

We review dynamics of mathematical models of the tear film as a single layer with insoluble surfactant present, and with and without a separate lipid layer. Evaporation and osmolarity are incorporated into the model in some cases. The focus will be on approximating the contribution of the lipid layer in various ways. Comparison with experiment is shown where possible.

Almut Burchard,
University of Toronto
Convergence to equilibrium for a thin-film equation on a horizontal cylinder

In this talk, I will discuss recent work with Marina Chugunova and Ben Stephens on the long-term evolution
of a thin liquid film on a horizontal cylinder, modeled by the degenerate parabolic equation

u_t + [u^3(u_xxx + u_x - sin x)]_x=0 .

For each given mass, we find that the unique steady state is a droplet that hangs from the bottom of the cylinder
and meets the dry part of surface at zero contact angle. The steady state attracts all strong solutions, but the distance decays no faster than a power law. The proofs rely on energy and entropy estimates.

Erin Byrne,
Harvey Mudd College
The biofilm as a thin film: assumptions, advantages, and limitations

Biofilms form virtually everywhere, from riverbeds to the surface of our teeth. Comprised of individual bacteria and a gel-like extracellular polymeric substance (EPS), biofilms form complex three-dimensional structures that make modeling their morphology a challenge. In this talk, we discuss the insights gained by employing the thin film approximation in modeling biofilm development, as well as the model's assumptions and limitations.

Joshua Dijksman,
Duke University
Spreading of thin rotating films: Competition of thermal Marangoni, centrifugal, and gravitational forcing.

A rotating bucket can easily create a thin film whose dynamics can be influenced by gravitational, thermo capillary (Marangoni) and centrifugal forces. This "Newton's bucket" experiment therefore can serve as a test bed for thin film studies. We experimentally study this system by probing the height dynamics of a thin film in the center of the bucket. We use silicone oil on a prewetted silicon wafer and use interferometry for the height measurements. We recover classic EBP scalings and study the effect of Marangoni forces. We give an outlook on future work, notably experiments related to the recent theory developed by Bostwick et al.

Lorenzo Giacomelli,
Sapienza Universita di Roma

This talk is concerned with the spreading of a liquid droplet on a plane solid surface in the regime of lubrication approximation. The focus is on effective conditions which relate the speed of the contact line (where liquid, solid and vapor meet) to the microscopic contact angle. One such condition has been recently proposed by Weiqing Ren and Weinan E [Phys. Fluids 19 (2007), 022101]: it includes into the model the effect of frictional forces which arise at the contact line from unbalanced components of the Young's stress, leading to an additional dissipation term in the energy balance. For speed-dependent contact angle conditions of rather general form, a matched asymptotic study is worked out, relating the macroscopic contact angle to the speed of the contact line. Here, well-posedness for a class of traveling-wave solutions, which does not seem to have been observed so far, is proved and used. For the specific model of Ren and E, ODE arguments are then applied to infer the intermediate scaling laws and their timescales of validity: in complete wetting, they depend crucially on the relative strength of surface friction (at the liquid-solid interface) versus contact-line friction; in partial wetting, they also depend on the magnitude of the static contact-angle. The results have been obtained jointly with Maria Chiricotto (Sapienza University of Rome).

Dan Ginsberg,
University of Toronto
Analytical and Numerical Results on the Positivity of Steady State Solutions of a Thin Film Equation

We consider an equation for a thin-film of fluid on a rotating cylinder and present several new analytical and numerical results on steady state solutions. First, we provide an elementary proof that both weak and classical steady states must be strictly positive so long as the speed of rotation is nonzero. Next, we formulate an iterative spectral algorithm for computing these steady states. Finally, we explore a non-existence inequality for steady state solutions from the recent work of Chugunova, Pugh, & Taranets

James Grotberg,
Biomedical Engineering Department, University of Michigan
A 3D Model of Surfactant and Liquid Delivery into the Lung

An important treatment for surfactant deficiency in prematurely born neonates is surfactant replacement therapy (SRT). It consists of instilling a surfactant/liquid mixture directly into the airways via the endotracheal tube. Clinicians and research scientists rely on trial and error to adjust the delivery parameters, because they have no suitable model of the process. We have developed a 3D multiscale model of liquid and surfactant delivery into the lung. The instilled liquid forms a plug which propagates distally from forced inspiratory air flow. The plug encounters the tracheobronchial tree, which is a branching network of tubes. Our novel approach is to model this tree as a sequence of two types of compartments: (i) each straight tube (airway) is a compartment where plug volume is decreased from deposition on the wall into the plug’s trailing film. The ratio of deposited plug volume to the airway volume is the “deposition ratio, RD”; and (ii) each bifurcation is a compartment where plugs split unevenly due to gravity favoring one of the two daughter tubes. The split volume ratio, lesser over greater, of the two progeny plugs we call the “split ratio”, RS. Our approach will be to solve the relevant computational fluid dynamics (CFD) problems for (i) and (ii) separately determining the parametric dependences for RS and RD as the small scale contribution. Then we will apply the compartmental results to a skeleton of the compartments in a 3D Airway Network. This
is the large scale result built from the small scale.

This is a joint work with Cheng-Feng Tai Ph.D (Biomedical Engineering Department, University of Michigan, USA), Benjamin L. Vaughan (Department of Mathematics, University of Cincinnati, USA), Marcel Filoche, Ph.D. (Department of Physics, Ecole Polytechnique, Palaiseau, France)

Shilpa Khatri,
University of North Carolina at Chapel Hill
A Numerical Method for Two Phase Flows with Insoluble and Soluble Surfactants

In many real world multiphase flow problems, there are surfactants present. These are surface reacting agents that modify the strength of the surface tension. The concentration of the surfactant on the interface separating the fluids can be modeled with a time dependent differential equation defined on the time dependent and deforming interface. For soluble surfactants, this is also coupled to a PDE for the concentration of surfactants in the bulk. We present a second order numerical method to model this problem. We will discuss the details of this implementation and show results in two dimensions.

Ewen King-Smith,
Ohio State University
Structure, function and dynamics of the tear film lipid layer

Evaporative dry eye is a common disorder of older people, which is caused by excessive evaporation through the superficial lipid layer of the tear film. This evaporation causes increased osmolarity (concentration of salts, etc.) of the underlying aqueous tears, which in turn causes irritation and inflammation of the ocular surface. The lipid layer is secreted from “Meibomian glands” in the lids. What characteristics are needed for the lipid layer to form a good evaporation barrier? First, lipid should be sufficiently fluid within the glands that it can be secreted by pressure on the glands. Second, the lipid layer should be stable and relatively thick and uniform (e.g., about 10 or more molecular layers); thus it should resist any tendency to dewetting (e.g., from van der Waals’ forces) and a less fluid, gel-like characteristic may aid in this respect. Third, evaporation resistance may be aided by the very long hydrocarbon chains found in Meibomian lipids which form an energy barrier to the penetration of water molecules. Fourth, the lipid layer structure should withstand the repeated compression-expansion cycles caused by blinking. These four characteristics will be discussed, with consideration of possible changes in dry eye conditions.

After a blink, there is a rapid upward movement of the lipid layer of the tear film which has been ascribed to the Marangoni effect and a concentration gradient of surfactant (e.g., at the interface between lipid and aqueous layers). This movement may help to make the lipid thickness more uniform along a vertical section, but it may also be a consequence of mechanisms which preserve lipid structure during the blink cycle (the fourth characteristic above).

Georgy Kitavtsev
, Max Planck Institute for Mathematics in the Sciences
Coarsening rates for the dynamics of polymer droplets in the presence of strong slippage

In this talk the coarsening dynamics of liquid polymer nanoscopic droplets will be described in the framework of the lubrication approximation for different slip regimes considered at the solid substrate. Starting from the work of Glasner and Witelski 03' it was shown that in the weak slip regime the coarsening rates obey the exponent $-2/5$. In our study we investigate rather the strong slip regime characterized by much larger slip lengths comparable with the averaged size of the film.The lubrication system for this case was derived recently in M\"unch et al. 06'. Our further derivation of the corresponding reduced ODE models governing the coarsening dynamics of weakly interacting droplets in the strong slip regime allows for an effective investigation of the coarsening rates. We identify and subdivide here two main limiting coarsening regimes, namely the intermediate slip regime and the suspended free film regime. The first one reproduces the known exponent $-2/5$, whereas the second one shows quite different coarsening slopes and behavior completely dominated by migration of droplets.

Jim Lewis,
Lawson Health Research Institute
Clinical Use of Surfactant

Pulmonary surfactant is a lipoprotein complex synthesized and secreted from type 2 cells within the alveolar space. It functions to lower surface tension at the air-liquid interface thereby optimizing lung compliance and allowing spontaneous respiration with minimal effort. The specific composition of surfactant mixtures (both endogenous and exogenous) as well as its functional aspects and in vivo metabolic pathways will be discussed. In particular, the effects of respiratory motion on surfactant subfractions, resulting in the conversion of functional large aggregates to dysfunctional small aggregate forms within the airspace, will be outlined.

In addition to its biophysical function, other functional aspects of surfactant will be discussed. These include, the role of surfactant in airway (bronchial tubes) stability, its effect on mucous and particle clearance from the airways, and is effect on ciliary function. The role of surfactant in the host’s inflammatory and immune response will also be briefly mentioned.

The latter parts of the discussion will explore the role of exogenous surfactant administration in the clinical setting of acute lung injury. The rationale for administering surfactant to patients with lung injury stems from the clinical experience demonstrating an improved mortality in neonatal pre-term babies who are born deficient in surfactant. Lung lavage obtained from adult patients with acute lung injury show dysfunctional surfactant and the various attempts to restore adequate surfactant pools within these patients’ lungs will be summarized. Factors that ultimately influence a host’s response to exogenous surfactant will be discussed including the surfactant preparation itself, the delivery method utilized to administer the surfactant (aerosol versus instillation versus bronchoscopic administration), the optimal dose required for beneficial responses and finally, the timing of surfactant administration over the course of the injury.

Kara Maki, Rochester Institute of Technology

When a coffee droplet dries on a countertop, a dark ring of coffee solute is left behind, a phenomenon often referred to as the coffee-ring effect.'' A closely related yet less-well-explored phenomenon is the formation of a layer of particles, or skin, at the surface of the droplet. In this work, we explore the behavior of a mathematical model that can qualitatively describe both phenomena. We consider a thin axisymmetric droplet of a colloidal suspension on a horizontal substrate undergoing spreading and rapid evaporation. The lubrication approximation is applied to simplify the mass and momentum conservation equations, and the colloidal particles are allowed to influence droplet rheology through their effect on the viscosity. By describing the transport of the colloidal particles with the full convection-diffusion equation, we are able to capture depthwise gradients in particle concentration and thus describe skin formation, a feature neglected in prior models of droplet evaporation. Whereas capillarity creates a flow that drives particles to the contact line to produce a coffee-ring, Marangoni flows can compete with this and promote skin formation. Increases in viscosity due to particle concentration slow down droplet dynamics, and can lead to a significant reduction in the spreading rate.

Omar Matar, Imperial College London
(slides)

The dynamics of surfactant laden-lenses spreading over liquid substrates are examined. Lubrication theory is used in conjunction with models for the spreading process, the effects of surfactant at the contact line and the sorption kinetics above and below the critical micelle concentration. This model is solved numerically using a finite-element formulation. We carry out a full parametric study to demonstrate a wide range of interesting behaviour: from complete spreading, spreading followed by retraction, to sustained pulsating oscillations.

Nades Palaniyar, The Hospital for Sick Children
Surfactant deficiency and neonatal lungs diseases

Pulmonary surfactant is a mixture of lipids (90% w/w) and surfactant-associated proteins (10% w/w). Fetal Type II epithelial cells located in the alveoli start to secrete surfactant into the fluid-filled lungs during the later stages of pregnancy. Surfactant forms a thin film at the air-liquid surface of the alveoli in the newborn infants. This surfactant film reduces surface tension to allow the expansion and contraction of alveolar sacs necessary for breathing. Surfactant film also protects the lungs from microbial infection. Premature infants have little or no surfactant in their lungs. Failure to administer surfactant and/or provide mechanical ventilation is fatal in premature babies. Mechanical ventilation often leads to lung injury resulting in the deposition of serum components including proteins and immune cells into the alveoli. These components inactivate surfactant film and compromise gas exchange. Surfactant-associated proteins (SP-) SP-A and SP-D bind DPPC and PI of the pulmonary surfactant lipid components, respectively. These proteins reorganize lipid structures (e.g., tubular myelin, liposome aggregate) in the alveoli and regulate lipid homeostasis in the lungs. These two proteins are also involved in clearing dying immune cells and regulating lung inflammation. SP-B and SP-C are hydrophobic proteins and are essential for regulating surface activity of the surfactant. A deficiency of SP-B is fatal in infants and in experimental animals. Natural surfactant preparations currently used in clinics contain both SP-B and SP-C. Presence of these proteins is considered to be important for successfully reducing the surface tension of these surfactants. Further modifications of surfactant preparations are necessary to reach their full potential for treating various neonatal lung diseases.

Ellen Peterson, Carnegie Mellon University
Behavior of a Droplet on a Thin Fluid Film
(slides)

Cystic Fibrosis is a disease of the lung, which is generally treated with an aerosol medicine. In order to better understand and improve this treatment we explore the spreading of a droplet (the medicine) on a thin liquid film (the lung lining). We
assume both fluids are Newtonian, incompressible, and immiscible. We formulate a system of coupled fourth order partial differential equations that models the spreading of this simplified physical system. When a drop is placed on another
fluid it may either completely spread over the underlying fluid or it may form a static lens. This behavior is predicted by he spreading parameter, which is a relation of the surface tensions of the fluids. In the case when a static lens forms, we explore the existence and structure of the equilibrium solution. Experimentally, we find that in some cases the spreading parameter predicts complete spreading however a static lens is observed. We compare the size of the static lens from the experiment to that predicted mathematically. The results of this investigation suggest that the lens resists flowing over the escaped layer of the same fluid - the mechanism of autophobing.

Mary Pugh, University of Toronto
Coating flows on slowly rotating cylinders

We consider a horizontal cylinder, rotating about its center. A viscous fluid is on the outside of the cylinder, coating the cylinder as it rotates. We consider a lubrication approximation of the Navier Stokes equations for the regime in which the fluid film is relatively thin and the surface tension is relatively large. The resulting lubrication model may have no steady state, a unique steady state, or more than one steady state. Using both numerics and analysis, we consider the dynamics of this flow, including whether or not solutions can become singular in finite time.

This is joint work with Marina Chugunova (University of Toronto) and Roman Taranets (University of Nottingham).

Michael Shearer, North Carolina State University
Two-phase flow in porous media: sharp fronts and stability

Plane waves for two phase flow in a porous medium are modeled by the one-dimensional Buckley- Leverett equation, a scalar conservation law. We analyze stability of sharp planar interfaces to two-dimensional perturbations, which involves a system of partial differential equations. Linear stability analysis results in a description of the dispersion relation to leading order in the wave number, leading to a criterion that distinguishes between interfaces that are long-wave stable and those that are not. Numerical simulations of the full nonlinear system of equations, including dissipation and dispersion, illustrate the analytical results.

This is joint work with Kim Spayd and Zhengzheng Hu.

Linda Smolka, Bucknell University
Fingering instability down the outside of a vertical cylinder
(slides)

We present an experimental and numerical study examining the dynamics of a gravity-driven contact line of a thin viscous film traveling down the outside of a vertical cylinder of radius $R$. Experiments on cylinders with radii ranging between 0.159 and 3.81~cm show that the contact line is unstable to a fingering pattern for two fluids with differing viscosities, surface tensions, and wetting properties. The dynamics of the contact line is studied and results are compared to
previous studies of inclined plane experiments in order to understand the influence substrate curvature plays on the fingering pattern. A lubrication model is derived for the film height in the limit that $\epsilon = H/R \ll 1,$ where $H$ is the upstream film thickness, and in terms of a Bond number $\rho g R^3/(\gamma H),$ and the linear stability of the contact line is analyzed using traveling wave solutions. Curvature controls the capillary ridge height of the traveling wave and the range of unstable wavelength when $\epsilon = O(10^{-1}),$ whereas the shape and stability of the contact line converge to the behavior one observes on a vertical plane when $\epsilon \leq O(10^{-2}).$ The most unstable wave mode, cutoff wave mode for neutral stability and maximum growth rate scale as $\widehat{\mathrm{Bo}}^{0.45}$ where $\widehat{\mathrm{Bo}} = \rho g R^2/\gamma \ge 1.3$ and the contact line is unstable to fingering when $\widehat{\mathrm{Bo}} \ge 0.56.$ Using the experimental data to extrapolate outside the range of validity of the thin
film model, we estimate the contact line is stable when $\widehat{\mathrm{Bo}} < 0.56.$ Agreement is excellent between the model and the experimental data for the wave number (i.e., number of fingers) and wavelength of the fingering pattern that forms along the contact line.

This work is in collaboration with Marc SeGall.

Roman Taranets, University of Nottingham
Nonnegative weak solutions for a degenerate system modelling the spreading of surfactant on thin films
(slides)

Depending on the parameter range, we prove local and global in time existence of nonnegative weak solutions to a coupled system of two degenerate parabolic equations. This system models the spreading of an insoluble surfactant on a thin liquid film. This model includes gravity, surface tension, capillarity effects, and van der Waals forces. The surface diffusion coefficient is not assumed constant and depends on the surfactant concentration.

This is joint work with Marina Chugunova.

Thomas Witelski, Duke University
The influence of surfactants on transient behaviors in forced film thinning Droplets of Colloidal Suspensions

During the opening of phase of an eye blink, a fluid film is spread over the exposed surface of the eye by the motion of the eyelid. Studies of this moving boundary problem have shown that the presence of surfactants can affect the thinning of the film and potentially reduce the occurrence of rupture and dry eye''. Motivated by this we carry out a numerical study of dynamics of the nonlinear partial differential equations describing forced thinning films with and without insoluble surfactants in related problems.