Full text of DOI:10.1007/BF02881679

B. D. Shaw and M. J. Harrison: Influences of support fibers on shapes of heptane/hexadecane droplets in reduced gravity
B. D. Shaw and M. J. Harrison
Influences of support fibers on shapes of
heptane/hexadecane droplets in reduced
gravity
extend completely through a droplet. In this case, the fiber pas-
ses through the droplet surface in two locations. Because gravi-
ty levels are very small in microgravity environments, it is pos-
sible to use very thin fibers as droplet supports. For example,
fibers as small as 10 - 15 μm have been used in drop tower
experiments with droplets initially about 1 mm in diameter [1,
2]. For larger droplets up to several millimeters in diameter, 80
to 150 μm fibers have been used. Such is the case in the space-
based droplet combustion experiments during the STS-73 and
STS-94 Spacelab missions [3-5].
This paper describes analytical and experimental results rela-
ted to the effects of support fibers on shapes of heptane/hexa-
decane mixture droplets (both burning and evaporating) in
reduced gravity. The experimental results were obtained from
large droplets (a few mm) investigated during the MSL-1 Flight
of Spacelab. Theoretical (asymptotic) analyses are developed to
predict droplet shapes. These analyses, which predict droplet
shapes very well, illustrate important aspects of droplet shapes
in a transparent fashion. The asymptotic theory shows that for
small droplet-fiber contact angles, two spatial zones exist
where droplet shapes behave differently. Away from a fiber, a
droplet is essentially spherical. As the fiber is approached,
however, deviations from spherical symmetry are significant.
Previously developed analytical theory to predict macroscopic
droplet shapes also compares well with experimental results. In
addition, the experiments indicate that thin liquid films can
form on support fibers. In the present experiments, these films
apparently lead to transient formation of small droplets/bubbles
on the support fibers at locations near the surface of a droplet.
The intent of the present paper is to present information on
certain aspects of the effects of support fibers on shapes of large
bi-component droplets in reduced gravity. In particular, we con-
sider the following: (1) influences of fibers on macroscopic sha-
pes of droplets (including development of asymptotic theory to
predict droplet shapes and presentation of experimental results
obtained from the STS-94 Spacelab mission); and (2) formation
of thin films on the support fibers, which can lead to generation
of satellite droplets/bubbles at locations along the fiber (micro-
scopic effects). We consider the situation where a fiber passes
through a droplet in an axisymmetric fashion. A schematic of
the droplet geometry considered is shown in Fig. 1 (the thin
films and satellite droplets/bubbles shown in Fig. 1 will be
discussed later in the paper).
1 Introduction
Support fibers are commonly used in droplet combustion expe-
riments, both in normal gravity and reduced gravity. The role of
a support fiber is to stabilize a droplet in order to prevent exces-
sive movement during an experiment. In normal-gravity experi-
ments, droplets are often suspended on the ends of fibers that
are sometimes beaded. In reduced-gravity experiments, droplets
are sometimes suspended on the ends of fibers, or on fibers that
Mail address: B. D. Shaw, MAE Department, University of California,
Davis, CA 95616, USA
E-mail: bdshaw@ucdavis.edu, Phone: +01-530-752-4130
Paper submitted: August 05, 2001, Submission of final revised version:
May 13, 2002, Paper accepted: May 15, 2002
Fig. 1. Schematic of a droplet supported on a fiber.
30
© Z-Tec Publishing, Bremen
Microgravity sci. technol. XIII/4 (2002)

B. D. Shaw and M. J. Harrison: Influences of support fibers on shapes of heptane/hexadecane droplets in reduced gravity
It is noted that previous researchers have studied analytically
and computationally how droplet shapes are affected by support
(a)
fibers (and in the absence of gravity) [e.g., 6-15]. One paper,
which included computational results as well as experiments,
considered burning droplets [12]. However, these previous eff-
orts did not yield simplified expressions to clearly illustrate the
effects of support fibers on droplet shapes. This was the goal of
the asymptotic analysis developed in the present paper. In addi-
tion, the previous paper that studied burning droplets [12] did
not report on the potential formation of thin films and satellite
droplets/bubbles on the fiber, and different fuels were employ-
ed in contrast to the present paper.
2 Experimental methods
The experiments were performed during a flight of the Space
Shuttle Columbia (STS-94, the MSL-1 Flight of Spacelab).
Twenty-two heptane/hexadecane droplets were burned in the
(b)
experiments. Eleven droplets had initial hexadecane mass frac-
tions (Y) of 0.058 while the remainder had Y = 0.201. The expe-
riments utilized the Glovebox Facility located in the Spacelab
module of the Shuttle. The Glovebox Facility is a general-pur-
pose facility that provides power and video capabilities to
small-scale experiments. The crew conducted several small-
scale experiments inside this facility during the STS-94
(USML) mission. One of these experiments was the Fiber
Supported Droplet Combustion-2 (FSDC-2) experiment.
In the FSDC-2 apparatus, modified airtight commercial syrin-
ge cartridges stored the fuels. The crew inserted a fuel cartridge
into the experiment module to start an experiment. They then
turned a plunger screw, forcing fuel through two opposed need-
les to a deployment site on a fiber. Support fibers were com-
mercially obtained (Textron) with a carbon monofilament as the
(c)
core and silicon carbide as cladding. Each fiber had small beads
at the deployment sites to reduce droplet drift along the fiber.
The beads were located between and perpendicular to the need-
les. After formation of a droplet on a fiber, the needles were
slowly retracted (manually) to minimize the contact area bet-
ween the liquid and the needles. The needles then retracted
rapidly to deploy the stretched droplet on the fiber. Ignition was
accomplished via a hot-wire igniter which retracted to the bot-
tom of the test chamber after the crew visually detected ignition.
The igniter was a loop of aluminum wire about 0.25 mm in dia-
meter and 35 mm in length, and was heated with a current of
about 3 amps. The crew waited until droplet motions from
deployment were reduced before activating the igniter. Droplets
burned in environments composed of Spacelab cabin air. The air
pressure was about 0.1 MPa and the ambient oxygen mole frac-
tion was about 0.21. Representative images showing fiber-sup-
ported droplets are shown in Fig. 2.
Fig. 2. Video images of heptane/hexadecane droplets (also visible are the fiber
The FSDC-2 image data were obtained from two video views
and a small bead on the fiber that was used to minimize droplet drift along the
of the combustion processes. One view was a backlit view of
the droplet and the other was an orthogonal view of the flame.
The backlight was produced by 5 LEDs (620 to 670 nm) that
fiber): (a) Y = 0.058, A = 40 μm (prior to ignition - the igniter wire is also
shown); (b) the same droplet as in (a), but after ignition; and (c) Y = 0.201, A
= 75 μm (after ignition).
Microgravity sci. technol. XIII/4 (2002)
31

B. D. Shaw and M. J. Harrison: Influences of support fibers on shapes of heptane/hexadecane droplets in reduced gravity
reflected from a mirrored lens in the bottom of the experiment image of a droplet (which has been ignited) with the larger fiber
module. The lens for the video camera was on the Glovebox diameter (150 μm).
facility microscope. An optical filter preceding the microscope
Experimental data on droplet outlines are shown as the solid
transmitted only the LED wavelength range, blocking almost all points in Figs. 3 to 6, where the droplet shape data in Figs 3 to
of the flame radiation. The second video camera recorded the 5 correspond to the droplet images in Figs 2a, 2b and 2c, respec-
droplet, the fiber, and the flame. This camera also had a filter in tively. The measurements showed that droplets generally were
front of the lens to block light from the LED backlight. Except elongated along the fiber relative to the dimension normal to the
for the addition of radiometers, the FSDC-2 apparatus was simi- fiber. Figures 3 to 6 also show plots of a circle (the dashed lines)
lar to the apparatus used in an earlier series of Fiber Supported as well as theoretical results derived in the next section (the
Droplet Combustion (FSDC) experiments [3].
solid lines). Figures 3 to 6 show that the droplets are basically
spherical away from the fiber and that deviations from spherici-
ty grow larger as the fiber is approached. The deviation of the
shape of a fiber-supported droplet from a sphere is considered
3 Data analysis methods
Images from the droplet experiments were transferred from in the next section.
videotape to computer using a commercial frame grabbing
board. Droplet outlines were evaluated for burning and non-bur- 5 Theory for macroscopic aspects of droplet shapes
ning droplets. Image analysis was performed using the NASA
image analysis program Tracker [16]. Droplet outlines were It is recognized that droplets studied in microgravity experi-
measured from the images. At the droplet border, the transition ments often have internal flows present and that the surface ten-
of pixel intensity from the dark droplet to the light background sion can vary along the surface of a droplet even under reduced-
occurred over two to six pixels, depending on image quality. gravity conditions that promote gas-phase spherical symmetry
Each data point defining the droplet border was taken to be in [18]. It is assumed here that droplet shapes are axisymmetric
the middle of this range. In order to determine the range, a about the fiber and are negligibly affected by droplet internal
histogram of a set of pixels defined by a straight-line region of flows, surface-tension gradients and gravity. The validity of
interest was used. First, the line was placed perpendicular to the these assumptions can be estimated by comparing surface ten-
image border, and from the resulting histogram the intensity sion stresses, which promote symmetry about the fiber, to
range and mean pixel intensity were found. Second, the pixel effects that promote asymmetry, for example viscous stresses
corresponding to the average intensity was determined. from flows both within and outside of droplets. There are seve-
Because Tracker displays the coordinates of the endpoints of the ral dimensionless parameters that are relevant to these estima-
line on the histogram, it was straightforward to move the line tes, for example the Bond number, the Weber number, and
until an endpoint matched the pixel of average intensity. The capillary numbers based on characteristic conditions in the
coordinates for the droplet edges were found in this manner.
liquid and gas phases. Estimates of the values of relevant
Estimates indicate that the uncertainties in droplet size mea- dimensionless parameters are given in Appendix A, where it is
surements ranged from 0.025 mm to 0.075 mm. This uncer- shown that it is reasonable to assume that droplet shapes are
tainty range is especially significant in the area of the fiber, dominated by surface tension with the droplets being axisym-
where the uncertainty is of the order of the fiber diameter.
metric about the fiber. It is noted that the present experiments
deal with liquid mixtures, and the potential exists for surface
tension variations to occur as a result of variations in the com-
position of the liquid surface. This issue is also addressed in
4 Experimental results on droplet shapes
In the experiments, droplet movements were observed along the Appendix A, where it is shown that these effects should be
axis of the fiber as well as normal to the fiber. These movements small.
were sometimes significant enough to induce observable chan-
When surface tension is dominant and droplets are axisym-
ges in droplet shapes. For comparison with the droplet-shape metric about the fiber, the outline of a droplet in the upper-half-
theory employed in this paper, which assumes that droplet plane is described by the following integral [7]
movements are negligible, we report here only representative
experimental results for droplets with small movements that did
not appear to influence quasisteady droplet shapes.
B
Figure 2 shows representative images of droplets supported
by fibers (the numerals in all of the droplet images refer to
times). The droplet in Fig. 2a has not yet been ignited (the igni-
ter wire is visible below the droplet). Figure 2b shows the same
droplet at a later time, shortly after ignition (note that the igni-
y2 +φ AB
x
=
dy
(1)
1
2
∫
y
[
(
B2
−
y2 )(y2 −φ 2 A2
)
]
ter wire has been retracted out of the field of view). The fiber where x, y, A and B are defined in Fig. 1, φ = [B cos(θ )- A]/[B
diameter for the droplet in Figs. 2a and 2b was 80 μm, and the - A cos(θ )], and θ is the contact angle between the droplet and
initial hexadecane mass fraction was 0.058. Figure 2c shows an the fiber. In general, the contact angle is not known apriori and
32
Microgravity sci. technol. XIII/4 (2002)

B. D. Shaw and M. J. Harrison: Influences of support fibers on shapes of heptane/hexadecane droplets in reduced gravity
Fig. 6. Plot showing experimental data (the solid points), the asymptotic theo-
ry (Eq. (20), shown as a solid line), the theory of Ref. [7] (Eq. (6)) which is
shown as a solid line), and the equation of a circle (Eq. (21), which is shown as
the dashed line). The asymptotic and exact theories agree very closely (basi-
cally to within the width of the lines) and cannot be distinguished on the graph.
The initial hexadecane mass fraction in the droplet (Y) was 0.201 and the fiber
radius (A) was 75 μm.
Fig. 3. Plot showing experimental data (the solid points), the asymptotic theo-
ry (Eq. (20), shown as a solid line), the theory of Ref. [7] (Eq. (6)) which is
shown as a solid line), and the equation of a circle (Eq. (21), which is shown as
the dashed line). The asymptotic and exact theories agree very closely (basi-
cally to within the width of the lines) and cannot be distinguished on the graph.
The initial hexadecane mass fraction in the droplet (Y) was 0.058 and the fiber
radius (A) was 40 μm.
Fig. 4. Plot showing experimental data (the solid points), the asymptotic theo-
ry (Eq. (20), shown as a solid line), the theory of Ref. [7] (Eq. (6)) which is
shown as a solid line), and the equation of a circle (Eq. (21), which is shown as
the dashed line). The asymptotic and exact theories agree very closely (basi-
cally to within the width of the lines) and cannot be distinguished on the graph.
The initial hexadecane mass fraction in the droplet (Y) was 0.058 and the fiber
radius (A) was 40 μm.
Fig 7. Plot showing Eq. (22) (shown as a solid line), the theory of Ref. [7] (Eq.
(5), shown as a solid line), and a circle with unity radius (shown as the dashed
line). The parameter values ε = 0.04 and φ = 2ε were used. The asymptotic and
exact theories agree very closely (basically to within the width of the lines) and
cannot be distinguished on the graph. The curves are plotted only for w > ε.
needs to be determined such that the condition x(A) = L is satis-
fied. In applying Eq. (1) to gasifying droplets, it is recognized
that the contact angle that applies to this equation is the appa-
rent macroscopic dynamic contact angle [12].
Because of symmetry, we will analyze Eq. (1) in the first qua-
drant such that we will drop the " " sign in the following ana-
lysis. For the analysis, we will define the dimensionless varia-
bles ε = A/B, z = x/B, and w = y/B. Use of these variables trans-
forms Eq. (1) to the form
Fig. 5. Plot showing experimental data (the solid points), the asymptotic theo-
ry (Eq. (20), shown as a solid line), the theory of Ref. [7] (Eq. (6)) which is
shown as a solid line), and the equation of a circle (Eq. (21), which is shown as
the dashed line). The asymptotic and exact theories agree very closely (basi-
cally to within the width of the lines) and cannot be distinguished on the graph.
The initial hexadecane mass fraction in the droplet (Y) was 0.201 and the fiber
radius (A) was 75 μm.
(2)
z
= E + F
where
Microgravity sci. technol. XIII/4 (2002)
33

B. D. Shaw and M. J. Harrison: Influences of support fibers on shapes of heptane/hexadecane droplets in reduced gravity
1
w2
integrands have different asymptotic behaviors as w is varied
E
=
dw
(3)
over the range of integration (ε ≤ w ≤ 1). This is exploited to
provide simplification of the problem in various regions. For
brevity, in the following text cos(θ) is sometimes denoted by the
letter "c".
1
2
∫
w
[
(
1
−
−
w2 )(w2 −ε 2φ2
)
]
We will first look at behaviors for w >> ε, which corresponds
to regions away from the fiber. Expanding the integrands of
Eqs. (3) and (4) for ε << 1 yields the following integrals.
and
1
εφ
F
=
dw
.
(4)
1
2
∫
⎡
⎢
⎢
⎤
⎥
1
w
2c2
w2
[
(
1
w2 )(w2 −ε 2φ2
)
]
w
ε
E
F
=
=
+
+
!
dw
⎥
1
2
1
2
∫
(
1
−
w2
)
2
w
(
1
ε 2
−
)
⎢
⎥
w
⎣
⎦
It is noted that φ = [cos(θ ) - ε]/[1 - ε cos(θ)]. Equation (2) can
be expressed in terms of elliptic integrals [7]. For example, Eq.
(2) can be written as
⎡
⎢
⎢
⎤
1
ε
c
(
c2
−
1
)
⎥
+
+
!
dw
⎥
1
2
1
2
∫
w
(
1
−
w2
)
w
(
1
−
w2
)
⎢
⎣
⎥
w
⎦
(5)
z
=
i
{(1
+εφ
)
[
J
(l
)−
J
(s
,
l
)]+εφ
[K
(s
,
l
)−
K
(l)]}
Integration yields the following equations.
where J(l) and J(s, l) are complete and incomplete elliptic inte-
grals of the first kind, respectively, K(l) and K(s, l) are comple-
te and incomplete elliptic integrals of the second kind, respecti-
vely, s = sin (w), and l = 1/(ε φ ). The factor i appearing in Eq.
(5) is necessary in order to cause z to be a real number. In terms
of dimensional variables Eq. (5) becomes as follows.
1
2
w2
)
1
+
O
(
ε 2
)
)
(7)
(8)
E
F
=
(
1
−
2 2
-1
1
2
⎡
⎤
⎥
⎥
+
(
1
−
w2
⎢
⎢
= ε
c
ln
+
O
( )
ε 2
w
⎢
⎣
⎥
⎦
⎡
⎤
⎥
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
A
B
B2
A
y
B
B2
⎛
⎜
⎞
⎠
⎛
⎜
⎞
⎟
−
1
x
=
+
iL
1
+
φ
J
−
J
sin
,
(6)
⎟ ⎢
Examination of Eqs. (3) and (4) shows that Eqs. (7) and (8)
break down when w = O(ε). We will develop solutions to Eqs.
(3) and (4) for w = O(ε) by splitting the region of integration
into two parts, as shown in Eqs. (9) and (10). Motivation for
splitting the region of integration is described elsewhere [19].
2φ 2
A2φ2
⎝
⎝
⎠
⎢
⎣
⎥
⎦
y
B
B2
B2
⎡
⎢
⎤
⎛
⎞
⎛
⎜
⎜
⎝
⎞
⎛
⎜
⎞
−
1
⎜
⎜
⎝
⎟
⎟
⎠
⎟
⎟
⎠
iLεφ
K
sin
,
− K
⎟
⎠
⎥
A2φ 2
A
2φ 2
⎝
⎢
⎣
⎥
⎦
It is difficult to physically interpret Eq. (5) or Eq. (6) in a trans-
parent fashion. To remedy this, Eq. (2) will be analyzed asymp-
totically for the case where the fiber is thin relative to the dro-
plet diameter, i.e., ε = A/B << 1, which corresponds to typical
droplet combustion experiments. Two cases will be considered.
The first case is where cos(θ) = O(1), leading to φ = O(1), and
the second case is where cos(θ) << 1, leading to φ << 1.
δ
1
w2
w2
E
F
=
=
dw
dw
+
+
dw
1
2
1
2
∫
∫
(9)
w
δ
[
(
1
−
−
w2 )(w2 −ε 2φ2
εφ
)
]
[
(
1
−
−
w2 )(w2 −ε 2φ 2
εφ
)
]
δ
1
dw
(10)
1
2
1
2
∫
∫
w
δ
[
(
1
w2 )(w2 −ε 2φ2
)
]
[
(
1
w2 )(w2 −ε 2φ 2
)
]
6 Asymptotic analysis of eq. (2) for cos(θ) = O(1)
The integrals in Eqs. (3) and (4) will be evaluated asymptoti-
cally assuming that ε << 1 and cos(θ) = O(1), leading to φ =
O(1). This situation applies to typical droplet combustion expe-
riments, at least until droplets become very small (on the order
of the fiber diameter). In the analysis, it is recognized that the
When making this split, the variable δ is selected such that ε <<
δ << 1.
In Eqs. (9) and (10), we will let the integrals between w and δ
be denoted as E and F , respectively, and the integrals between
1
1
34
Microgravity sci. technol. XIII/4 (2002)

B. D. Shaw and M. J. Harrison: Influences of support fibers on shapes of heptane/hexadecane droplets in reduced gravity
1
2
δ and 1 will be denoted as E and F , respectively. From Eqs.
2
2
2
2
⎡
⎢
⎢
⎢
⎢
⎤
⎥
⎥
⎥
⎥
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
δ
ε
δ
ε
+
−
c2
(7) and (8), it is evident that
(16)
F1 = ε
c
ln
+!
1
2
β +
(
β 2 − c2
)
⎢
⎣
⎥
⎦
1
2
E2
=
(
1
−δ 2
⎡
)
1
+
!
(11)
(12)
1
2
⎤
⎥
⎥
⎢
⎢
+
(
1
−δ 2
δ
)
Adding Eqs. (11) and (12) to Eqs. (15) and (16), respectively,
and dropping the smallest terms yields the following approxi-
mations for E and F in the vicinity of the fiber.
F2 = ε
c
ln
+!
⎢
⎣
⎥
⎦
For analysis of E and F , we will define the stretched variable
(17)
1
1
E
F
=1+!
β = w/ε, yielding the following equations.
⎡
⎤
⎥
⎥
⎥
⎥
⎥
⎢
⎢
⎢
⎢
⎢
4
(18)
= ε
c
ln
+!
1
2
δ
⎛
⎞
⎟
⎜
ε
β 2
ε β +
(
β 2 − c2
)
1 = ε 2
dβ
⎜
⎝
⎟
⎠
(13)
(14)
E
⎢
⎣
⎥
⎦
1
2
∫
[(
1
−ε 2β 2 )(β 2 −φ 2 )]
β
δ
ε
Equations (17) and (18) are valid for w = O(ε), while Eqs. (7)
and (8) are valid for w = O(1). These equations may be combi-
ned to form uniformly valid solutions. When this is done and the
results are substituted into Eq. (2), the following asymptotic
solution is obtained.
1
F1 = εφ
dβ
1
2
∫
[(
1
−ε 2β 2 )(β 2 −φ 2 )]
β
When the integrands of Eqs. (13) and (14) are expanded for ε
<< 1, the following integrals result.
1
⎡
⎤
⎥
⎥
1
2
1
+
(
1
−
w2
)
2
⎢
ln 2
⎢
z
=
(
1
−
w2
)
+εc
+!
(19)
1
2
δ
ε
w2 −ε 2c2
)
⎡
⎢
⎢
⎤
⎥
⎢
⎣
⎥
⎦
w
+
(
β 2
β 2
β 2 − c2
E
1 = ε 2
−ε
c
(
1
−
c2
)
+
!
d
β
⎥
1
2
3
2
∫
2
⎢
⎣
⎥
β
(
β −
c
)
(
)
⎦
δ
ε
Equation (19) is uniformly valid over the range ε ≤ w ≤ 1.
⎡
⎢
⎢
⎤
⎥
⎛
⎜
⎞
⎟
2
c
c2
−
1
c
(
c2
−
1
)
½
Analysis of Eq. (19) shows that the term (1 − w ) is dominant
for w = O(1), which means that droplets are essentially spheri-
cal away from the fiber. As the fiber is approached, however, the
second group of terms grows logarithmically as w → ε. For w =
ε, Eq. (19) can be expressed as
F1 = ε
+ε
+
+
!
d
β
⎜
⎜
⎟
⎟
⎠
⎥
1
2
1
2
3
∫
β
(
β 2 − c2 ) (β 2 − c2 ) (β 2 − c2
)
⎢
⎣
⎥
2
⎝
⎦
Evaluating these integrals yields the following expressions.
z
=
1
−
cε
lnε +
O
(ε )
1
2
⎧
⎪
⎨
⎪
⎩
⎫
⎪
⎬
⎪
⎭
⎡
⎢
⎢
⎤
⎥
⎥
1
2
2
2
2
2
c2
2
δ
ε
δ
ε
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
δ
2ε
δ
ε
(15)
E
1 = ε 2
−
c2
+
ln
+
−
⎤
c2
where an expansion for ε << 1 and cos(θ) = O(1) has been app-
lied. This expression indicates that deviations from sphericity
(along the fiber) decrease slowly (i.e., as εlnε) as the fiber dia-
meter is decreased.
When expressed in terms of dimensional variables, Eq. (19)
becomes as follows.
⎢
⎣
⎥
⎦
1
2
1
c2
2
⎧
⎪
⎫
⎪
⎡
β
−ε 2
(
β 2 − c2
)
+
ln ⎢β +
(
β 2 − c2
)
2 ⎥ +
!
⎨
⎬
2
⎢
⎣
⎥
⎪
⎩
⎪
⎭
⎦
Microgravity sci. technol. XIII/4 (2002)
35

B. D. Shaw and M. J. Harrison: Influences of support fibers on shapes of heptane/hexadecane droplets in reduced gravity
1
addition, Figs. 3 to 6 show that droplets approach a spherical
shape away from the fiber with appreciable deviations from a
spherical shape in the vicinity of the fiber. In general, it was
found that agreement between theory and experiment was good
for relatively stationary droplets regardless of whether a droplet
had been ignited or not.
⎡
⎢
⎤
⎥
⎥
1
2
B
+
(
B2
−
y2
A2 cos2
)
2
B2
−
y2
)
+
A
cos
(
θ
)
ln 2
(20)
+!
x
=
(
1
2
⎢
⎢
⎣
⎥
⎦
y
+
(
y2
−
(
θ
)
)
We will compare Eq. (20) with data obtained from experiments
θ
on combustion of fiber-supported droplets as well as with the 8 Asymptotic analysis of eq. (2) for cos( ) << 1
results found using Eq. (6). Equation (20) will also be compa-
red with the projection of a sphere onto the x-y plane (i.e., a cir- We also consider theory related to the case where φ << 1, cor-
cle), which is described by Eq. (21).
responding to large contact angles. The experiments did not
exhibit these conditions. We consider it here, however, to show
that droplets should be nearly spherical everywhere, even near
the fiber, when φ << 1. This can be demonstrated analytically by
treating φ as a small parameter and expanding Eq. (2) for φ <<
1. When this is done, it is found that only a regular perturbation
analysis is needed, yielding the following expression, which is
uniformly valid over the range ε ≤ w ≤ 1.
1
2
x
=
(
B2
−
y2
)
(21)
7 Comparison of theory and experiment for cos(θ) = O(1)
Shown in Figs. 3 to 6 are plots of Eqs. (6), (20) and (21). The
droplet shape data shown in Figs. 3, 4, and 5 correspond to the
droplet images shown in Figs. 2a, 2b, and 2c, respectively.
Equation (21) is shown as the dashed line in Figs. 3 to 6, and
Eqs. (6) and (20) are shown as solid lines. Also plotted in these
figures are experimental data points (the solid points) determi-
ned from analysis of droplet outlines. Equation (21) is plotted
only until it intersects the experimental fiber diameter. In apply-
ing Eq. (20) as well as Eq. (6), cos(θ) was varied until the dro- a circle with unity radius (i.e., z = (1 − w ) , which is shown as
plet length along the fiber calculated using Eq. (20) matched the dashed line) are shown in Fig. 7 for the parameter values ε
experimental data on the droplet length (the same value of θ was = 0.04 and φ = 2ε. The theory of Ref. [7] (Eq. (5)) is also shown
used in Eq. (20) as well as Eq. (6)). This is because the contact as a solid line; this line lies very near the plot of Eq. (22) and
angle could not be accurately measured from the images. It is cannot be distinguished on the graph. As can be seen in Fig. 7,
also noted that the ε values were obtained from the experiments. droplet shapes are very nearly spherical when φ << 1.
1
⎡
⎢
⎢
⎤
⎥
⎥
1
2
ε 2φ2
2
1
+
(
1
−
w
w2
)
2
⎛
⎜
⎝
⎞
⎟
⎟
⎠
w2
)
+ εφ +
ln
+!
(22)
⎜
z
=
(
1
−
⎢
⎣
⎥
⎦
A plot of Eq. (22) (shown as a solid line) and the equation of
2 ½
For the data in Figs. 3 to 6, ε values were 0.035, 0.034, 0.021
and 0.023, respectively.
If we set φ = 0, it turns out that droplets are spherical eve-
rywhere away from the fiber. It should be noted, however, that
In general, the inferred contact angles were found to vary the axisymmetric droplet shape shown in Fig. 1 can become
from experiment to experiment. For the droplets shown in Figs. unstable and change into an asymmetric shape if φ is too small
3 to 6, contact angles were found to lie in the range 4.7° ≤ θ ≤ (or the contact angle is too large). Specifically, it has been
44°. Differences in contact angles likely result from uncertain- shown [8] that the axisymmetric droplet shape is stable if Eq.
ties in the measurements as well as variations in experimental (23) is satisfied (if Eq. (23) is not satisfied, droplets are expec-
conditions (e.g., variations in fiber surface conditions). Because ted to assume an axisymmetric shape about the fiber).
of the cos(θ) term that appears in the equations, the droplet
length L is relatively insensitive to θ for the range of θ investi-
gated here. However, for this range of contact angle, θ depends
B3
3
B2
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
2
sensitively on L such that small changes in L will lead to large
changes in θ. It is noted that for the variation in θ that applies to
these experiments, cos(θ) is in the range 0.997 to 0.719 such
that cos(θ) >> ε.
(23)
cos
(
θ
)
−
+
1
>
0
A3
A2
The data in Figs. 3 to 6 show that the asymptotic theory (Eq. Equation (23) can be rearranged to yield the following stability
(20)) provides an excellent approximation to the theory of Ref. condition.
[7]. As can be seen in these figures, the asymptotic and exact
theories agree very closely (basically to within the width of the
lines) and cannot be distinguished on the graph. These figures
also show that theory and experiment agree quite closely. In
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
3
ε −ε3
2
π
2
3ε
2
ε 3
16
1
θ < cos−
=
−
−
+
O
( )
ε5
(24)
36
Microgravity sci. technol. XIII/4 (2002)

B. D. Shaw and M. J. Harrison: Influences of support fibers on shapes of heptane/hexadecane droplets in reduced gravity
Using the definition of φ, Eq. (23) can also be expressed as fol- ved between a droplet and a fiber is an "apparent" or "macro-
lows.
scopic" contact angle that is at the location where the three-
phase contact line appears to reside.
The appearances of the satellite droplet/bubbles in the present
experiments may be related to thin films that may have formed
on the fibers. Based on this idea, we suggest the following
mechanism for the appearance of satellite droplets/bubbles.
Because the droplets were composed of two species (heptane
ε
φ >
(25)
2
−ε 2
Equations (24) and (25) give limits on θ and φ in order for fiber- and hexadecane), it is reasonable to assume that the films were
supported droplets to exhibit stable axisymmetric shapes, i.e., θ also composed of these same components. Since the fiber (and
cannot be too large and φ cannot be too small.
gas) temperature will increase with increasing distance from the
droplet surface (between the droplet and the flame), heptane
would be preferentially depleted from a film, with larger mass
fractions of hexadecane existing in a film as the distance from
9 Formation of satellite droplets/bubbles
In some (but not all) experiments, small satellite droplets/bub- the droplet surface increases. The heptane depletion would be
bles appeared on the fiber. Representative images are shown in strongest at the film surface exposed to the gas phase, leaving a
Fig. 8, where the satellite droplets/bubbles are located symme- heptane-rich mixture close to the fiber surface. Eventually, a
trically on both sides of the parent droplets. The parent droplets homogeneous or heterogeneous nucleation temperature could
in Figs. 8a and 8b are about 4.6 and 3.3 mm in diameter (nor- be exceeded in the heptane-rich portion of the film, which
mal to the fiber), respectively, and the satellite droplets/bubbles would lead to bubble formation and growth. This could easily
in Fig. 8 are all about 0.2 mm in diameter (normal to the fiber). lead to the satellite droplets/bubbles observed in these experi-
We use the terminology "droplets/bubbles" because it was not ments.
clear from the images whether these objects were composed
only of liquid or whether they included appreciable amounts of
gas. The opacity of the satellite objects suggests that they were
mostly liquid. However, these objects sometimes disappeared
very rapidly (within about 0.03 s or less), suggestive of the bre-
akage of a bubble.
The characteristics of the film (and whether it exists at all) can
(a)
The satellite droplets/bubbles usually appeared in pairs loca-
ted symmetrically about the parent droplet. They would appear
after ignition of the parent droplet, generally growing for seve-
ral seconds and then rapidly disappearing, with both satellite
droplets/bubbles usually disappearing at essentially the same
time. This cycle could repeat several times during the lifetime
of a droplet. If the parent droplet moved along the fiber, the
satellite objects generally moved with it, remaining symmetri-
cally located about the parent droplet. The shapes of the parent
droplets were not noticeably altered by the appearance or disap-
pearance of the satellite objects. This is reasonable since the
parent droplet shapes are determined by surface tension and the
droplet/fiber contact angle, and the satellite droplets/bubbles
appeared to not be in direct contact with the parent droplets. In
addition, volumes of the parent droplets were much larger than
volumes of the satellite droplets/bubbles. As such, the appea-
rance or disappearance of satellite droplets/bubbles would chan-
ge parent droplet dimensions negligibly.
(b)
We have not found any references to the appearance of thin
films or satellite droplets/bubbles in fiber-supported droplet
combustion experiments. For other types of situations that
involve fiber-supported droplets, however, it is known that thin
films (e.g., in the 100 to 300 angstrom range [17]) can someti-
mes be formed on fiber sections outside of the droplets [14, 15,
17]. When a film is present, the droplet blends smoothly into the
film (at small length scales) and the contact angle that is obser-
Fig. 8. Video images of heptane/hexadecane droplets after ignition, showing
satellite droplets/bubbles to either side of the parent droplets: (Y = 0.058, A =
40 μm for both parent droplets).
Microgravity sci. technol. XIII/4 (2002)
37

B. D. Shaw and M. J. Harrison: Influences of support fibers on shapes of heptane/hexadecane droplets in reduced gravity
3
depend on the fiber surface as well as the interactions between
the fiber and the liquid comprising the droplet. Because the
fiber surfaces were not characterized in a detailed fashion in the
present experiments, it is difficult to predict characteristics of
any films that might be present. As a result, we have presented
analyses only for macroscopic droplet shapes and have not con-
sidered theory related to the film or the satellite droplets/bub-
bles.
droplet is f = gρ πD /6, where g is the gravitational accelera-
g
L
tion and ρ is the liquid density. The surface tension force f
L
σ
exerted on the droplet is characterized by f = 2πdσ, where σ is
σ
a characteristic surface tension. The Bond number is the ratio of
3
f to f , i.e., Bo = gρ D /(12dσ). It is estimated that Bo << 1 for
g
σ
L
the microgravity experiments under consideration here such
that gravitational effects should not cause significant asymme-
try of a droplet about a support fiber, i.e., only small deflections
of a droplet from its location of symmetry about a fiber are
required to balance gravitational forces normal to the fiber.
Momentum effects might also influence droplet shapes. For
example, internal flows are typically present within droplets
burning in microgravity, and there is also a momentum change
associated with vaporization at the gas-liquid interface. The
ratio of inertial forces to surface tension forces is the Weber
number (We). Inertial forces related to flows within droplets are
Conclusions
This research has shown that previous theory [7] and the asymp-
totic theory developed in this paper can be used to predict sha-
pes of burning or evaporating heptane/hexadecane droplets sup-
ported on thin fibers in reduced gravity. For the theory to be
accurate, however, it appears that droplets need to maintain
nearly steady positions on support fibers. This implies that sha-
pes of nearly stationary fiber-supported burning droplets are
dominated by surface tension and the contact angle between the
liquid and the fiber, which is in agreement with computational
analyses [12].
The asymptotic analyses provided a simplified expression
that can be used to calculate shapes of droplets supported on
thin fibers in the limit of having fiber diameters much smaller
than droplet diameters (measured perpendicular to the fibers).
The asymptotic analyses illustrate that for cos(θ) = O(1) and ε
<< 1, two spatial zones exist where droplet shapes behave dif-
ferently. Away from a fiber, a droplet is essentially spherical. As
the fiber is approached, however, deviations from spherical
symmetry can be significant.
It is also noted that for cos(θ) << 1 and ε << 1, theory sug-
gests that droplets should be nearly spherical everywhere, even
near the fiber. As a result, it seems worthwhile to seek droplet-
fiber combinations that will promote large contact angles (sub-
ject to Eqs. (24) and (25)). Finding such combinations may
allow gas-phase spherical symmetry to be more closely approa-
ched in microgravity droplet experiments that require support
fibers. Because ε increases as a droplet burns (i.e., as a droplet
reduces in size), Eqs. (24) and (25) should be applied to the
smallest droplet size of interest in an experiment.
This research has also shown that satellite droplets/bubbles
can be formed on support fibers during combustion of hepta-
ne/hexadecane droplets. This is an interesting topic that is not
well understood at present, and it would be of interest to per-
form further studies of this phenomenon. Such studies would
likely require experiments with well-characterized fiber surfa-
ces.
2
2
characterized by ρ U D , where U is a characteristic flow velo-
L
city within a droplet. Surface tension forces are characterized by
the product Dσ. The ratio of these forces yields the Weber num-
2
ber (We ) for liquid flows We = ρ U D/σ. By using data on U
L
L
L
values from microgravity droplet experiments [20] it is estima-
ted that We << 1 for the experiments discussed here indicating
L
that liquid momentum effects should not cause significant
asymmetry of a droplet about a support fiber.
A gas-phase Weber number (We ) may also be defined as the
g
ratio of the pressure change across the liquid-vapor interface
associated with momentum changes at the droplet surface (cha-
2
2
racterized by u ρ /ρ ) to the pressure change across the
s
L
g
liquid-vapor interface associated with surface tension (charact-
erized by σ/D). Here, u is the negative of the time-rate-of-
s
change of the droplet radius and ρ is the gas density at the
g
liquid-gas interface (it is assumed that ρ >> ρ ). The velocity
L
g
u is characterized as u = k/D, where k is the burning-rate con-
s
s
2
stant based on the droplet diameter (i.e. k = -d(D )/dt where t is
time). The ratio of these pressure variations yields
2
2
We =k ρ /(Dσρ ). Interestingly, this expression predicts that
g
L
g
We ∝ 1/D such that momentum changes at the liquid surface
g
may become important for small droplets. The droplets under
consideration here were large enough such that We was much
g
smaller than unity. As a result, vapor momentum effects should
not cause significant asymmetry of a droplet about a support
fiber.
Viscous stresses may also affect droplet shapes. The liquid-
phase capillary number Ca is the ratio of the characteristic
L
viscous stress μ U/D (where μ is the liquid viscosity) to the
L
L
characteristic surface tension stress (σ/D), i.e., Ca = μ U/σ.
L
L
We may also define a gas-phase capillary number Ca as the
g
ratio of gas-phase viscous stresses normal to the droplet surfa-
2
Appendix A
ce, which are characterized as μ u /D = μ ρ k/(ρ D ), to surfa-
g g
g L
g
ce tension stresses, which are characterized as σ/D, i.e., Ca =
g
The Bond number (Bo) is the ratio of gravitational forces to
capillary forces. To estimate Bo, consider a droplet with an ave-
rage diameter D (based on the droplet volume) that is supported
μ ρ k/(ρ Dσ) (μ and u are the gas-phase viscosity and velo-
g L
g
g
g
city on the gas side of the gas-liquid interface, respectively). It
is estimated that Ca << 1 and Ca << 1 for the droplets consi-
L
g
on a fiber of diameter d. The gravitational force f acting on the
dered here such that viscous effects should not cause significant
g
38
Microgravity sci. technol. XIII/4 (2002)

B. D. Shaw and M. J. Harrison: Influences of support fibers on shapes of heptane/hexadecane droplets in reduced gravity
asymmetry of a droplet about a support fiber.
ce of a droplet
Finally, it is noted that surface-tension gradients along the
surface of a droplet will cause local changes in radii of curvatu-
re, which will influence droplet shapes. However, if the ratio
u
Negative of the time-rate-of-change of the droplet
radius
s
U
Characteristic fluid velocity in a droplet
2
2
Δσ/σ is small relative to unity, where σ is a representative value We
of the surface tension and Δσ a representative change in surfa- We
Gas phase Weber number (= k ρ /(Dσρ ))
g
L
L g
2
Liquid phase Weber number (= ρ U D/σ)
L
ce tension along the droplet surface, then droplet shapes should
w
x
X
Dimensionless distance normal to the fiber (= y/B)
Distance along the fiber from the droplet center
Mole fraction of component 1 at the droplet surface
Representative change in the mole fraction of compo-
nent 1 along the droplet surface
be influenced only minimally from changes in σ. For purposes
of making estimates, we may write Δσ ≈ (∂σ/∂T) ΔΤ + (∂σ/∂X )
1
1
ΔX where ΔT and ΔX are representative changes in the tem- ΔX
1
1
1
perature T and mole fraction X of species 1 along the surface
1
-4
of a bi-component droplet. Using the values (∂σ/∂T) = 10
y
Y
z
Distance normal to the fiber
Initial mass fraction of hexadecane in a droplet
Dimensionless distance along the fiber (= x/B)
-5
N/(m K), (∂σ/∂X ) = 10 N/m, and σ = 0.02 N/m (these are
1
representative values for hydrocarbons) yields Δσ/σ << 1 if ΔT
2
3
<< 10 K and ΔX << 10 . The requirement for ΔT should be
1
easily met for microgravity droplet combustion experiments, Greek
and the requirement for ΔX will always be satisfied. As a result,
1
it is expected that surface tension gradients will not significant-
ly influence shapes of fiber-supported droplets of the types
investigated here.
β
δ
Δσ
Stretched variable (= w/ε)
Variable introduced in Eqs. (9) and (10)
Representative change in surface tension along the sur-
face of a droplet
Small parameter of expansion (= A/B)
Apparent contact angle
Gas viscosity
Liquid viscosity
Gas density at the gas-liquid interface
Liquid density
Characteristic surface tension
Variable defined after Eq. (1)
Nomenclature
ε
θ
μ
μ
ρ
ρ
A
Fiber radius
g
L
g
L
B
Droplet dimension normal to the fiber
3
Bo
c
Ca
Ca
d
Bond number (= gρ D /(12dσ))
L
Cosine of the apparent contact angle (= Cos (θ))
Gas phase capillary number (= μ ρ k/(ρ Dσ))
σ
φ
g
L
g L
L
g
Liquid phase capillary number (= μ U/σ)
Fiber diameter
D
E
F
Average droplet diameter
Variable defined by Eq. (3)
Variable defined by Eq. (4)
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1
2
Dietrich, D. L., Haggard, J. B., Jr.: Combustion of droplet arrays in a
3
f
f
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σ
L
g
E
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E
F
i
Variable defined after Eq. (10)
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3
4
1
1
2
2
½
(-1)
J(s, l) Incomplete elliptic integral of the first kind
J(l)
k
Complete elliptic integral of the first kind
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2
(= -d(D )/dt)
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ΔT
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Microgravity sci. technol. XIII/4 (2002)
39

B. D. Shaw and M. J. Harrison: Influences of support fibers on shapes of heptane/hexadecane droplets in reduced gravity
8
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13 McHale, G., Rowan, S. M., Newton, M. I., Kab, N. A.: Estimation of contact
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We also thank the many individuals at the Payload Operation
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Without the dedicated efforts of the Engineering Team at the
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B. Shaw would like to thank fellow FSDC-2 investigators R.
15 Bauer, C., Dietrich, S.: Shapes, contact angles, and line tensions of droplets Colantonio, F. Dryer, J. Haggard, V. Nayagam and F. Williams
on cylinders. Physical Review E, vol. 62, p. 2428 (2000).
16 Klimek, R. B., Wright, T. W., Sielken R. S.: Color image processing and Ross also provided valuable suggestions and support during the
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for their help in the Shuttle experiments. Drs. M. King and H.
MSL-1 mission.
40
Microgravity sci. technol. XIII/4 (2002)