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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

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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

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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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