The capillary
instability that occurs on an annular film lining a tube is studied as
a model of airway closure. Small waves in the film can amplify and form
a plug across the tube. This dynamical behavior is studied using
theoretical models and bench-top experiments. Our model predicts the
initial growth rate of the instability and its dependence on surfactant
effects. In experiments, an annular film is formed by infusion of water
into an initially oil-filled glass capillary tube. The thickness of the
oil film varies with the infusion flow rate. The instability growth
rate and closure time are measured for a range of film thicknesses. Our
theory predicts that a thinner film and higher surfactant activity
enhance stability; surfactant can decrease the growth rate to 25% of
its surfactant-free value. In experiments, we find that surfactant can
decrease the growth rate to 20% and increase the closure time by a
factor of 3.8. Functional values of a critical film thickness for
closure support the theory that it increases in the presence of surfactant.
pulmonary surfactants; surface tension; closing volume; airway
liquid lining; microgravity
 |
INTRODUCTION |
THE LUNG'S AIRWAYS are liquid-lined
flexible tubes. It is well recognized that the liquid lining of the
lung can close off a small airway by forming a liquid plug and/or by
provoking the collapse of the flexible airway wall (17, 32). The
formation of a plug blocks airflow and can further induce the complete
collapse of an airway, since liquid continuously drains into the plug. This is a postclosure filling flow, which extends the axial length of
the plug. This happens most frequently near the end of expiration, when
the airway diameters are small, i.e., at the closing volume. During
inhalation the airways expand, which may result in the rupture of the
plug and the reestablishment of airflow to the periphery (13, 23, 34).
The closure and subsequent reopening of these airways contribute to the
shape of the pressure-volume curve in cycled lungs, which characterizes
the mechanical response of the lung. Understanding the phenomena that
determine it is a fundamental inquiry in pulmonary physiology in normal
and microgravity environments. Measuring the occurrence of airway
closure is one component of a standard pulmonary function test: the
single-breath nitrogen-washout test. When this test shows early airway
closure, it is often interpreted that there is inhomogeneous
ventilation distribution within the lung, which may be caused by a
number of normal and pathological conditions (6). Surfactants tend to
stabilize the lining from closure, as has been shown qualitatively (31).
There are likely several contributing forces that lead to airway
closure. Studies by Halpern and Grotberg (20, 21) have modeled closure,
including the effects of a flexible wall. We focus on modeling closure
of smaller airways, typically beyond and including generation
16, in which gravitational effects are small. Our present study
investigates one of the key mechanisms believed to cause airway
closure: a surface tension-driven instability. By isolating the
experimental system from gravitational and flexible-wall effects, we
are able to focus on the effects of the surface tension instability on
closure and the relative effects of surfactant.
A capillary instability occurs on the wavy interface between two
fluids. A circular-cylindrical tube is lined with a uniform layer (the
film) of viscous fluid surrounding a less viscous fluid (the core). It
is well known that this configuration is unstable to finite-wavelength
disturbances. We use a rigid-tube experimental model and compare the
experimental results with rigid-tube theory. Depending on the initial
film thickness, such disturbances may culminate in stable, steady,
finite-amplitude waves or lead to a topological transition in which the
film completely pinches off the core fluid, forming a liquid plug. We
examine the effects of surfactant on the surface tension-induced
instability that contributes to airway closure.
Glossary
We find it helpful to convert some dimensional variables into
dimensionless variables. We use the following guide to define which are
dimensional and dimensionless. Greek symbols representing the fixed
fluid parameters (µ and 
) are dimensional, and changing fluid
parameters are dimensionless unless indicated with an asterisk. Roman
variables representing the fixed system parameters (a,
b, and g) are dimensional, and all other Roman
variables are dimensionless unless indicated with an asterisk.
| a |
Inner radius of the rigid tube
|
| b |
Radius of the undisturbed cylinder of fluid (core radius)
|
| C* |
Bulk concentration of surfactant
|
| Ca |
µfilmU*/ *, capillary number for film formation
|
| D*s |
Surface diffusivity of surfactant
|
| D |
µfilmD*s/(h*0*m )
|
 |
  3D
|
| |
 ( *m/*m)d */d *|*=*m,
Marangoni number
|
 |
2, surface activity
parameter
|
| |
(a b)/a, dimensionless thickness of the
undisturbed film
|
| crit |
Critical dimensionless film thickness
|
| f |
Functional value of crit (taken at
tc = 2 × 104)
|
| g |
Gravitational constant
|
| * |
Surface concentration of surfactant
|
| *m |
Mean surface concentration of surfactant
|
| |
*/*m = 1 + 2
|
| |
Variation of surface concentration from the mean
|
 |
2
|
| h*0 |
Undisturbed film thickness
|
| h* |
Amplitude of the film disturbance
|
| h |
h*/a
|
| h1 |
Initial amplitude (perturbation parameter)
|
| k* |
2 / *, wave number of the disturbance
|
| k |
k*a
|
| K |
k*b
|
| k*1 |
Surfactant adsorption constant
|
| k*2 |
Surfactant desorption constant
|
| * |
Wavelength of the disturbance
|
| |
*/a
|
| µcore |
Viscosity of the core fluid
|
| µfilm |
Viscosity of the film fluid
|
| q* |
Initial growth rate of the disturbance
|
| q |
q*aµfilm/*
|
 |
q*3aµfilm/*
|
| r |
r*/a, radial distance from the tube axis
|
| core |
Density of the core fluid
|
| film |
Density of the film fluid
|
| * |
Surface tension at the interface between the fluids
|
| *m |
Mean value of surface tension
|
| t*f |
Final time of the disturbance (at closure)
|
| t*0 |
Start time of the disturbance
|
| t*c |
t*f t*0, total time for closure
to occur
|
| t |
t*/(aµfilm/*)
|
 |
t*/(3aµfilm/*)
|
| U* |
Mean velocity of the infused core fluid (during film formation)
|
| z |
z*/a, distance along the tube axis
|
| |
PREVIOUS RESEARCH |
Mechanical phenomena related to airway closure can be drawn from
several sources. From the physiological literature, the stabilizing effects of surfactants on models of airway closure have been
demonstrated. Liu et al. (31) conducted experiments where air is forced
through a narrow glass or esophageal tube segment initially filled with saline or a surfactant-saline solution. They used a constant tube size
and driving pressure, with a constant liquid volume localized to the
narrow, constricted segment of the tube. The liquid in the tube gives
way to airflow but reforms a plug in systems with little surfactant or
surfactant weakened by albumin and fibrinogen. However, the plug does
not reform if adequate surfactant is present, for the conditions
explored. This demonstrates the important clinical effects of exogenous
surfactant replacement therapy (11, 30, 44), which can help stabilize
airways in patients who have a surfactant deficiency. Liu et al.
attributed this additional stability to the reduction of surface
tension caused by the surfactant polar molecules. Saline lung lavage in
isolated rat lungs reduces the endogenous surfactant concentration,
causing an increase in airway resistance due to rapid plug formation
(7, 8). However, instillation of exogenous calf lung surfactant extract
after the lavage can increase the duration of unobstructed airway time
by a factor of 3 (8) to 7 (7), determined by measuring resistance via
airway pressure. It would be interesting to examine the airway closure
and film amplification for a range of initial film thicknesses and
compare behavior before and after addition of exogenous surfactant.
From the engineering literature, the study of capillary instabilities,
such as the breakup of a cylindrical column of fluid or a thin film
coating a rigid tube, continues to be an area of ongoing investigation.
Plateau (37) considered the static stability of a cylindrical column of
fluid with undisturbed radius b. He proved, using energy
arguments, that a perturbation to the interface is unstable if its
wavelength (*) exceeds the circumference of the column
(2b). The perturbation causes axial waves in the interface, similar to waves in the interface of our system, which we show and
discuss in Experimental methods. Rayleigh (40) studied the dynamic stability of an initially cylindrical liquid jet. He found that
waves at the surface amplify until the jet breaks up into a stream of
droplets, provided Plateau's criterion holds, i.e., that K < 1, where K = k*b = 2/* is the disturbance
wave number. Rayleigh derived a relationship between the disturbance
growth rate (q*) and K. The q* attains a
maximum at some critical wave number (Kcrit). He
found the least-stable mode at Kcrit = /4.5 
0.698 for a liquid jet in an inviscid or a viscous fluid. Rayleigh (41)
modified this analysis for a jet of air immersed in a liquid and found
the same criterion for the maximum growth rate
(qmax): Kcrit = /4.5.
Goren (16) studied the stability of an annular liquid film lining a
tube otherwise filled with air. A rigid tube of radius a, lined
with a uniform film at radial distance b, has a dimensionless film thickness = (a b)/a. A
disturbance of * causes axial waves in the interface, which can
amplify and eventually cause film closure. Goren performed experiments
for 0.06 < < 0.33 and allowed the systems to reach closure.
Plug formation occurred at multiple, spatially periodic locations
defined by Kcrit, which was shown to depend weakly
on . In his accompanying theoretical analysis for the initial growth
rate, q = q*aµfilm/*, (where µfilm is viscosity of the fluid film), he predicts how
qmax depends on Kcrit and .
Goldsmith and Mason (15) performed closure experiments using immiscible
liquids of similar density for the film and core fluids to minimize
buoyancy. They measured disturbance wave numbers that match Goren's
theory well and initial growth rates that were slightly greater than
predicted by theory (16). Sample results are shown tracing the
logarithm of the amplitude of the growth vs. time for two experiments,
at = 0.19 and = 0.25, on a surfactant-free system. The
disturbance amplitude initially grows exponentially with time, at a
constant q, but as the wave amplifies, q accelerates. Also, the closure times (tc) of the two sample
experiments can be depicted from this graph.
The must be larger than a critical value (crit)
for closure to occur. Disturbance waves in a film that is thinner than crit may amplify to form undulate collars spaced
throughout the tube. However, not enough liquid volume is present for
the collars to bridge the tube. Everett and Haynes (9) found that a
collar of surfactant-free fluid wetting a rigid tube would undergo a topological transformation to a liquid plug, provided the collar volume
exceeds 5.47a3 enclosed within one axial
wavelength. Similar results were also obtained by Kamm and Schroter
(28), who conducted experiments where oil is dripped into small
vertical rigid tubes. By assuming an initial film thickness of 10 µm
at total lung capacity (TLC), they estimated that airway closure first
occurs in the terminal bronchioles (about generations
16-17) at a lung volume of 23% TLC. For an initial film
thickness of 5 µm, airway closure first occurs at 12% TLC (28).
These closing volumes are commonly seen in pulmonary function tests.
Gauglitz and Radke (12), extending theoretical work by Hammond (22),
computed an crit = 0.12 necessary for closure, whereas their experiments indicate an average crit = 0.09. Johnson et al. (27) analyzed the stability of a viscous annular film,
including the effects of inertia. They found their approximation of the initial growth rate to match Goren's exact solution well. Closure of
liquid-lined, linearly elastic tubes has been investigated by Halpern
and Grotberg (20). They found that flexibility enhances the growth of
the unstable film. Theoretical models developed to examine the effects
of surfactant on film stability (21, 35) show that plug formation in a
system containing surfactant is slowed by a factor of ~4 over the
surfactant-free model. Halpern and Grotberg (21) found that
crit is increased in a system containing surfactant, and
crit decreases with increasing ratio of surface forces
to elastic forces.
In EXPERIMENTS, we describe the materials and methods for
performing experiments on annular film closure in a capillary tube. We
consider a range of initial film thicknesses and examine systems without and with surfactant. The experimental system uses liquids for
the core and film fluids that approximately match the viscosity ratio
and Reynolds numbers (Re) in the lung system. Gravitational effects are
negligible, inasmuch as the density difference between the core and
film liquids is small. This allows us to investigate the closure
phenomena induced by nongravitational forces. Also, film closure occurs
much more slowly in a liquid-liquid system than in an air-liquid
system, allowing the film dynamics to be viewed and recorded in real
time. Then we report the experimental results of
tc, q, and crit; we also
report inflow results of the annular film formation. Next we outline
our theoretical model for the initial growth rate in a system
containing surfactant and show theoretical results. Then we discuss
some observed behavior, compare the experimental data with theory, and
make conclusions about the effects of surfactant on film formation and stability.
| |
EXPERIMENTS |
Materials.
Four fluid systems are used in these sets of experiments (Table
1). We use high-viscosity oils as the film
fluid and water as the core fluid. In systems 1 and 2,
the oil used for the annular film is LB-1715 (Union Carbide Chemical,
South Charleston, WV), which is a propylene glycol monobutyl ether.
System 1 is surfactant free, and system 2 contains a
small amount of the surfactant
1-palmitoyl-2-lauroyl-sn-glycero-3-phosphocholine (Avanti Polar
Lipids, Alabaster, AL; PLPC). It is an asymmetric fatty acid that is
insoluble in water. System 2 contains 103 mM
bulk concentration of PLPC dissolved in the oil phase before contact
with the water phase. In systems 3 and 4, the oil used for the annular film is LB-400-X (Union Carbide Chemical). System 3 is surfactant free, and system 4 contains a small amount
of the surfactant Tween 20 (Sigma Chemical, St. Louis, MO). Tween 20 is
polyoxyethylenesorbitan monolaurate and is made up of various fatty
acids; it is soluble in water and insoluble in oil. It is initially
dissolved in the water phase at a bulk concentration of 0.01% by
weight. In Table 1 the film viscosities were measured at room
temperature (24-25°C) with an Oswald bulb viscometer, and the
film densities are from manufacturer specifications. The surface
tension of the LB-1715 and LB-400-X in air is 33-38 dyn/cm. Mean
values for the surface tension of an oil-water interface were measured
using a ring tensiometer and are displayed in Table 1. In calculations
that require a value for the surface tension (*) for these
experiments, we use the mean values for each of the four systems.
Similitude: matching parameters.
The system parameters are organized into dimensionless groups (Table
2). We attempt to model the ranges found in
the pulmonary airways, with some important differences. We assume that
at generation 16 in normal lungs the film is the aqueous phase
with a mean airway diameter of 0.049 cm (46). In calculating lung
properties in Table 2, we use the density and viscosity values of water
and air for the film and core fluids, respectively. The surface tension varies by region of the lung because of the local quantity of surfactant and by lung volume because of dynamic surface concentration of surfactant. The tracheal surface tension was directly measured to be
31 dyn/cm in normal horse lungs (24) and 33 and 32 dyn/cm in normal rat
and guinea pig lungs, respectively (29). The equilibrium surface
tension in the alveolar region of rabbit lungs was measured to be 7 and
30 dyn/cm at 50 and 100% TLC, respectively (43). We expect the surface
tension in normal pulmonary airways to be ~20-25 dyn/cm at
functional residual capacity, which agrees with previously published
values (14, 18). We use * = 20 dyn/cm in Table 2 for calculations of
normal lung parameters at generation 16.
In normal lungs the film lining is very thin, but many diseases as well
as the state of lung inflation can affect , the ratio of film lining
thickness to airway radius. Yager et al. (48) measured the airway
surface liquid thickness in the small airways of the guinea pig. With
use of data from their Fig. 4 on the airway perimeter and film
thickness, and the assumption that the airways are approximately
circular in cross section, the average ratio of film thickness to
airway radius is 0.02. In a study by Codd et al. (4), in a sheep
trachea, the mean = 0.04. Excessive bronchial secretions in
individuals with unhealthy lungs can commonly produce airway surface
liquid films that are 10% of the airway diameter (42); this
corresponds to = 0.20. In our experiments we model airways in which
the film is thick enough to close: > 0.12.
The Bond number (Bo) = 
g a2/* (where is fluid density and g is gravitational constant) indicates
where gravitational effects are significant. Here, = |film core|,
the difference in densities between the two fluids. Gravity can be
neglected, provided 1 Bo 
1 (22), which generally
holds in smaller airways. Under normal-gravity conditions, for = 0.15, we calculate that 1 Bo < 0.3, starting at
generation 16 in adults and starting at generation 9 in
infants. In the case of lessened gravity (such as the microgravity
environment) or higher surface tension (which can occur in premature
infants born with insufficient pulmonary surfactant) the Bo values
would be smaller. In our experiments we wish to examine the dynamics of
closure as contributed by forces other than gravitational. By using a
liquid-liquid system, where the fluid densities are similar, we can
create systems with low Bo values by reducing . The time scale of
these experimental systems is larger than the air-liquid system in the
airways by a factor of 104. By slowing the closure process,
data capture is simplified, with improved accuracy in the results.
These fluids are chosen, oil for the film and water for the core fluid,
such that the viscosity ratio 
= µcore/µfilm 1, which matches the lungs.
Fluid inertia is negligible, provided 2
Refilm = (5film*a)/µ2film
1 (22), which is true in the lungs (Table 2). Similarly, fluid
inertia in the core is neglected, since 2
Recore = [(5*)/µfilm][(corea)/µcore]
1, as in the lungs.
To create a surfactant monolayer between the two fluid layers, we first
dissolve the surfactant into one of the bulk fluids. When the interface
between the two fluids is formed, we assume that the surfactant rapidly
adsorbs to the surface and does not rapidly desorb back into the bulk
of the fluid. By calculating the solubility parameter 
= (k*1C*)/(k*2h*0),
we can estimate the ratio of time scales for desorption to adsorption
(26), where k*1 is the
adsorption coefficient, k*2
is the desorption coefficient, C* is the bulk concentration of
surfactant, and h*0 is the
film thickness. Jensen and Grotberg (26) estimate that the pulmonary
surfactant component dipalmitoylphosphatidylcholine (DPPC) dissolved in
an aqueous lung surface layer 1 µm thick will have = 5 × 107. At generation 16 the airway surface liquid is
~6-12 µm thick (using 0.02 
0.03), yielding 4.2 × 106 8.5 × 106. The surfactant
PLPC used in our experiments has adsorptive/desorptive properties
similar to DPPC. Using the range of film thicknesses found in our
experiments (3.5 µm h*0 10.4
µm), we estimate 1.4 × 106 4.8 × 106 for the system containing PLPC. Because 
1 in
the lung and in our experiments, we assume that surfactant
concentration at the interface is initially uniform. Surface
diffusivity of surfactant is measured by the diffusivity parameter = 3D, where D = [(µfilmD*s)/(h*0*m)]
and D*s is the dimensional surface diffusivity. The
Marangoni number, defined as = (*m/*m)d*/d*|*=*m,
is a measure of the ability of the surfactant to lower the surface
tension of a monolayer. Here, *m is the mean
value of *, the surface concentration of surfactant, and
*m is the mean value of *. Agrawal and Neuman
(1) directly measured the surface diffusivity of DPPC in myristic acid
and found 107 < D*s < 7 × 105 cm2/s. Jensen and Grotberg (25)
approximate that D is on the order of 106 for
experiments and 104 in an airway. For = 0.1, this
still makes quite small: 103 101. Therefore, the current theory neglects surface diffusivity.
Experimental methods.
A thin cylindrical borosilicate glass capillary tube with an inner
diameter of 0.058 cm and length of 20 mm is viewed under a
microscope-camera configuration (Fig. 1).
The glass capillary is cleaned with chromic acid, rinsed with acetone
and deionized water, and dried before the experiment. Fluids are
delivered to the capillary tube through flexible Teflon tubing having
the same inner and outer diameters as the rigid capillary tube and held together by a sleeve of silicone tubing and glue. A three-way stopcock
is used to direct the fluids from respective syringes through the
Teflon tubing. The film fluid is injected manually, and a
positive-displacement syringe pump (model 341-A, Sage Instruments) drives the core fluid through the tube. The capillary tube, immersed in
glycerol and held between two glass slides perpendicular to the
microscope to correct for visual distorsion, is then placed under the
microscope and into a pipette holder, which is attached to a Leitz
micromanipulator. A camera is attached to the top of a Leitz Laborlux D
microscope; the capillary tube is viewed at ×200 or ×500
magnification. All experiments are recorded from a Javelin JE3362
camera connected to a JVC SR-S360 video recorder. A Sony PVM-2030
20-in. color monitor is also connected, which provides for viewing
capabilities during experiments and later for data acquisition.
To measure the value of the mean interfacial surface tension between
the film fluid and the core fluid, we use a standard 6-cm
platinum-iridium ring tensiometer. Measuring accurately the surface
tension between liquids of similar densities can be difficult. In the
tensiometer test, a thin ring is pulled up through the horizontal
liquid-liquid interface, creating a vertical force on the ring, which
is then measured by the tensiometer scale. However, a ring pulled
through liquids of similar density may not produce a vertical
interface, particularly on the inner edge of the ring. The inaccuracy
may be quite large, since the ring tensiometer measures only the
vertical component of the force. For contact angles deviating from
vertical by up to 30°, the true surface tension may be underestimated
by as much as 15%. Assuming a vertical interface, we estimate the mean
surface tension of LB-1715 with deionized distilled water to be 4.59 dyn/cm. The addition of the surfactant PLPC added to the oil phase in
molar bulk concentrations of 104-101
mM causes a slight decrease in the mean surface tension to 4.23 dyn/cm
(<8% change). The mean surface tension of LB-400-X and water,
estimated at 4.77 dyn/cm, exhibits a dramatic decrease with the
addition of 0.01% (by weight) Tween 20, to 2.88 dyn/cm (~40%).
As part of the preliminary steps in performing stability experiments,
it is necessary to create the two-phase system of an annular film
lining a tube with a fluid core. This system is established by first
filling the tube with the film fluid from a 10-ml syringe. The core
fluid is then pumped into the center of the tube, driven by a
positive-displacement syringe pump. This core fluid flows through the
center of the tube like a long finger (Fig.
2A) and leaves an axisymmetric
annular film on the tube walls (Fig. 2B). The thickness of
the film depends on the capillary number
(Ca = µfilmU*/*, where µfilm
is the film viscosity, U* is the average core fluid velocity,
and * is the interfacial surface tension). Previous studies (10, 15,
45) support an empirical equation for
|
(1)
|
for
104 < Ca < 101. Results from several
experimental studies support Bretherton's theoretical analysis (2),
indicating
|
(2)
|
for
104 < Ca < 102. Martinez and Udell
(33) conducted a numerical analysis on a finite-length bubble to
describe the shape at the leading and trailing menisci. Their
predictions of (Ca) are in excellent agreement with experimental
data by Taylor (45) for 102 Ca 2 and support
Eq. 1 for 102 Ca 101. Goldsmith and Mason (15) also performed experiments
on two sets of fluid systems with and without the surfactant Tween 20. Without surfactant, they concluded that varies like
Ca0.5 for small Ca. However, when 0.5% Tween 20 is added
to the water phase of the two systems, they find that varies like
Ca0.28 and Ca0.44, respectively.
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Fig. 2.
Schematic of inflow (A), uniform annular film (B),
and disturbed film (C). See Glossary for definition
of abbreviations.
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|
The average core fluid velocity is defined as the infusion flow rate
divided by the measured cross-sectional area of the core. Using this
relationship, we can directly control the formed film thickness by
varying the flow rate settings on the syringe pump. In creating the
annular film, syringe pump settings range from 0.35 ml/h to 0.49 ml/min. Here, films are formed with ~0.12-0.36. It is
necessary to record before waves develop. Measurements of are
taken at several locations between about two and five tube diameters
from the leading meniscus of the infused core fluid, then averaged.
Once the uniform film is established throughout the tube, the inflow is
stopped and the downstream end of the tube is plugged. The tube is then
scanned to search for small waves at the interface with use of a finely
controlled micromanipulator, and wave amplification (Fig. 2C)
is viewed and recorded.
Figure 3 shows six photographs taken from a
sample experiment in its entirety; Fig. 3, A and B, are
from the inflow part; Fig. 3, C-F, correspond to
the film instability. We use Targa-Plus software and Frame-Grabber PC
board. In Fig. 3A the film fluid is displaced by a finger of
the core fluid, which leaves a uniform annular film, as shown in Fig.
3B. The infusion flow is shut off, then a small disturbance is
located (Fig. 3C), which then amplifies (Fig. 3D).
Measurements of the film interface evolution are taken at the
longitudinal position where the radius of the interface is the
smallest. Interface positions are recorded periodically, and the time
at closure is noted. We see the film as it amplifies just before
closure in Fig. 3E, then just after closure in Fig. 3F.
The recorded data are analyzed to determine the initial, constant growth rates and the closure times as a function of and system parameters. For this sample experiment shown on system 2,
recorded values are as follows: inflow rate = 2.7 ml/h, = 0.29, q* = 0.155 s1, and closure time
(t*c) = 24 s.
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Fig. 3.
Photographs of experimental procedure. Inflow begins (A) and
a uniform film is established (B); then inflow is stopped, a
small disturbance is located (C), and disturbance amplifies
(D). E: just before closure. F: just after
closure.
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|
Determining t*c and initial growth rate.
To extract t*c from an experimental trial,
it is necessary to determine the start time
(t*0) of the disturbance.
However, a wave has already begun to amplify
(t* > t*0)
before it is large enough to be visually detected. Below, we describe a
method that allows the dimensional values of
t*0 and the initial growth
rate q* to be estimated. A small wave disturbance initially
grows exponentially with time. A normal mode expression for h = h1eq*t*+ik*z*
is used where h = h*/a is a
sinusoidal film amplitude. By plotting ln(h) vs. t*
from our data, the abscissa represents an arbitrary time corresponding
to the time of the videotape recording. A straight-line curve fit to
the early data results in the t* = 0 intercept at h = h1. The
t*0 of the perturbation is
taken to be the time at which the disturbance amplitude is 5% of the
undisturbed film thickness. The slope of the initial portion of this
graph also gives q* (in s1). Then
t*c = t*f t*0, where
t*f is the final time documented at closure (in s). Values for the dimensionless growth rate
and closure time are then computed.
| |
EXPERIMENTAL RESULTS |
In Figs. 4-6, involving our experimental data, symbols represent
average values for a set of experiments. A "set" constitutes a
group of film systems that were created by using a specific inflow rate
to provide consistent values of for the set. There is some
deviation within a set for , as well as the values of tc
[t*c/(aµfilm/*)],
q [q*aµfilm/*], and
Ca. For this reason, bidirectional scatter bars, which are based on the
standard deviation of the set, are shown. We used 13 flow settings in
the range 0.35 ml/h to 0.49 ml/min. For each set, up to 20 experimental
trials were performed and averaged. For the experiments on stability, useful data were extracted from two to eight trials per set. For inflow
experiments establishing the uniform film thickness, the number of
trials providing useful data varied from 2 to 20 per set.
Closure time and minimum film thickness.
Figure 4 indicates how
tc decreases as increases and how the
presence of surfactant delays closure. In Fig. 4A, the addition of PLPC to this system increases tc by up to a
factor of 3.8 (the value at = 0.19). Note that for increasingly
thick films, the difference between the surfactant-free and the
surfactant data becomes less significant. In Fig. 4B, the
addition of Tween 20 causes an increase in tc by up
to a factor of 3.0 (at = 0.24). The thinnest films we found to
close were at = 0.123 for system 1, at = 0.154 for
system 2, at = 0.143 for system 3, and at = 0.156 for system 4.
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Fig. 4.
Closure time
[tc = t*c/(aµfilm/*)]
vs. undisturbed film thickness () for systems 1 ( ) and
2 ( ; A) and systems 3 () and 4 (; B). Surfactant delays closure, and thicker films take
longer to close than thinner films. Note how closure time
tends to infinity as approaches crit. See text for
theoretical curves.
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|
The solid lines in Fig. 4 are computed from the surfactant-free theory
of Halpern and Grotberg (20) in the rigid-tube limit. The dashed lines
are computed from the theory of Halpern and Grotberg (21) that predicts
closure in a system containing surfactant. We simultaneously
solve their Eq. A1 numerically. Values are chosen for the
surface activity ( = 8) and for the
disturbance wavelength ( = 2
)
corresponding to a system with normal surface activity and a
disturbance with the most dangerous wavelength. We calculated
crit = 0.13 for a surfactant system using these
values (20).
Initial growth rate of the disturbance.
The q of the capillary instability is found to increase as increases and to decrease in the presence of surfactant. In Fig. 5A, the presence of PLPC in
system 2 decreases the growth rate to as little as 21% (at
= 0.19) of surfactant-free system 1 values. In Fig.
5B, Tween 20 causes q to decrease to as little as 24%
(at = 0.20) of the corresponding surfactant-free values for these
experiments.
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Fig. 5.
Initial growth rate
(q = q*aµfilm/*) vs.
for systems 1 () and 2 (; A) and
systems 3 () and 4 (; B). Surfactant
decreases growth rate, and thicker films grow faster than thinner
films. See text for theoretical curves.
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|
In Fig. 5, the solid lines are a reproduction of Goren's theory (16)
for the maximum growth rate (qmax) of a
surfactant-free system of arbitrary thickness . We have calculated
the theory line by rescaling the wave number on the tube radius, or
k = k*a. The three broken lines are from our
linear theory of qmax, which is valid for thin
films; we have shown lines representing the theory up to = 0.20. The short-dashed line is our current theory in the limit of no
surfactant, or = 0. Derivation of our present theory is detailed in LINEAR STABILITY ANALYSIS FOR THE AIRWAY CLOSURE MODEL WITH SURFACTANT. Details on the theoretical results follow the derivation in THEORETICAL RESULTS. The
long-dashed and long-and-short-dashed lines show our theory for an
interface with a small amount of surfactant
= 0.01) and saturated with surfactant
= 
), respectively. Normal surface
activity = 8 lies in the narrow margin
between the very small and very large surfactant activity values. As
the surface activity increases, the growth rate decreases. Note how
even a small amount of surfactant can cause a significant decrease in
the growth rate compared with the same system without surfactant.
Thickness of the annular film.
The of the formed annular film is shown to increase with Ca for the
inflow procedure in Fig. 6. In Fig.
6A, some experimental results of Goldsmith and Mason's
surfactant-free system 6a (15) are also shown for comparison.
There is a smooth transition between their data and ours. We find the
asymptotic behavior as predicted by others (5, 33), with our
experimental value near = 0.34 at Ca = 10. In Fig. 6B, at a
given Ca, decreases slightly in the presence of Tween 20, from
system 3 to system 4. However, the change in with
the addition of surfactant is comparable to the experimental scatter,
which is <10%. The long-dashed line is Eq. 2, Bretherton's
theory (2), and the short-dashed line is Eq. 1, from
Fairbrother and Stubbs (10). Equation 2 is inaccurate for Ca > 102 and overpredicts for all our experiments in
this study. Equation 1 matches well for Ca < 0.2 but is
invalid for large Ca and overpredicts for Ca > 0.2. We reproduce
the numerical results of Martinez and Udell (33) (solid line) for a
surfactant-free system, in the range 102 < Ca < 10. There is little variation in between systems 1 and 2, the deviation being comparable in size to the experimental scatter, which ranges from 2 to 13%.
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Fig. 6.
= (a b)/a vs. capillary number
(Ca = µfilmU*/*) of infused fluid for
systems 1 () and 2 () and Goldsmith and Mason's
system 6a ( ; A) and systems 3 () and
4 (; B). increases with Ca, with little
variation between surfactant-free and surfactant systems. See text for
theoretical curves.
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LINEAR STABILITY ANALYSIS FOR THE AIRWAY CLOSURE MODEL WITH
SURFACTANT |
A linear stability analysis is performed on the airway closure model
with surfactant. The model is based on the inertia-free analysis
developed by Halpern and Grotberg (20), who derived a system of
nonlinear evolution equations (Eq. A1) for the film thickness
and surface concentration of surfactant (see APPENDIX). To
investigate linear stability of the system, we use normal-mode sinusoidal disturbances
|
(3)
|
where = q*3aµfilm/* and
= t*/(3aµfilm/*). The
dimensionless surface concentration = */*m = 1 + 2, where = 2 and
is the variation of surface
concentration from the mean. For Eq. 3,
|h1| 1 and
, and
substitute Eq. 3 into Eq. A1. We define the
dimensionless wave number k = k*a and the
surface activity parameter = 2. The dispersion
relation for is
|
(4)
|
If = 0, the constant surface tension result is
recovered
|
(5)
|
Equation 5 can also be obtained by using normal
modes with Hammond's analysis (22) or in the rigid-tube limit of
Halpern and Grotberg (20). In the limit 
|
(6)
|
This limit can also be obtained by applying a no-slip
condition at the fluid interface. Hence, in relatively large
concentrations, surfactant slows closure by up to a factor of 4. This
is consistent with results of Otis et al. (35), who discussed these two limits.
| |
THEORETICAL RESULTS |
Figure 7A shows
max vs. . This comes from
Eq. 4 by use of the corresponding most dangerous wave number
kcrit at each value of . The
max decreases with increasing surface
activity. In the limit of vanishing surface activity, the growth rate
approaches the constant surface tension value,
max = 
. As the interface becomes
saturated with surfactant, the growth rate asymptotes to one-fourth of
the surfactant-free value. This corresponds to the rigid-surface limit.
Figure 7B shows kcrit vs. ;
kcrit varies weakly with . In the
limit of zero surface activity, kcrit = 21/2 is recovered. There is a minimum value of
kcrit at = 0.2, and the critical wave
number then asymptotes to kcrit = 21/2 for an interface saturated with surfactant.
Theoretically, typical values range from
0 (no surfactant) to 8 (normal surfactant activity);
represents an interface saturated
with surfactant. With the assumption that = 0.1-0.4, the
corresponding values of range up to 800 (for
0 8). In Fig. 7, for
> 1, kcrit and
max rapidly approach their respective
asymptotic limits for an interface saturated with surfactant.
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Fig. 7.
Linear film stability, including surfactant effects. Maximum growth
rate (max) vs. surfactant activity
() (A), critical wave number
(kc) vs. (B), and
vs. k (C) for (top to
bottom) = 0, 0.1, 1, and , respectively.
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Figure 7C shows vs. k for = 0, 0.1, 1, and . The max decreases with
increasing surface activity. Because the growth rate appears to be more
sensitive to changes in the at small values of
, we focus on smaller values of for
demonstration. The kcrit at which the growth rate
is maximum shifts slightly as changes, according to Fig.
7B. At any given wave number, the growth rate of a disturbance
is larger in a surfactant-free system than in a system with nonzero
surface activity.
| |
DISCUSSION |
General observations.
The growth of the disturbance is related to the in general and to
the relation of to crit in particular. Figure
8 shows sample data from three different
experiments. We observe that for > crit (Fig.
8A; = 0.152 compared with predicted
crit = 0.12), the disturbance initially grows at a
constant rate, which then accelerates until closure. When the
disturbance amplifies, it draws fluid from the thinning region near the
wall into the bulge of the film. For thin films where is just
slightly larger than crit (Fig. 8B;
= 0.131, predicted
TOP
ABSTRACT
INTRODUCTION
