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Clemson University Clemson University
TigerPrints TigerPrints
All Theses Theses
May 2020
Asymmetric Instability of Thin-Film Flow on a Fiber Asymmetric Instability of Thin-Film Flow on a Fiber
Chase Tyler Gabbard Clemson University, cgabbar@g.clemson.edu
Follow this and additional works at: https://tigerprints.clemson.edu/all_theses
Recommended Citation Recommended Citation Gabbard, Chase Tyler, "Asymmetric Instability of Thin-Film Flow on a Fiber" (2020). All Theses. 3279. https://tigerprints.clemson.edu/all_theses/3279
This Thesis is brought to you for free and open access by the Theses at TigerPrints. It has been accepted for inclusion in All Theses by an authorized administrator of TigerPrints. For more information, please contact kokeefe@clemson.edu.
Asymmetric instability of thin-film flow on a fiber
A Thesis
Presented to
the Graduate School of
Clemson University
In Partial Fulfillment
of the Requirements for the Degree
Master of Science
Mechanical Engineering
by
Chase Gabbard
May 2020
Accepted by:
Dr. Joshua Bostwick, Committee Chair
Dr. John Saylor
Dr. Gregory Mocko
Abstract
The flow of a thin film down a vertical fiber is valuable for many industrial applications, such
as fiber coating, microfluidics, heat exchangers, and desalination processes. It is well known that
such flows give rise to a number of instabilities that define the bead-on-fiber morphology including
Plateau-Rayleigh breakup, isolated bead formation, and convective instabilities, all of which leave
the interface shape axisymmetric. A new asymmetric instability in the flow of liquid on a fiber is
observed and its dependence upon liquid properties, flow rate, and fiber diameter is documented.
The transition between symmetric and asymmetric morphology depends critically upon the fiber
diameter and surface tension. The instability dynamics are described by the bead spacing and
bead velocity/frequency and are used to contrast the behavior of the symmetric and asymmetric
morphologies. The asymmetric morphology displays more regular dynamics than the symmetric
morphology. For example, in the asymmetric morphology the transition from the Plateau-Rayleigh to
convective regime occurs when the nozzle velocity is equal to the bead velocity and this dimensionless
velocity scales with the capillary number. This prediction, in addition to a full description of the
instability dynamics of thin film flows down fibers, can be used as a design tool for novel desalination
processes.
ii
Acknowledgments
Thank you to Dr. Joshua Bostwick, who has dedicated countless hours to the success of
this project and provided valuable guidance and wisdom. Thank you to Dr. John Saylor and
Dr. Gregory Mocko, for their involvement in this work and their insightful comments. Thank
you to Taylor Nichols, Xingchen Shao, Saiful Tamim, Phillip Wilson, Caleb Wilson, Bailey Basso,
Shivuday Kala, Jennifer Shaffer Brown, and Carlos Perez for constructive conversations which have
undoubtedly shaped and improved this work. Lastly, I want to thank my family and friends for the
support and encouragement they have provided.
iii
Table of Contents
Title Page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Plateau-Rayleigh Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Thin-Film Flow Down a Fiber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Symmetry of Fluid Beads on Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Motivation and Organization of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.1 Experimental Set-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Image Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3 Experimental Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.4 Dimensionless Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.5 Experimental Protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3 Bead Morphology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.1 Symmetric Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.2 Asymmetric Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4 Bead Symmetry and Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.1 Independent Variable (Df ,Dn,Q) Effects . . . . . . . . . . . . . . . . . . . . . . . . . 275.2 Viscous Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.3 Frequency Trends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.4 Scaled Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406.1 On Transition of Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406.2 On Transition of Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
iv
Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47A Bead Spacing Trends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48B Bead Frequency Trends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
v
List of Tables
2.1 Fluid properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Experimental variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3 Dimensionless groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
vi
List of Figures
1.1 Plateau-Rayleigh breakup of water from a dripping faucet. . . . . . . . . . . . . . . . 21.2 Beading pattern regimes observed by Kliakhandler et al. for fluid flow down a fiber [34] 41.3 The “Roll-up Process” describes the transition of a drop on a fiber from a symmetric
(top) to an asymmetric (bottom) profile [7]. . . . . . . . . . . . . . . . . . . . . . . . 6
2.1 Symmetric (right) and asymmetric (left) morphologies observed. . . . . . . . . . . . 82.2 Experimental set-up. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 The initial image (left), black and white image (middle), and final image (right)
achieved using image processing techniques. . . . . . . . . . . . . . . . . . . . . . . . 112.4 Light intensity against location on the fiber with fitted spline. . . . . . . . . . . . . . 132.5 Peak locations of the the fitted spline used to calculate the movement and spacing of
beads from image-to-image. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.6 Close-up of nozzle showing the flow rate Q, nozzle diameter Dn and fiber diameter Df . 142.7 Bead morphology properties showing the bead spacing Sb, bead velocity Vb and bead
diameter Db. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.1 Symmetric bead morphology exhibits isolated (left), Plateau-Rayleigh (middle), andconvective (right) regimes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 Illustration of bead coalescence in the symmetric convective regime. . . . . . . . . . 203.3 Phase diagram for the symmetric morphology delineates the isolated, Plateau-Rayleigh,
and convective symmetric morphologies in the nozzle diameter (Dn) – flow rate (Q)parameter space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.4 Asymmetric bead morphology exhibits isolated (left), Plateau-Rayleigh (middle), andconvective (right) regimes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.1 Phase diagram of the bead symmetry, as it depends upon the surface tension andfiber diameter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.2 Bead morphology for glycerol/water mixtures as Df is increased. . . . . . . . . . . . 254.3 Bead morphology for glycerol/water mixtures as σ is increased. . . . . . . . . . . . . 26
5.1 Bead diameter Db against flow rate Q, as it depends upon fiber diameter Df (left)and nozzle diameter Dn (right) for the symmetric morphology. . . . . . . . . . . . . 28
5.2 Bead velocity Vb against flow rate Q, as it depends upon fiber diameter Df (left) andnozzle diameter Dn (right), for the symmetric morphology. . . . . . . . . . . . . . . 29
5.3 Bead velocity Vb against fiber diameter Df at the onset of the convective regime,contrasting the symmetric and asymmetric morphologies. . . . . . . . . . . . . . . . 30
5.4 Bead velocity Vb against nozzle velocity Vn contrasting the symmetric and asymmetricmorphologies at the onset of the convective regime. . . . . . . . . . . . . . . . . . . . 31
5.5 Bead velocity Vb against flow rate Q, as it depends upon viscosity µ for the asymmetric(left) and symmetric (right) morphology. . . . . . . . . . . . . . . . . . . . . . . . . . 32
5.6 Bead velocity Vb against viscosity µ at the onset of the convective regime contrastingthe symmetric asymmetric morphology. . . . . . . . . . . . . . . . . . . . . . . . . . 33
vii
5.7 Bead spacing Sb against flow rate Q, as it depends upon viscosity µ for the asymmetric(left) and symmetric (right) morphology. . . . . . . . . . . . . . . . . . . . . . . . . . 34
5.8 Bead spacing Sb against viscosity µ at the onset of the convective regime contrastingthe symmetric and asymmetric morphology. . . . . . . . . . . . . . . . . . . . . . . . 35
5.9 Viscosity µ against flow rate Q at the onset of the convective regime contrasting thesymmetric and asymmetric morphologies. . . . . . . . . . . . . . . . . . . . . . . . . 36
5.10 Frequency f against flow rate Q contrasting the symmetric and asymmetric morphologies 375.11 Frequency ratio against Weber number contrasting the symmetric and asymmetric
morphologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385.12 Velocity ratio against Capillary number contrasting the symmetric and asymmetric
morphologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
1 Bead spacing Sb against flow rate Q, as it depends upon nozzle diameter Dn, for theasymmetric (left) and symmetric (right) morphology. . . . . . . . . . . . . . . . . . . 48
2 Bead spacing Sb against flow rate Q, as it depends upon fiber diameter Df , for theasymmetric (left) and symmetric (right) morphology. . . . . . . . . . . . . . . . . . . 49
3 Frequency f against flow rate Q, as it depends upon fiber diameter Df , for theasymmetric (left) and symmetric (right) morphology. . . . . . . . . . . . . . . . . . . 50
4 Frequency f against flow rate Q, as it depends upon nozzle diameter Dn, for theasymmetric (left) and symmetric (right) morphology. . . . . . . . . . . . . . . . . . . 51
viii
Chapter 1
Introduction
Rich dynamics are observed in the thin film flow down a fiber and these instabilities are
utilized in a large number of applications where heat and mass transfer across a liquid-gas interface
is desirable, such as gas absorption [28] [25] [11], heat exchangers [57] [58], microfluidics [23], drop
capturing [37], and desalination [50]. Shape-change instabilities resulting in the bead-on-fiber mor-
phology, with associated high surface area to volume ratio, that are desirable for optimal interfacial
mass and heat transfer. These beading patterns are surface tension driven and the result of the
well-known Plateau-Rayleigh hydrodynamic instability [43] [46].
1.1 Plateau-Rayleigh Instability
Capillary breakup is readily observed in the disintegration of liquid columns into spherical
drops in leaky faucets, as shown in Figure 1.1. This instability represents a balance between fluid
inertia and the restorative force of surface tension acting at the liquid/gas interface, which wants
to minimize the surface energy. For liquid columns with length L greater than the circumference
2πR, L > 2πR, the configuration is unstable and the column will reconfigure itself into a series
of drops separated by an optimal wavelength. Plateau was the first to connect the instability and
surface tension and used a static stability analysis to derive the stability threshold described above.
However, he wasn’t able to successfully predict the critical wavelength. [43]. Lord Rayleigh used a
hydrodynamic stability analysis to correctly predict the wavelength and growth rate of the critical
disturbance [46]. For this reason, this instability has come to be known as the Plateau-Rayleigh
1
instability. More recent research includes investigation of nonlinear effects, the role of viscosity on
the breakup time, the stability of liquid bridges [38] [4] and the role of liquid/solid contact. With
regard to the latter, Bostwick & Steen (2010) have shown that pinning the liquid column with a
wire along its length can stabilize the PR limit by 13% [3]. Recent reviews on the subject include
[22] [20] [5].
Figure 1.1: Plateau-Rayleigh breakup of water from a dripping faucet.
A thin liquid film coating a fiber similarly forms a liquid column but in the ‘bead-on-fiber’
morphology. Here one needs to account for the interaction of the fluid with the solid fiber in
describing the instability dynamics. Simon L. Goren, investigated the instability of a cylindrical
column of fluid symmetrically wrapped around a wire[24] and concluded the critical wavelength is
(2πa/λ) where a is the initial radius of the liquid cylinder and is the wavelength of the disturbance.
Mead-Hunter uses computational fluid dynamics to analyze the breakup of a fluid column on a
thin fiber [42]. Droplet motion and displacement under fluid flow were simulated, and Mead-Hunter
2
determined that airflow around the droplet has little influence on the stable film thickness but reduces
the time required for droplet formation. More recently, Haefner et al. analyzed the influence of slip
on the Rayleigh-Plateau instability on a fiber [26] showing the hydrodynamic boundary conditions
at the solid-liquid interface does not affect the dominating wavelength but does affect the growth
rate of the undulations.
1.2 Thin-Film Flow Down a Fiber
Thin-film fluid flow down a fiber produces a number of temporal beading patterns related
to Plateau-Rayleigh instability. The constant base flow rate leads to a number of spatially-evolving
patterns. The emergence of a convective instability in the fluid flow results in both steady and
unsteady beading patterns. Kliakhandler et al. experimentally highlighted this and defined three
primary regimes [34], as illustrated in Figure 1.2. The convective regime (a) is characterized by
faster-moving fluid beads, which coalesce with smaller, slower beads. The Plateau-Rayleigh regime
(b) is characterized by a stable traveling wave in which no bead coalescence is observed. Lastly, the
isolated regime (c) is where an array of smaller drops separates widely spaced traveling beads. Recent
work by Sadeghpour et al. experimentally examined the effects that nozzle size has on the observed
regime [51]. Interestingly, it is seen that when the flow rate and fiber size were held constant, all
three regimes could be observed by altering the nozzle size. Experimental work performed by Smolka
et al. attempted to explore the effects of altering the fluid properties by using three different fluids
and analyzing their resulting beading pattern [54]. Experimental studies of convective and absolute
instabilities have shown a transition from the absolute instabilities observed in the isolated and
Plateau-Rayleigh to the convective regime occurs when the characteristic time scale τm of the most
amplified wave is balanced with the characteristic time of advection τ of the wave [18] [17].
Quere examines under what conditions drops cease to form and shows how it depends on
the film thickness and fiber diameter [44]. These are volume effects. Quere later revisits the subject
and proposes a description for fluid coating a fiber that is slowly or quickly withdrawn from a
bath for both pure and complex fluids [45]. Frenkel developed a nonlinear theory which showed
good agreement with Quere’s experimental work [21]. Using the scaling arguments presented by
Frenkel, Kalliadasis and Chang found solitary wave solutions using a matched asymptotic analysis
and determine a critical thickness hc, which must be exceeded for beads to develop [33]. Chang and
3
Figure 1.2: Beading pattern regimes observed by Kliakhandler et al. for fluid flow down a fiber [34]
Demekhin study the stability of the thin film [9] and show that for fluid films where h > hc, where
hc is a critical thickness, fluid films evolve into continually growing pulses and enter the convective
regime [9]. Craster and Matar investigated a weakly nonlinear thin film model [14]. In each of these
investigations, the primary assumption is that the fiber radius is much larger than the film thickness.
Kliakhandler et al. proposed an alternative model for a thick-film which assumed creeping-
flow which was able to predict two of the three primary regimes: the Plateau-Rayleigh and isolated
regimes [34]. While this model was capable of predicting the existence of the Plateau-Rayleigh
regime, it shows a slight discrepancy in the bead spacing and velocity which they compare with
experimentation. Several additional studies have addressed these discrepancies by considering slip-
enhanced drop formation [27], different scalings [56], streamwise viscous diffusion [48], and disjoining
pressure effects [31]. A comprehensive review of the current models used to describe thin-film flows
down a fiber is found in [49] [31].
4
1.3 Symmetry of Fluid Beads on Fibers
When a drop is placed on a fiber, the fluid can spread into a film or morph into one of
two beading profiles depending upon the energetics, which have been investigated for a number
of different spreading factors [6]. When a drop profile is energetically preferred, both asymmetric
and symmetric profiles are possible. The asymmetric configuration is termed “clam shell” and the
symmetric “barrel”, as shown in Figure 1.3. Carrol experimentally showed the existence of these two
configurations in a study on the effects of the fiber roughness on the resulting drop symmetry [8].
Carrol showed that both profiles are achievable on a rough or smooth fiber, but the geometry of the
roughness plays a significant role in determining the behavior of the fluid. Revisiting the problem,
Carrol establishes the first theory on the transition from a symmetric to an asymmetric drop profile
and labels the transition the “Roll-up Process” [7] showing the axial symmetric configuration tends
to lose its stability and transform into an asymmetric profile as the contact angle exceeds a critical
value. The critical contact angle is shown to depend on the fiber radius and the volume of the drop.
The dependence of the drop profile on the contact angle makes the ability to measure the
contact angle vital to validating the theory proposed by Carrol. Kumar and Hartland investigated the
measurement of contact angles on fibers and determined that the difference between the advancing
and receding contact angles increased as the drop radius is increased [35]. This result shows that
gravitational forces affect the drop shape and, thus, the observed contact angles. The high curvature
of the drop profile makes a direct measurement of the apparent contact angle difficult. McHale et
al. investigate the theoretical profile of a symmetric drop on a fiber [41] and present an argument
that the inflection angle can be determined using the droplet profile, therefore bypassing the need
to measure the contact angle.
McHale et al. used the theoretical inflection angle of a drop profile on a fiber to propose a
new theory for the roll-up process, which states that a transition from a symmetric to asymmetric
profile occurs as the inflection angle disappears [40]. This occurs as the inflection point approaches
the contact line at the fiber surface. A finite element approach is used to evaluate this new theory for
the roll-process. Evaluating the surface energies of the two possible profile symmetries for a broad
range of contact angles and volumes, McHale et al. validate the correlation between symmetric bead
profiles and the existence of an inflection angle by showing all finite element results for symmetric
profiles also possess a positive inflection angle [39].
5
Figure 1.3: The “Roll-up Process” describes the transition of a drop on a fiber from a symmetric(top) to an asymmetric (bottom) profile [7].
1.4 Motivation and Organization of the Thesis
Asymmetric beading patterns in thin-film flow down a fibers have not been observed experi-
mentally. Thus, our primary motivation for this work involves the experimental comparison between
asymmetric beading patterns with the classically studied symmetric beading patterns. In order to
do this we perform experiments to address two main points. First, we wish to understand under
what conditions is an asymmetric or beading pattern observed and when does the transition occur.
Secondly, we want to understand how the dynamics of an asymmetric pattern compare to a symmet-
ric pattern. To do this we conduct a large number of experiments, varying multiple experimental
parameters and using multiple working liquids to compare the asymmetric and symmetric beading
patterns. In addition, our symmetric data expands the parameter space of existing research.
The outcomes and conclusions of this experimental study have direct application in many
areas previously discussed. These application areas include processes such as water desalination [50],
6
which are critical in addressing global issues that are sure to shape the coming century of scientific
endeavors. This fact further stimulates the research performed in this work. Many vital areas of
research are unexplored for thin-film flow down fibers and could be fruitful investigation topics to
advance this work. The commonly explored topics of transition from dripping to jetting [12], the
route to chaos in dripping water [16], and non-Newtonian fluid mechanics have all yet to be explored
for systems which include a cylinder geometry modifying the falling fluid flow.
7
Chapter 2
Experiment
Figure 2.1: Symmetric (right) and asymmetric (left) morphologies observed.
We perform experiments on the beading patterns which form from applying a thin-film flow
of Newtonian liquid onto a vertically hung fiber. Figure 2.1 illustrates the typical bead symmetries.
Imaging processing techniques are used to quantify the instability dynamics, as they depend upon
8
the experimental parameters.
2.1 Experimental Set-Up
The experimental setup consists of a structural testing apparatus, syringe pump, camera,
adjustable camera mount, fiber, nozzle, fluid collection basin, and a calibration tool, as shown in
Figure 2.2. Fluid flows at volumetric flow rate Q through a nozzle of diameter Dn onto a fiber with
a diameter of Df . The flow rate of the fluid is controlled by a NE-1000 series programmable syringe
pump and we orient the nozzle such that it applies the fluid orthogonal to the fiber. Two pinning
devices are located at the top and bottom of the testing apparatus and hold the fiber vertical.
The fluid evolves into beading patterns as it flows down the fiber until it reaches the collection
basin. The testing apparatus design allows the fluid to flow down the fiber for 22 inches before it
reaches the basin. This distance allows the beading patterns of all experiments performed to become
fully-developed.
Three liquids, i) glycerol/water mixture, ii) silicone oil, iii) mineral oil, were used with
material properties given in Table 2.1. These liquids were intentionally chosen in order to explore
large viscosity and surface tension ranges. Surfactant was used to vary the range of surface tension
for glycerol/water mixtures from 30 to 60 mN/m. The weight percentage of glycerol to water changes
the viscosity of the fluid mixture. Silicone oils of differing viscosities were chosen to overlap those
of the glycerol/water mixtures. The viscosity range explored is 10 to 1000 mPa · s. Experiments
using mineral oils allows us to capture the effects of density. Liquid density is measured using a
Anton Paar DMA 35 density meter, viscosity using a shear rate test with an Anton Paar MCR 302
rheometer, and surface tension with a Kruss K100 surface tensiometer using the Wilhelmy plate
method.
Fluid Density, ρ [kg/m3] Viscosity, µ [mPa · s] Surface Tension, σ [mN/m]Glycerol/Water 1,040 - 1,243 85 - 787 30 - 60
Silicone Oil 936 - 974 10 - 1000 22Mineral Oil 848 17.3 25
Table 2.1: Fluid properties.
High-speed videos are used to capture the bead dynamics. A camera, recording at 960
frames per second, is mounted onto the test apparatus via two horizontal and one vertical threaded
9
Figure 2.2: Experimental set-up.
rod. The threaded rods allow for the camera to be adjustable in the x and y-direction. The threaded
rods allow for optimal positioning to capture clear, focused images of any part of the beading pattern.
Before recording a beading pattern, proper image analysis requires the use of an image calibration
tool. The image calibration tool contains a series of vertically oriented, default images of known size
that allow for the calibration of one pixel to its corresponding length. Image analysis can calculate
important measurable parameters in terms of pixels and then convert them into a standard unit of
length. The following section further details the image processing and analysis used.
10
2.2 Image Analysis
Image analysis of each experiment allows for the measurement of the bead velocity Vb, bead
spacing Sb, and bead diameter Db. These quantities define the instability dynamics. Here our images
are processed and analyzed in MATLAB. Optimizing certain factors before recording an experiment
allows for high-quality images to be captured. These factors include ensuring the video is focused,
lighting is optimized to highlight the bead profile, and the camera is stable. Custom lighting, a
reliable and sturdy camera mount, and manually focusing the camera for each experiment ensure
these factors are ideal. Videos recorded for each experiment result in 150 individual images. Image
processing of each of the 150 images puts them into a standard, black and white format, which is
evaluated using a specialized Matlab code consisting of three steps. First, the code extracts all of
the individual images from the slow-motion recording. Second, each image is converted into a binary
(black and white) image. The resulting image has a black background with the outline of the fluid
being white. However, this step often does not fully convert all pixels inside of the bead profile to
match the white outline of the fluid. The last step uses a MATLAB function to fill in the rest of the
fluid so that the entire fluid body is colored white. Figure 2.3 illustrates this three-step process.
Figure 2.3: The initial image (left), black and white image (middle), and final image (right) achievedusing image processing techniques.
Each image consists of a matrix of 0’s and 1’s where 0 and 1 represent a pixel colored black
and white, respectively. Summing up the number of 1’s across each row in the matrix allows for
11
the thickness of the fluid at that position to be calculated. Performing this operation for each row
of the image matrix produces a vector that contains the width of the fluid at that location. The
largest value within the vector is considered the bead diameter. This method produces reliable
values for the diameter of the largest bead in each image. However, this method does not allow for
the measurement of more than one bead diameter in the image.
We next apply a smooth spline to the light intensity versus vertical location plot shown in
Figure 2.4. A built-in MATLAB function that optimally fits a spline to a data set generates a best
fit to the the light intensity values. It is important to note that while the spline gives a reliable
method for tracking the location of a bead, the maximum bead thickness it calculates is often less
than the actual value. Thus, we use the raw data to calculate the bead diameter, as mentioned
above. Figure 2.4 displays the resulting light intensity versus the vertical location plot. Each local
maxima of the spline represent the location of a bead’s thickest cross-section and we use the points
to track its movement from image-to-image. Custom MATLAB code generates the location of each
of these maximum points in an image file. A concern using this method is the false recognition of
small, secondary beads that emerge in the isolated regime as primary beads. We customize our code
such that it only registers maximum points higher than a set value to assure the code registers the
primary beads. This setting is modified on a video-to-video basis to ensure the number of maximum
locations output correctly matches the number of beads observed in the image.
Bead spacing is determined by counting the number of pixels between each bead. The bead
velocity is extracted using the bead location from image-to-image and the number of frames per
second the video recording captured. This method is reliable as long as the code correctly tracks
the same bead down the entire length of the image. Velocity values from this method become
inaccurate if two bead locations are confused for each other, or a bead spline profile is close to the
bead recognition value that we previously discussed.
Figure 2.5 shows the final plot produced in the image analysis process. The bead spacing
and bead velocity values calculated using this plot, along with the bead diameter calculated for each
image, are averaged to give the final bead spacing, velocity, and diameter values for the experiment.
Last, we define the bead frequency as
f = Vb/Sb (2.1)
where Vb is the bead velocity and Sb is the bead spacing.
12
Figure 2.4: Light intensity against location on the fiber with fitted spline.
2.3 Experimental Variables
The three independent variables in our experiment include the fiber diameter Df the nozzle
diameter Dn and the volumetric flow rate Q as shown in Figure 2.6. The fiber material plays a
notable role as differing the material used will result in a different contact angle between the fluid
and fiber. Different fiber material can also result in different material roughness, which affects the
resulting fluid/fiber interaction [2]. An investigation of the fiber wetting properties is not a goal of
this work and we use nylon thread exclusively to minimize these issues. We are only concerned with
the fiber diameter Df and we explore a wide range Df = 0.101 − 0.5018mm. We use stainless steel
nozzles with circular cross section oriented orthogonal to the vertical fiber. Our interest is in the role
of nozzle diameter. A constant volumetric flow rate Q is used to avoid any transient effects. The
range of values tested for the fiber diameter, nozzle diameter, and volumetric flow rate are given in
Table 2.2.
The instability dynamics present in our experiments creates a large pool of measurable
parameters that can be explored. We primarily concern ourselves with the dynamics of the beading
13
Figure 2.5: Peak locations of the the fitted spline used to calculate the movement and spacing ofbeads from image-to-image.
Figure 2.6: Close-up of nozzle showing the flow rate Q, nozzle diameter Dn and fiber diameter Df .
patterns that develop as measured by the bead velocity Vb, bead spacing Sb, and bead diameter
Db. Bead velocity is specified as the steady-state velocity of each bead as it travels down the
14
Parameter Range
Fiber Diameter (Df ) 0.101 - 0.5018 [mm]
Nozzle Diameter (Dn) 0.4 - 1.6 [mm]
Volumetric Flowrate (Q) 5 - 600 [mL/hr]
Table 2.2: Experimental variables.
fiber. In most cases, the bead velocity that initially forms at the nozzle is not consistent with the
steady-state velocity observed further down the fiber and quantitative measurements are taken a
significant distance down the fiber such that nozzle effects are no longer observable. Bead spacing
is also measured once it has attained a constant value. Lastly, we define the bead diameter as the
maximum thickness of each bead profile, perpendicular to fiber. The bead diameter of each fluid
bead develops as the Plateau-Rayleigh instability saturates the fluid behavior; thus, we only take
bead diameter values once they have stabilized. It is worth noting that since asymmetric beading
patterns allow beads to rotate, exact values for bead diameter are much harder to obtain. For that
reason, we do not analyze bead diameter values for asymmetric bead profiles.
We restrict measurements of each of the three dependent variables to those experiments
that produce beading patterns categorized as isolated or Plateau-Rayleigh. In these regimes, bead
spacing, velocity, and diameter are steady values that are repeatable in experiments. However, in
the convective regime, beading patterns have properties that vary significantly from moment-to-
moment due to coalescence events occurring at irregular distances down the fiber. Thus, all beading
patterns in the convective regime do not have values recorded for their flow characteristics. Figure
2.7 illustrates our three dependent variables from a symmetric beading pattern.
2.4 Dimensionless Groups
Our data can be scaled and plotted against several dimensionless numbers to provide insight
in the dominant physical mechanisms at play. Table 2.3 gives the dimensionless numbers we use
to analyze our experimental data. The nozzle Weber number We gives a comparison of inertial to
capillary forces. Its historical use in thin-film problems and drop formation relates well with our
experiment. Here the characteristic length, L, is given the value of the fiber diameter used in the
experiment. The capillary number Ca provides a comparison of viscous to inertial forces and we
define the capillary number for the fluid at the exit of the nozzle. Unlike the Weber number, the
15
Figure 2.7: Bead morphology properties showing the bead spacing Sb, bead velocity Vb and beaddiameter Db.
capillary number takes into account how the viscosity of the fluid may govern certain aspects of
the fluid flow. The velocity ratio V ∗ is defined as the velocity of the beads flowing down the fiber
divided by the velocity of the fluid as it exits the nozzle, V ∗ = Vb/Vn. The scaled frequency f∗ is
defined as the frequency times the viscous time scale Tv where Tv is defined as D2f/ν.
Dimensionless Group Definition
Weber Number (We) ρQ2L/A2σ
Capillary Number (Ca) νρQ/Aσ
Velocity Ratio (V ∗) Vb/Vn
Frequency Ratio (f∗) f × Tv
Table 2.3: Dimensionless groups.
2.5 Experimental Protocol
In this section, we thoroughly describe the experimental protocol used during experimen-
tation laying out all procedures conducted during an experiment, as well as the order to complete
16
them. We strictly follow the same experiment protocol for all experiments conducted during this
investigation.
First the camera is secured onto the experimental apparatus via the camera mount. After
positioning the camera so that it captures the desired portion of the fiber, we calibrate the camera.
We do this by placing a calibration tool where the fluid flow will pass and recording a video. We
establish a relationship between each image pixel and a physical unit of length. Next, the fiber
diameter, nozzle diameter, and flow rate of our experiment are selected and setup. The fiber is then
secured vertically. Next, we load a syringe with the liquid being tested and attach the desired nozzle.
We load the syringe into the syringe pump and set the pump to the desired flow rate Q. Lastly, we
ensure that the collection basin is clean of any previous testing liquid and is located directly below
the fiber. Ensuring all previous steps were fulfilled, we then begin the experiment by turning the
syringe pump on so that fluid begins to flow down the fiber.
The fluid flows down the fiber until the beading pattern approaches a steady state. Constant
bead velocities and bead spacing signify the flow has arrived at a steady state. We typically allow
fluid to flow for a minimum of 20 seconds before taking a video recording of the beading pattern.
We manually focus the video camera onto the beading pattern once the flow has reached a steady
state and record the beading pattern at 960 frames per second.
Each experiment was repeated five times. The experimental output values obtained through
image processing are averaged and appointed the final value of that parameter for the experiment.
Finally, we clean up the experiment, discard the fiber used, and ensure the laboratory is free from
any liquid or debris spilled during the experiment. We take exceptional care to ensure we thoroughly
wash the nozzle and collection basin after each experiment. If we need to wash the nozzle using soap,
then we apply a flame to the nozzle to ensure we have removed all surfactant from the surface of the
nozzle. Using these procedures, we ensure that the data presented in this document is consistent
and repeatable.
17
Chapter 3
Bead Morphology
Kliakhandler et al. qualitatively categorized fully-developed symmetric beading patterns
into three primary regimes, isolated, Plateau-Rayleigh, and convective [34]. We show that these three
regimes are also seen in asymmetric beading patterns and exhibit similar qualitative characteristics.
In this section, we compare the traditional symmetric regimes with the newly discovered asymmetric
regimes.
3.1 Symmetric Flow
Performing experiments using silicone oil, we observe the classically studied beading patterns
for symmetric flows shown in Figure 3.1. The leftmost image shows the isolated regime which
corresponds to the creation of primary beads that travel with equal spacing and velocities and are
broken up by smaller, secondary beads. The primary beads and secondary beads are both results
of the Plateau-Rayleigh instability, and we observe that at both scales, the beads remain symmetric
about the fiber. The Plateau-Rayleigh regime, shown in the middle image, results in primary beads,
which also flow down the fiber with equal spacing and velocities. However, unlike the isolated
regime, the Plateau-Rayleigh regime no longer includes smaller secondary beads separating the larger
primary beads. Here the thin film does not have time for a secondary Plateau-Rayleigh instability to
develop. Lastly, we observe the convective regime, shown as the far-right image in Figure 3.1. The
convective regime is characterized by the coalescence of primary beads into large, dominant beads
which progress down the fiber with increasing volume and velocity as they coalesces with the smaller
18
primary beads. Figure 3.2 illustrates a typical coalescence event observed for convective beading
patterns. We note that the majority of applications which utilize beading patterns benefit from
increased interfacial surface area to volume ratio. Thus, the convective regime, in which beads move
with increasing velocity down the fiber clearing out all previously existing primary beads yields no
benefit compared to the Plateau-Rayleigh regime. Therefore the transition from Plateau-Rayleigh
to convective is a critical point, which we discuss in detail later.
Figure 3.1: Symmetric bead morphology exhibits isolated (left), Plateau-Rayleigh (middle), andconvective (right) regimes.
We are interested in how the nozzle diameter, fiber diameter and flow rate affect the observed
regime. Flow rate effects are best illustrated in Figure 3.1. Moving from left-to-right corresponds to
increasing fluid flow rate. Here the emergence of the isolated regime happens at a low range of flow
rates and upon increasing the flow rate, the Plateau-Rayleigh regime develops. Further increases in
flow rate lead to bead coalescence and thus the convective regime. Interestingly, the evolution of a
Plateau-Rayleigh pattern into a convective one is difficult to observe at a single critical flow rate.
Instead, the transition into convective occurs can occur within a small range of flow rate.
Similarly increasing the fiber diameter while fixing the nozzle diameter and flow rate results
in movement from the isolated to Plateau-Rayleigh to the convective regime. Thus, the fiber diameter
significantly alters the range of flow rates where a transition in regime can occur. Fixing the flow rate
19
Figure 3.2: Illustration of bead coalescence in the symmetric convective regime.
and fiber diameter, while changing the nozzle diameter has a more complex effect on the resulting
regime. Figure 3.3 is a phase plot of the three regimes in the Dn − Q space that illuminates this
complicated relationship. When the flow rate is fixed, we can observe all three of the primary regimes
by merely changing the nozzle diameter. Interestingly, we see that for a large nozzle diameter, the
Plateau-Rayleigh regime no longer exists, and patterns transition directly to the convective regime
from the isolated regime.
3.2 Asymmetric Flow
Asymmetric beading patterns are observed using glycerol/water mixtures. The regimes
that emerge from asymmetric thin-film flow parallel those observed for symmetric flow as shown in
Figure 3.4. The key characteristics are analogous to the previously discussed symmetric regimes.
Observing the far left image of the isolated regime, we see that the small secondary beads that form
have the same bead symmetry as the primary beads. Thus, the isolated regime is characterized by
primary and secondary beads of the same symmetry for both symmetric and asymmetric flow. In the
convective regime, asymmetric flow produces a large dominant bead, which progressively increases
in volume with each coalescence event it causes.
20
Flowrate [m3=s]150 200 250 300 350
Noz
zle
Dia
met
er[m
m]
0
0.5
1
1.5
2
2.5
3
3.5Isolated Plateau-Rayleigh Convective
Figure 3.3: Phase diagram for the symmetric morphology delineates the isolated, Plateau-Rayleigh,and convective symmetric morphologies in the nozzle diameter (Dn) – flow rate (Q) parameter space.
With regards to the fiber diameter, volumetric flow rate, and nozzle diameter, regime de-
pendency on each parameter reflects that observed for symmetric beading. Transition from isolated
to Plateau-Rayleigh to the convective regime occur as we increase the flow rate or as we increase the
fiber diameter. However, this pattern only holds for fiber diameters above a critical diameter since
we observe a transition in bead symmetry from asymmetric to symmetric as the bead diameter is
decreased. We will expand upon this observation further in Chapter 6. Interestingly, fiber diameters
close to the critical diameter for transition can show beading patterns that alternate between sym-
metric and asymmetric. During such flows, the regimes seen during the symmetric portion of the
flow is typically not equivalent to the one observed in the asymmetric portion. A typical example
of this flow pattern involves the transition of asymmetric flow in the isolated regime transitioning
to symmetric flow in the Plateau-Rayleigh regime back to asymmetric flow in the isolated regime
again, all happening at arbitrary distances down the fiber. Lastly, we remark that there exists a
more complicated regime dependency on nozzle diameter, as was seen with the symmetric flow.
21
Figure 3.4: Asymmetric bead morphology exhibits isolated (left), Plateau-Rayleigh (middle), andconvective (right) regimes.
22
Chapter 4
Bead Symmetry and Transition
The classically studied symmetric instabilities play a crucial role in the morphology of the
beading patterns as they flow down the fiber, but we have identified a symmetry-breaking instability
that also manipulates the interface shape. We previously showed how the regimes for asymmetric
flows parallel those of symmetric flows and how the bead dynamics compare. We further analyze
this asymmetric instability by experimentally defining when symmetry is broken. Figure 4.1 plots
the observed bead symmetry for a range of fiber diameter and surface tension values. Interestingly, a
narrow range of fiber diameters and surface tension values allow for a “transitioning” bead symmetry
in which the fluid is transitioning between symmetries down the length of the fiber.
The results presented in Figure 4.1 come from experiments where the fluid is administered
orthogonal to the fiber instead of from directly above. Thus, the orthogonal application starts the
fluid in an asymmetric configuration that must transition to symmetric while fluid applied from
directly above starts symmetric and transition to asymmetric. The difference in the initial fluid
symmetry likely affects the observed transition point, but a comparison between the transition points
for each fluid application technique has not yet been conducted. Figure 4.1 reveals the underlying
dependence of the bead symmetry on fiber diameter and surface tension. Fixing the fiber diameter,
and adjusting only the surface tension, a transition from symmetric to asymmetric occurs as the
surface tension is increased. The critical surface tension, which causes a transition in symmetry,
decreases as the fiber diameter is increased. Fixing the surface tension and modifying the fiber
diameter, a transition from symmetric to asymmetric occurs as the fiber diameter is decreased.
We also report that no symmetry-breaking transitions occur when other independent vari-
23
Fiber Diameter [mm]0 0.1 0.2 0.3 0.4 0.5 0.6
Surfac
eTen
sion
[mN
/m]
20
30
40
50
60
70Symmetric Transitioning Asymmetric
Figure 4.1: Phase diagram of the bead symmetry, as it depends upon the surface tension and fiberdiameter.
ables are altered. Altering the nozzle diameter significantly affects the flow regime developed for
both symmetric and asymmetric flows but has no effect on the flow symmetry is observed. Viscosity
shows no effect on the final symmetry of the flow but instead significantly alters the time scale for
symmetry transition. This is observed when comparing the perpendicular application of silicone oils
with different viscosities. Applying a low viscosity silicone oil, a quick transition into a symmetric
configuration occurs near the application site. However, for high viscosity silicone oils, the fluid
makes a slow transition to a symmetric configuration. This transition time increases with viscosity
and increases the distance from the nozzle where the fluid becomes symmetric.
The transitioning region separating a symmetric interface from an asymmetric one only ap-
pears for small windows of surface tension and fiber diameter. Figures 4.2,4.3 capture the transition
from a symmetric to transitioning to asymmetric interface profile. In Figure 4.2, the transition from
24
symmetric (left image) to asymmetric (right image) occurs when all parameters are fixed, and only
fiber diameter is increased. Here the middle image shows a transitioning profile at a critical fiber
diameter of 0.2921 mm. Figure 4.3 shows the same transition but for increasing interfacial surface
tension with all other parameters fixed. Again, the transitioning profile is illustrated by the middle
image, which occurs at a surface tension of 40 mN/m.
Figure 4.2: Bead morphology for glycerol/water mixtures as Df is increased.
25
Figure 4.3: Bead morphology for glycerol/water mixtures as σ is increased.
26
Chapter 5
Results
In this section we experimentally quantify and compare the symmetric and asymmetric fluid
morphologies by determining their bead spacing, velocity, diameter, and frequency values for flows
in the isolated and Plateau-Rayleigh regime. The data in this section is presented in two main ways.
The first way we present the data is by showing how the dynamics change with the flow rate, which
is often the easiest of the independent parameters to vary and allows for the data point with the
highest flow rate to represent the transition from the Plateau-Rayleigh to convective regime. In
many applications, the onset of the convective regime is the primary point of interest for a given
morphology, and thus plots illustrating data at this point provide application-focused insight into
how symmetric and asymmetric morphologies differ.
5.1 Independent Variable (Df ,Dn,Q) Effects
In this section, we evaluate the effects of fiber diameter, nozzle diameter, and flow rate.
We note that in our examination of the bead diameter, we only report and compare data for sym-
metric morphologies because asymmetric morphologies are able to rotate about the fiber freely, and
attempts to measure the diameter of such beading patterns accurately would require a 3D imaging
approach.
Figure 5.1 plots the bead diameter against the flow rate as it depends on both the fiber
diameter (left image) and nozzle diameter (right image). We observe a clear increase in the bead
diameter as the fiber diameter is decreased. The same trend is observed as the nozzle diameter is
27
increasing. An interesting note when comparing the two plots shown in Figure 5.1 is the appearance
of a steep increase in bead diameter for the first few data points in the left image that is not seen in
the right. This steep increase is the region of flow that produces isolated droplets that are separated
by great spacing due to the low flow rate. In the right plot, these flows are ignored by starting
experimentation at a higher flow rate. In both cases, it is observed that a slight decrease in bead
diameter occurs as the convective regime is approached, which could imply fluid draining from the
bead into the thin-film between beads in the Plateau-Rayleigh regime.
Flowrate [mL/hr]0 50 100 150 200 250 300
Bea
dD
iam
eter
[mm
]
2.2
2.4
2.6
2.8
3
3.2
3.4
3.6
Df = 0:101 mm
Df = 0:2032 mm
Df = 0:2921 mm
Df = 0:414 mm
Df = 0:508 mm
Flowrate [mL/hr]150 200 250
Bea
dD
iam
eter
[mm
]
2.5
2.6
2.7
2.8
2.9
3
3.1
3.2
3.3
3.4
Dn = 0:41 mmDn = 0:61 mmDn = 0:84 mmDn = 1:2 mmDn = 1:6 mmDn = 2:3 mmDn = 2:9 mm
Figure 5.1: Bead diameter Db against flow rate Q, as it depends upon fiber diameter Df (left) andnozzle diameter Dn (right) for the symmetric morphology.
Figure 5.2 plots the bead velocity against the flow rate as it depends on both the fiber diam-
eter (left image) and nozzle diameter (right image). The striking resemblance of the relative change
in bead velocity in relation to its analogous change in bead diameter reveals the clear association the
two have for symmetric morphologies. We observe an increased bead velocity as the fiber diameter
is decreased, and as the nozzle diameter is increased. Similar to the bead spacing as the convective
regime is approached, the bead velocity also undergoes a noticeable decrease when approaching the
convective regime. The highest flow rate for each data set in Figures 5.1,5.2 correspond to the
28
Plateau-Rayleigh regime with highest flow rate, i.e. further increase in Q leads to the convective
regime.
Flowrate [mL/hr]0 50 100 150 200 250
BeadVelocity
[mm/s]
0
20
40
60
80
100
120
140
160
180
Df = 0:101 mm
Df = 0:2032 mm
Df = 0:2921 mm
Df = 0:414 mm
Df = 0:508 mm
Flowrate [mL/hr]120 140 160 180 200 220 240 260 280
BeadVelocity
[mm/s]
60
80
100
120
140
160
180
Dn = 0:41 mm
Dn = 0:61 mm
Dn = 0:84 mm
Dn = 1:2 mm
Dn = 1:6 mm
Dn = 2:3 mm
Dn = 2:9 mm
Figure 5.2: Bead velocity Vb against flow rate Q, as it depends upon fiber diameter Df (left) andnozzle diameter Dn (right), for the symmetric morphology.
In the previous chapter, we showed the effect fiber diameter has on regime transition and
now focus on comparing how the fiber diameter affects the bead velocity at the onset of the con-
vective regime for the symmetric and asymmetric bead morphologies. Figure 5.3 plots the bead
velocity against fiber diameter at the onset of the convective regime for both symmetric and asym-
metric transition points. For the symmetric morphology, the bead velocity decreases with increasing
fiber diameter. Interestingly, the asymmetric morphology exhibit a near-constant bead velocity at
transition for all fiber diameters with average value 26.6 mm/s.
Figure 5.4 plots bead velocity against nozzle velocity at the onset of the convective regime for
all experiments with a nozzle diameter of 0.16mm. We observe a collapse of the asymmetric transition
points along at Vb = Vn and report that symmetric transition points show no such collapse. A clear
trend is observed for experiments with different nozzle diameters but for different relationships
between Vb and Vn. We will expand upon the importance of this in the following section.
29
Fiber Diameter [mm]0.2 0.25 0.3 0.35 0.4 0.45 0.5
Bea
dVelocity
[mm
/s]
10
20
30
40
50
60
70
80
90
Average Vb = 26:26mm/s
#
Asymmetric
Symmetric
Figure 5.3: Bead velocity Vb against fiber diameter Df at the onset of the convective regime,contrasting the symmetric and asymmetric morphologies.
5.2 Viscous Effects
In this section, we evaluate the effects of viscosity on the bead morphology. Figure 5.5 plots
the bead velocity against the flow rate as it depends upon the viscosity for the asymmetric (left)
and symmetric (right) morphologies. Aligned with intuition, increased viscosity results in a decrease
in the bead velocities observed for both bead morphologies. We observe that for similar viscosity
fluids, the bead velocity for an asymmetric morphology is much higher than symmetric morphology.
Figure 5.6 portrays the bead velocity plotted against the viscosity at the onset of the con-
vective regime for the asymmetric and symmetric morphologies. In asymmetric and symmetric
morphologies, rapidly increasing bead velocity is observed as the viscosity is decreased with larger
asymmetric velocities.
Next, we show the relationship between the fluid viscosity and the bead spacing. Figure
5.7 plots of the bead spacing against the flow rate as it depends upon viscosity. Both morphologies
30
Nozzle Velocity [mm/s]101 102
Bea
dVelocity
[mm
/s]
101
102
Vn = VbA
Asymmetric
Symmetric
Figure 5.4: Bead velocity Vb against nozzle velocity Vn contrasting the symmetric and asymmetricmorphologies at the onset of the convective regime.
show a decrease in bead spacing as the convective regime is approached and we see a fundamental
difference in the rate of change of the bead spacing when the convective regime is being approached.
For the asymmetric morphology bead spacing approach a steady value prior to transitioning. We
also remark that while the rate of change remains roughly constant for symmetric morphologies,
increasing the viscosity results in a higher rate of change compared to a lower viscosity fluid. We
further contrast the bead spacing trends of each morphology in Appendix A and see that similar
trends are seen for all asymmetric morphologies.
In Figure 5.8, we further illuminate the role of viscosity at the transition point by plotting
the bead spacing against the viscosity at the point of transition for the asymmetric and symmet-
ric morphologies. Contrasting the two symmetries, we see a considerable variation in the bead
spacing for the symmetric morphologies while we observe a near-constant bead spacing value for
the asymmetric morphologies. The average bead spacing observed for asymmetric morphologies is
31
Flowrate [mL/hr]0 100 200 300 400 500
BeadVelocity
[mm
=s]
20
30
40
50
60
70
80
90
100
110
120
7 = 191 mPa " s
7 = 331 mPa " s
7 = 404 mPa " s
7 = 590 mPa " s
7 = 630 mPa " s
7 = 787 mPa " s
Flowrate [mL/hr]0 100 200 300 400
BeadVelocity
[mm/s]
0
50
100
150
200
250
300
350
400
7 = 9:4 mPa " s7 = 48 mPa " s7 = 96:5 mPa " s7 = 194 mPa " s7 = 340 mPa " s7 = 486 mPa " s7 = 974 mPa " s
Figure 5.5: Bead velocity Vb against flow rate Q, as it depends upon viscosity µ for the asymmetric(left) and symmetric (right) morphology.
17.363 mm. Interestingly, the trend of asymmetric morphologies producing similar bead spacing
values compared to the significant variations observed for symmetric morphologies is seen as all
experimental parameters altered.
Looking once more at the plots in Figure 5.6, we recognize the significant role viscosity plays
in altering the flow rate at which transition into the convective regime occurs. In Figure 5.9, we
highlight this role by plotting the viscosity against the flow rate at the onset of the convective regime
for the asymmetric and symmetric morphologies. Decreasing the viscosity leads to an increase in the
critical flow rate to cause transition into the convective regime. The critical flow rate for asymmetric
morphology is always greater than that for symmetric morphology, irrespective of viscosity. The
trends are also different with the asymmetric morphologies display a linear trend, while a exponential
trend is observed for the symmetric morphologies.
32
Viscosity [mPa " s]0 100 200 300 400 500 600 700 800 900 1000
Bea
dVelocity
[mm
/s]
0
50
100
150
200
250
300
350 Asymmetric
Symmetric
Figure 5.6: Bead velocity Vb against viscosity µ at the onset of the convective regime contrastingthe symmetric asymmetric morphology.
5.3 Frequency Trends
Predicting the bead frequency is essential in developing high-efficiency heat and mass trans-
fer devices. An increased bead frequency produces a higher total surface area in which mass and
heat transfer can occur. We previously showed that asymmetric morphologies often exhibit near-
constant values for bead spacing and velocity. In this section, we further illustrate the predictable
and similar behavior that all asymmetric morphologies have in common by contrasting the frequency
values observed versus those observed for symmetric flows.
Asymmetric morphologies produce frequency trends that follow similar trajectories and
in Appendix B we contrast the asymmetric and symmetric frequency trends for fiber diameter and
nozzle diameter changes. In Figure 5.10 we plot the frequency against the flow rate for all asymmetric
and symmetric morphologies. This plot encompasses changes in nozzle and fiber diameter, as well as
in viscosity, density, and surface tension. We see dramatic contrast between the scattered symmetric
frequency data and the more tightly organized asymmetric data. While symmetric frequency trends
33
Flowrate [mL/hr]0 100 200 300 400 500
BeadSpacing[m
m]
10
20
30
40
50
60
70
80
7 = 191 mPa " s
7 = 331 mPa " s
7 = 404 mPa " s
7 = 590 mPa " s
7 = 630 mPa " s
7 = 787 mPa " s
Flowrate [mL/hr]0 50 100 150 200 250 300 350
BeadSpacing[m
m]
0
10
20
30
40
50
60
70
80
90
100
7 = 9:4 mPa " s
7 = 48 mPa " s
7 = 96:5 mPa " s
7 = 194 mPa " s
7 = 340 mPa " s
7 = 486 mPa " s
7 = 974 mPa " s
Figure 5.7: Bead spacing Sb against flow rate Q, as it depends upon viscosity µ for the asymmetric(left) and symmetric (right) morphology.
show no collapse along a single trend line, all asymmetric data collapses along the same trend.
This conclusion has powerful implications in applications in which the frequency must be accurately
predicted.
5.4 Scaled Data
Scaling our data, we find two non-dimensional relationships that exist for asymmetric flows
that are not present for flows of the symmetric type. The following results show the relationships
which exist among these terms for asymmetric flow patterns. Figure 5.11 plots the frequency ratio
versus the Weber number for all data. Recall that We has been defined using the fiber diameter as
the relevant length scale. The Weber number is a common dimensionless number in fluid mechanics,
which compares the inertial to capillary forces and has been historically used for fluid flows with
34
Viscosity [mPa " s]0 100 200 300 400 500 600 700 800 900 1000
Bea
dSpac
ing
[mm
]
5
10
15
20
25
30
35
40
Average Sb = 17mm
#
Asymmetric
Symmetric
Figure 5.8: Bead spacing Sb against viscosity µ at the onset of the convective regime contrastingthe symmetric and asymmetric morphology.
strongly curved surfaces similar to those produced by beading mythologies down a fiber. The
asymmetric data collapses and obeys a power-law relationship with an exponent of 0.4160. We also
note that the symmetric data does not display a similar collapse along a single trend line. These
results are valid for the Weber number range shown and only represent fluid flows that fall into the
isolated or Plateau-Rayleigh regimes.
The capillary number is commonly used in flows where the fluid interface is deformed due
to a competition between friction forces due to viscosity and capillary forces acting to minimize
the interface’s surface area. We use a broad range of viscosity and surface tension values in our
experiments and thus produce an extensive range of capillary numbers, which we compare to the
velocity ratio. Figure 5.12 illustrates the comparison of the velocity ratio to the capillary number
for all asymmetric and symmetric data.
The asymmetric data in Figure 5.10 shows a clear collapse along a single trend line which is
35
Flowrate [mL/hr]0 100 200 300 400 500 600 700
Visco
sity
[mPa"s]
0
100
200
300
400
500
600
700
800
900
1000
Asymmetric
Symmetric
Figure 5.9: Viscosity µ against flow rate Q at the onset of the convective regime contrasting thesymmetric and asymmetric morphologies.
not seen for the symmetric data. The trend is linear on a logarithmic plot and thus follows a power-
law characterized by exponent -0.7986. The regime transition from isolated to Plateau-Rayleigh
to convective occurs as the capillary number is increased which corresponds to a decrease in the
velocity ratio. This trend also illustrates how increasing the viscosity increases the tendency of a
flow towards a convective pattern. We see that no capillary number where the surface tension forces
are shown to dominate (Ca > 1) is produced for flows in the isolated or Plateau-Rayleigh regime.
This conclusion, combined with the regime analysis of the Weber number, tells us that the capillary
forces acting on the fluid interface play a role in retaining the stability seen in the non-convective
patterns.
36
Flowrate [mL/hr]0 100 200 300 400 500 600
Fre
quen
cy[1
=s]
0
2
4
6
8
10
12
Asymmetric
Symmetric
Figure 5.10: Frequency f against flow rate Q contrasting the symmetric and asymmetric morpholo-gies
37
Figure 5.11: Frequency ratio against Weber number contrasting the symmetric and asymmetricmorphologies
38
Figure 5.12: Velocity ratio against Capillary number contrasting the symmetric and asymmetricmorphologies
39
Chapter 6
Discussion
6.1 On Transition of Regime
We have documented that asymmetric morphologies evolve into the same three primary
regimes that have been previously observed and categorized for symmetric morphologies [34]. We
note that the qualitative descriptions and characteristics of these regimes remain constant between
the two morphology symmetries. It is also noted that the experimental variables, nozzle diameter
Dn, fiber diameter Df , and flow rate Q, all show similar effects on regime transition for the two
symmetries. Interestingly we note two interesting qualitative characteristics of asymmetric mor-
phologies. The asymmetric morphology produces bead profiles that are asymmetric regardless of
the volume of the drop and this observation is supported by the asymmetric profiles observed for
small-volume beads produced from secondary Plateau-Rayleigh breakup in the isolated regime as
well as the large-volume dominate beads that develop in the convective regime. Secondly, we note
that asymmetric morphologies consist of fluid motion which occurs along the same side of the fiber
and the first fluid bead that propagates down the fiber sets the path which all other subsequent fluid
beads follow.
We see the largest difference between how the morphologies transition from one regime to the
next in the transition from the Plateau-Rayleigh to the convective regime. Approaching this critical
transition, bead morphology properties such as the bead spacing Sb and bead velocity Vb behave
differently for the asymmetric and symmetric morphology. The bead spacing for all asymmetric
morphologies begins to level out to a constant value as the convective regime is approached, and
40
we note that the majority of bead spacing values where asymmetric morphologies experience this
transition are between 15-20mm. These observations are not observed for symmetric morphologies,
and the bead spacing values at the transition to convective vary significantly. The stabilization of the
bead spacing that occurs happens within the Plateau-Rayleigh regime and suggests that a unique
interaction between the fluid beads and the thin-film between the beads is at work. One possible
interaction that could explain the peculiar decreasing rate of change of the bead spacing is fluid
drainage between the beads and thin-film section of the morphology. A dynamic draining action
that increases as the bead spacing decreases could increase the fluid in the thin-film and regulate
the spacing of the beads.
Similar to the bead spacing, we note that the bead velocity also shows a unique behavior
at the transition point to the convective regime. In the symmetric morphology, the fiber diameter
greatly alters the bead velocity at the onset of the convective regime while the asymmetric mor-
phology shows a constant bead velocity at the transition point of 26.26mm/s. Interestingly, we can
again attribute this to fluid drainage between the bead and thin-film section of the morphology. As
discussed previously, fluid drainage would increase the amount of fluid volume within the thin-film
section while stabilizing the amount located within the fluid beads resulting in beads of similar
volume and traveling at similar velocities regardless of the fiber diameter. We also can conclude
that the diameter of the fiber is vital in determining the volume of the fluid beads for symmetric
morphologies but plays a lesser role for asymmetric flows. We show that for a given nozzle diame-
ter, asymmetric morphologies show consistent transition to the convective regime when the nozzle
velocity and bead velocity reach a critical ratio. We show in Figure 5.4 that for Dn = 0.16mm
this occurs at Vb/Vn ≈ 1. This valuable characteristic of asymmetric morphologies is not observed
for symmetric morphologies and allows the critical flow rate that will cause a convective flow to be
predicted using only the nozzle diameter.
6.2 On Transition of Symmetry
We have shown an asymmetric instability exists that causes a thin-film of fluid to alter its
interface symmetry, and we have shown that the symmetry transition is dependent upon surface
tension and fiber diameter. In many theories regarding the roll-up process, which we discussed in
Chapter 1, the apparent contact angle and volume of fluid within a drop determine the preferred
41
fluid symmetry. In our investigation, we focus on the surface tension and fiber diameter as the
apparent contact angle is dependent on each. Measuring the apparent contact angle on a thin-fiber
is a historically difficult task with results that are often difficult to validate. It has been shown
that for a drop on a fiber in the symmetric configuration, the interface shape is self-similar among
all drops regardless of volume, and thus the bead diameter and length can be used to provide an
approximate contact angle [41]. However, applying similar approximations to our experiments is a
post-experiment protocol that has not been validated for fluid beads in motion on a fiber. Thus, we
discuss the transition in terms of the surface tension and fiber diameter so that parameters that can
be precisely measured before an experiment can provide a predicted bead symmetry.
In describing the transition of symmetry of a stationary drop on a thin-fiber, the roll-up
process predicts that an asymmetric configuration becomes the preferred configuration as the volume
of the drop decreases or the diameter of the fiber increases [7]. In our experiments, we also see that
a transition from symmetric to asymmetric occurs as the fiber diameter is increased, and we note
that in this way, the conclusions from the roll-up process are applicable. However, we are unable to
precisely control or calculate the amount of volume which each fluid bead possesses since the beads
evolve from a thin-film instead of being directly placed on the fiber and do not report evidence that
bead volume affects transition. Future work is needed on this topic.
Although some similarities between the conclusions formed from the roll-up process and
thin-film flow on a fiber are present, we observe instances in which roll-up process predictions do
not match the behavior of thin-film flow. The transition of an asymmetric morphology into the
convective regime causes a drastic spike in the bead diameter of the dominant beads, which are
developed through continuous coalescence with smaller primary beads. According to the roll-up
process, the drastic increase in the bead diameter of the dominant convective bead should cause a
transition into a symmetric configuration while no such transition is observed, and all asymmetric
morphologies remain asymmetric in all regimes.
The inability of the previously developed theory on the roll-up process to predict the sym-
metry transitions for thin-film flow down a fiber shows that by allowing the fluid beads to move
along the fiber due to a thin-film of moving fluid alters how and when the bead transitions. Observ-
ing many asymmetric morphologies, one quickly observes that at all fluid bead scales, the fluid is
asymmetric and follows the exact path the previous bead took. We thus note that the asymmetric
morphologies possess a memory of what symmetry the previous bead was and what path it took
42
down the fiber thanks to its deposit of a thin-film or isolated droplets along its path and containing
the same symmetry. This memory is not accounted for in the roll-up process, and thus a clear need
for a new theory is evident to describe bead symmetry of moving beads on a fiber.
6.3 Applications
The morphologies produced from thin-film flow down a fiber are desirable for many applica-
tions because of their high surface area to volume ratio. Designing towards this high surface area to
volume ratio requires the ability to fine-tune the frequency to be as high as possible while also being
a safe distance away from the transition point into a convective regime. In the convective regime,
the benefits of a high frequency and large bead retention time are lost, and thus remaining in the
Plateau-Rayleigh regime is the desired regime.
We recall that the bead frequency is a calculated value obtained from the observed bead
spacing and velocity, and thus, optimizing the frequency requires knowledge on how these two
parameters behave. In Chapter 4, we illustrate the predictability and unique pattern of these two
parameters for asymmetric morphologies. The ability for fiber diameters to be switched out and a
consistent bead velocity as the convective regime is an excellent benefit of asymmetric morphologies
that can be exploited for many applications. The bead spacing at the onset of the convective
regime for asymmetric morphologies has two beneficial behaviors that make it better suited for
many applications. Bead spacing at the onset of the convective regime consistently falls within a
small range of values, making it known what range of bead spacing can be expected no matter
what fluid of experimental property is altered. Even more beneficial is the tendency for the bead
spacing to arrive and stabilize at a bead spacing value well before the onset of the convective regime.
This quality allows for the lowest possible bead spacing value to be achievable using a flow rate
significantly lower than that which would cause a transition into the convective regime.
Along with reliable and predictable bead spacing and bead velocity values, we show that
all asymmetric morphologies result in frequencies that follow the same linear trend. This allows
applications to alter the bead frequency by adjusting the flow rate actively. This linear trend
also allows one to use the easily measurable frequency to approximate the initial volumetric flow
rate being applied onto the fiber. We lastly provided evidence that all asymmetric data for the
frequency ratio and velocity ratio collapse with the Weber and Capillary number, respectively. The
43
data collapse observed reveals that through only altering the Weber and Capillary numbers, we can
produce optimal morphology properties for desalination devices and other applications.
44
Chapter 7
Concluding Remarks
An experimental investigation into the bead morphologies that develop in the flow of thin
liquid films down a fiber was conducted. We reveal and document a new asymmetric instability,
which breaks the fluid interface symmetry about the fiber. We document this instability by con-
structing an experimental apparatus that allowed the fiber diameter Df , nozzle diameter Dn, and
flow rate Q to be changed and perform 1,675 individual experiments. Image processing protocols
were utilized to quantify the bead morphology properties, while qualitative observation was used to
document interface symmetry and regime.
In Chapter 3, we illustrate and compare the morphology regimes that emerge in asymmetric
and symmetric morphologies. We show that the three primary regimes previously observed for
symmetric morphologies, the isolated, Plateau-Rayleigh, and convective regimes, also are seen in
asymmetric morphologies. We find that a transition in regime as fiber diameter and flow rate are
increased follows the same trend for the asymmetric and symmetric morphologies while the nozzle
diameter exhibits a more complex relationship with the regime observed for each symmetry. We
provide a phase diagram in Chapter 4 that illustrates the asymmetric instabilities critical dependence
upon the fluid surface tension and the fiber diameter. Images of the fluid interface as the fluid
transitions from an asymmetric to symmetric morphology are provided to illustrate how the bead
shape is altered during the transition. We also discover a thin transitioning region in which the fluid
is in a transitional motion between the asymmetric and symmetric configuration along the length of
the fiber.
In Chapter 5, we highlight the critical differences in bead morphology properties observed
45
between the asymmetric and symmetric morphology. We show that for changes in the nozzle and
fiber diameter, each asymmetric morphology has a similar bead spacing trend, while each symmetric
morphology displays its own unique trend. We show that for flows down fibers with diameters
between 0.2mm and 0.5mm, the asymmetric morphologies have nearly constant bead velocities of
26.26 mm/s at the onset of the convective regime while bead velocities for symmetric flows vary
greatly. We show that asymmetric morphologies display a consistent bead spacing of 17.363 mm
at the onset of transition to the convective regime regardless of the fluid viscosity. We show that
the flow rate at which a morphology transition to convective behaves linearly in regards to the
fluid viscosity for asymmetric morphologies but is described by a polynomial trend for symmetric
morphologies. We have shown that for all fluid flows resulting in an asymmetric morphology, a
single linear trend is observed for the bead frequency as the flow rate is increased. It is also seen
that for asymmetric morphologies, all experiments with the same nozzle velocity transition into the
convective regime at the same ratio of Vb/Vn and this ratio is approximately unity at transition for
all experiments with Dn = 0.16mm. Lastly, we non-dimensionally analyze the experimental bead
morphology data in and show a collapse of the frequency and velocity ratio data with the Weber and
Capillary numbers, respectively. The data collapse falls along a power-law trend with a power-law
exponent of approximately 2/5 and -4/5 for the frequency and velocity ratio, respectively.
Focusing on the application of these asymmetric morphologies in heat and mass transfer
devices, we provide an in-depth discussion regarding the results observed in this investigation and
offer insight into the underlying physics at play. The application and physics which describe this
new asymmetric instability remain highly untouched by the scientific community, and thus many
fruitful areas of future work involve this asymmetric instability. Novel desalination units have been
developed and investigated using symmetric thin-film flow down a fiber [50], but the application of
an asymmetric morphology has not been investigated. Previous theories on the roll-up process have
given light to when the transition between asymmetric and symmetric drop configurations occur for
a static bead on a fiber [39] [7]. However, these fall short in describing the transition of symmetry
for beads on a fiber subject to a thin-film base flow; thus, a theory that accurately captures the
symmetry-breaking transition of these beads is needed.
46
Appendices
47
Appendix A Bead Spacing Trends
We contrast the asymmetric and symmetric morphologies by evaluating the behavior of
the bead spacing as the fiber diameter and nozzle diameter is altered. Figure 1 plots the bead
spacing against the flow rate as it depends upon the nozzle diameter for the asymmetric (left) and
symmetric (right) morphology. The symmetric morphology shows an increasing bead spacing as
the nozzle diameter is increased whereas the analogous asymmetric plot shows nearly identical bead
spacing trends for the Dn = 0.84 mm and Dn = 1.6 mm with a slight shift in bead spacing trend
is seen for the largest nozzle diameter. For increasing flow rate, the bead spacing decreases as the
convective regime is approach. We also note the large discrepancy between the bead spacing values
at the onset of transition for the symmetric morphology while bead spacing values at transition are
within 3 mL/hr for asymmetric morphologies.
Flowrate [mL/hr]50 100 150 200 250 300 350
BeadSpacing[m
m]
20
25
30
35
40
45
50
55
60 Dn = 0:84 mm
Dn = 1:6 mm
Dn = 2:3 mm
Flowrate [mL/hr]100 150 200 250 300
BeadSpacing[m
m]
0
10
20
30
40
50
60
70
80
90
Dn = 0:41 mm
Dn = 0:61 mm
Dn = 0:84 mm
Dn = 1:2 mm
Dn = 1:6 mm
Dn = 2:3 mm
Dn = 2:9 mm
Figure 1: Bead spacing Sb against flow rate Q, as it depends upon nozzle diameter Dn, for theasymmetric (left) and symmetric (right) morphology.
Figure 2 plots the bead spacing against the flow rate, as it depends upon fiber diameter.
48
The symmetric plot shows that as the fiber diameter is decreased, the bead spacing is increased,
and the regime transition into the convective regime is significantly delayed. The asymmetric plot
shows similar conclusions to the asymmetric morphologies illustrated in the left plot of Figure 1.
Asymmetric morphologies tend to maintain similar bead spacing values as the fiber diameter is
altered. We also note that it is again evident that the rate of change of the bead spacing as
the convective regime is significantly different between the two pattern symmetries. While the
symmetric morphologies display a near-constant rate of change of the bead spacing as they move
into the convective regime, the asymmetric morphologies experience a decreasing rate of change that
appears to level off around a similar bead spacing value for all fiber diameters. These results match
the general trends previously illustrated in Figure 1, and we observe these characteristics of bead
spacing as the convective regime hold for all asymmetric and symmetric morphologies examined.
Flowrate [mL/hr]50 100 150
BeadSpacing[m
m]
10
20
30
40
50
60
70
80
Df = 0:2921 mm
Df = 0:414 mm
Df = 0:508 mm
Flowrate [mL/hr]0 50 100 150 200 250
BeadSpacing[m
m]
0
10
20
30
40
50
60
70
80
Df = 0:101 mm
Df = 0:2032 mm
Df = 0:2921 mm
Df = 0:414 mm
Df = 0:508 mm
Figure 2: Bead spacing Sb against flow rate Q, as it depends upon fiber diameter Df , for theasymmetric (left) and symmetric (right) morphology.
49
Appendix B Bead Frequency Trends
We examine how the frequency varies between the morphology symmetries as it depends
upon the fiber and nozzle diameters. In Figure 3, we plot the frequency against the flow rate as
it depends upon fiber diameter. The asymmetric plot shows all three frequency trends following
the same trajectory. whereas each symmetric morphology produces its unique trajectory based on
the fiber diameter tested.Increasing the fiber diameter leads to increased frequency for fixed flow
rate. Interestingly, for symmetric morphologies, the last few data points in each trend exhibit an
increased rate of change of the frequency. This is not observed for the asymmetric trends.
Flowrate [mL/hr]20 40 60 80 100 120 140 160 180
Fre
quen
cy[1
=s]
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Df = 0:2921 mm
Df = 0:414 mm
Df = 0:508 mm
Flowrate [mL/hr]0 50 100 150 200 250
Fre
quen
cy[1
=s]
0
1
2
3
4
5
6
7
Df = 0:101 mmDf = 0:2032 mmDf = 0:2921 mmDf = 0:414 mmDf = 0:508 mm
Figure 3: Frequency f against flow rate Q, as it depends upon fiber diameter Df , for the asymmetric(left) and symmetric (right) morphology.
Next, we evaluate how flow rate affects the frequency trends. In Figure 4 we plot the
frequency against the flow rate as it depends upon the nozzle diameter for the asymmetric and
symmetric morphology. The asymmetric frequency trends again show similar trajectories and appear
to fall along a linear trend line, albeit less strictly than we observed in the asymmetric plot in Figure
3. The symmetric morphologies do not display the same similarity.
50
Flowrate [mL/hr]50 100 150 200 250 300 350
Fre
quen
cy[1
=s]
1
1.5
2
2.5
3
3.5
4
4.5
Dn = 0:84 mm
Dn = 1:6 mm
Dn = 2:3 mm
Flowrate [mL/hr]100 150 200 250 300
Fre
quen
cy[1
=s]
2
3
4
5
6
7
8
9
10
Dn = 0:41 mm
Dn = 0:61 mm
Dn = 0:84 mm
Dn = 1:2 mm
Dn = 1:6 mm
Dn = 2:3 mm
Dn = 2:9 mm
Figure 4: Frequency f against flow rate Q, as it depends upon nozzle diameter Dn, for the asym-metric (left) and symmetric (right) morphology.
51
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