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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
Droplets spreading under contact-line friction
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
Fast Evaporation of Spreading
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
Spreading, retraction, and sustained oscillations of surfactant-laden lenses
(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.
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