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Mechanics of removing water from the ear canal: Rayleigh–Taylor instability

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Water stuck in the ear is a common problem during showering, swimming or other water activities. Having water trapped in the ear canal for a long time can lead to ear infections and possibly result in hearing loss. A common strategy for emptying water from the ear canal is to shake the head, where high acceleration helps remove the water. In this present study, we rationalize the underlying mechanism of water ejection/removal from the ear canal by performing experiments and developing a stability theory. From the experiments, we measure the critical acceleration to remove the trapped water inside different sizes of canals. Our theoretical model, modified from the Rayleigh–Taylor instability, can explain the critical acceleration observed in experiments, which strongly depends on the radius of the ear canal. The resulting critical acceleration tends to increase, especially in smaller ear canals, which indicates that shaking heads for water removal can be more laborious and potentially threatening to children due to their small size of the ear canal compared with adults.
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J. Fluid Mech. (2023), vol.963, A12, doi:10.1017/jfm.2023.309
Mechanics of removing water from the ear canal:
Rayleigh–Taylor instability
Seungho Kim1,2,3,AnujBaskota
1, Hosung Kang1,4and Sunghwan Jung1,
1Department of Biological and Environmental Engineering, Cornell University, Ithaca, NY 14853, USA
2School of Mechanical Engineering, Pusan National University, Busan 46241, South Korea
3Eco-friendly Smart Ship Parts Technology Innovation Center, Pusan National University, Busan 46241,
South Korea
4Department of Biomedical Engineering and Mechanics, Virginia Tech, Blacksburg, VA 24061, USA
(Received 12 August 2022; revised 24 January 2023; accepted 5 April 2023)
Water stuck in the ear is a common problem during showering, swimming or other water
activities. Having water trapped in the ear canal for a long time can lead to ear infections
and possibly result in hearing loss. A common strategy for emptying water from the ear
canal is to shake the head, where high acceleration helps remove the water. In this present
study, we rationalize the underlying mechanism of water ejection/removal from the ear
canal by performing experiments and developing a stability theory. From the experiments,
we measure the critical acceleration to remove the trapped water inside different sizes of
canals. Our theoretical model, modified from the Rayleigh–Taylor instability, can explain
the critical acceleration observed in experiments, which strongly depends on the radius of
the ear canal. The resulting critical acceleration tends to increase, especially in smaller ear
canals, which indicates that shaking heads for water removal can be more laborious and
potentially threatening to children due to their small size of the ear canal compared with
adults.
Key words: parametric instability, capillary flows, breakup/coalescence
1. Introduction
It is common for adults and children to accidentally get water stuck in their ear canal after
swimming or submerging their head underwater. The most common remedy for removing
water involves shaking the head or jumping up and down while tilting the head towards the
shoulder (Marken 2002). During this motion, head acceleration increases significantly due
to abrupt stopping and reciprocal shaking motions (Özgüven & Berme 1988; McNitt-Gray
1993; McKay et al. 2005). Even with high acceleration, we typically experience a
Email address for correspondence: sj737@cornell.edu
© The Author(s), 2023. Published by Cambridge University Press. This is an Open Access article,
distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/
licenses/by/4.0), which permits unrestricted re-use, distribution and reproduction, provided the original
article is properly cited. 963 A12-1
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S. Kim, A. Baskota, H. Kang and S. Jung
failure in removing water. In such instances, a more violent motion of shaking the head
is needed.
High-acceleration body movements can cause significantly negative impacts on the
head (Laksari et al. 2015). Accelerated brain can create a sharp pressure gradient in the
head, subsequently causing tissue damage because internal tissue and cerebrospinal fluids
are close to incompressible (Holbourn 1943). Especially, children and adolescents are
vulnerable to traumatic brain injuries during various school sports, high-risk behaviours,
vehicle accidents, and other incidents, which can cause long-term cognitive impairments
(Youngblut et al. 2000; Asemota et al. 2013). There is no single acceleration value that
causes brain injuries, however, acceleration approximately larger than 10gis dangerous
and possibly attributed to mild head trauma (Crisco et al. 2010; Sarmiento et al. 2021).
A physical representation of the problem of interest is the ear canal with water stuck
inside. When a person tilts their body to forcefully push water out, it would be better to
make the ear canal parallel to the gravitational direction. This configuration is when water
(a dense fluid) in the ear canal is located above the air (a light fluid) along the direction
of gravity, which is similar to the situation of the classical Rayleigh–Taylor instability
(Rayleigh 1882;Taylor1950). Thus, the dynamics of removing water could be related to
this classical instability.
In this study, we investigate the mechanism of ejecting lodged water through both
theoretical and experimental approaches. We first design a transparent replica of the ear
canal and a glass tube for a parametric study. High-speed camera image sequences are
shown to visualize the ejection process and quantify the critical acceleration to destabilize
and remove the lodged water. This critical acceleration is predicted by modifying the
Rayleigh–Taylor instability. Lastly, we find that shaking heads to remove water can be
more laborious for children or babies due to their small ear canal, which explains the
danger of shaking the head.
2. Ear anatomy and physiology of water lodging
The external ear canal of humans begins in the auricle and ends at the tympanic membrane
(TM), forming a cylinder-shaped structure as shown in the inset of figure 1(a). The isthmus
starts at the section of the ear canal where the cartilage in the ear exists and ends near the
bony section of the ear, and is the narrowest section of the ear canal (Feher 2012). The
average radius of the ear canal is shown to change from 1.6mm for infants to 3 mm for
adults as shown in figure 1(a)(Fels2008).
The ear canal is covered with cerumen, a hydrophobic waxy layer (Guest et al. 2004;
Feig et al. 2013) that presumably has high contact angle hysteresis. Wax helps to capture
water in the ear canal by pinning the contact line of a drop rather than allowing it to flow
through the skin surface of the ear canal. While water may lodge in the ear canal, more
laborious actions are required to dislodge water, especially between the isthmus and TM.
This is because the narrowest radius of the isthmus can hold water tightly as surface tension
is dominant over gravity. The area between the isthmus and TM, or the bony section of the
ear canal, does not include hair, unlike the auricle (Kumar et al. 2013). Thus, we focus on
how water can be dislodged from the smooth and narrow section of the ear canal.
3. Experiments
3.1. Sample preparation
Two different surrogates for the human ear canal were used. The first surrogate was a
polymer replica through a moulding process. The human ear was obtained from CT scans
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Mechanics of removing water from the ear canal
0 5 10 15
1.0
1.5
2.0
2.5
3.0
3.5
Age (years)
Radius of ear canal (mm)
Auricle
Temporal
bone
Cartilage Tempanic
membrane
0
15
1 cm
Ear canal
Isthmus
Slider
Substrate
Shaft
Spring
z
00.5 1.0 1.5
(×10–2)(×10–2)
(×10–2)
2.0
–1.2
–0.8
–0.4
0
Time (s)
z (m)
012
–3.0
–1.5
0
1.5
3.0
Time (s)
dz/dt (m s–1)
(a)(b)(c)
Figure 1. (a) Radius of human ear canal versusagefrom0to17(Fels2008). The dashed line represents a
third-order regression curve. The inset shows a schematic of the ear anatomy. Water is lodged in the middle
of the ear canal, which begins from an auricle and ends at a tympanic membrane. The tympanic membrane
physically divides the area between the ear canal and the eustachian tube. (b) Motion of the ear model during the
first impact at different dropping heights, H=100 mm (circles) and H=200 mm (squares). (c) Corresponding
velocity and solid and dashed lines of 175 and 240ms2, respectively.
of a human skull (by courtesy of Prof. Frank Gaillard, www.Radiopaedia.org, rID: 2630
(https://doi.org/10.53347/rID-2630), as shown in figure 6 in Appendix A). The CT scans
were used to create a computer three-dimensional (3-D) model of the human head and ear
canal using 3-D Slicer software ver. 4.11. Next, a positive mould of the human ear canal
was created using a 3-D printer (Formlabs form 3L). To create the negative mould of the
ear canal, a two-step process was used with polydimethylsiloxane (PDMS, SYLGARD
184 silicone). First, PDMS was poured to form an initial layer in a rectangular box for the
artificial ear canal. After the first layer of PDMS was cured for 2 h at 60 C, the 3-Dprinted
ear canal was placed on the first layer of PDMS. More uncured PDMS was poured onto
the ear canal. This was cured again in an oven heated to 60 C. After the cooling process,
the PDMS mould was cut vertically along the edge to remove the 3-D mould. By shrinking
the size of the human ear canal model, various sizes of the artificial human ear were made
(figure 1a).
Second, one-side closed glass tubes were used to further simplify the human ear canal,
where the inner diameter of the glass tubes varied from 2.4 to 5.5mm. Glass tubes
were coated with trichlorosilane (Sigma Aldrich, Model 448931), a hydrophobic coating.
This was done by placing the glass tubes and an open bottle of trichlorosilane in a
vacuum chamber for twenty-four hours. The trichlorosilane evaporated because of the
lower pressure in the chamber to evenly coat the tubes. Water was pipetted into the tubes
at various positions. This allowed the water volume, position of the water, acceleration
range and contact angle to be measured more accurately since the tube size was uniform
throughout.
3.2. Water ejection experiments
Amass–spring system was built and used to generate a high deceleration on an ear canal
sample (see the inset of figure 1b). We attached a transparent substrate on a heavy slider
as the closed side was facing upward, as shown in the inset of figure 1(b). After a certain
amount of water was placed inside the artificial ear canals using a syringe, the artificial
ear canal was dropped at different heights, H, along a vertical shaft. Then, the artificial
ear canal impacted a spring (Uxcell Die spring with spring constant K14 000 N m1)
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S. Kim, A. Baskota, H. Kang and S. Jung
hinged on the ground (see the inset of figure 1b). Here, special care was taken to make
the lower water–air interface symmetric. The asymmetry of the lower water–air interface
can be caused during the syringe withdrawal from an artificial ear canal after injection.
The impact velocity, V, was measured and found to be close to 2gH with gbeing
the gravitational acceleration. After impact, the artificial ear canal follows a simple
harmonic motion, and thus the acceleration of the artificial ear canal is approximately
constant (figure 1c). The time period of impact, Ti, should be independent of the impact
velocity as Ti=πM/Kwith Mbeing the mass of the substrate and shaft. The Tiwas
experimentally measured to be 18 ms after the initial collision with the spring. Then
the acceleration is calculated as a=V/(Ti/2)=22gH/Ti, which was confirmed by
experiments. Experimentally, the acceleration values ranged from 30 to 360 m2s1.
After an LED light was placed behind our experimental set-up, the dynamics of water
ejection from the ear canal was captured by a high-speed camera (Photron Nova S9) at
a frame rate of 5000 s1with a resolution of 1024 ×1024 pixels. This experiment was
conducted with different sizes of the ear canal.
4. Acceleration of shaking head for water removal
The most common remedy to remove water is to shake the head or jump up and
down repeatedly with the head tilted to one side (Marken 2002) (see supplementary
movie 1 available at https://doi.org/10.1017/jfm.2023.309). The typical acceleration value
of head-shaking motions ranges from 3 to 22 m s2(Funk et al. 2011). Thus, we performed
experiments including this range of acceleration values, where we varied the length and
position of liquid inside artificial ear canals.
Figure 2(a,b) shows the ejection of water from an artificial human ear depending on
the magnitude of acceleration, |a|. Water stably stays in the artificial human ear with low
acceleration (figure 2a), whereas the lower part of the water column becomes unstable and
flows down with higher acceleration (figure 2b). This implies that a critical acceleration
exists as the stable air–water interface breaks in the ear canal. We also characterize the
water-dislodging dynamics using a simple geometry, i.e. a one-side closed hydrophobic
glass tube. Figure 2(ce) illustrates the sequential phenomena of water ejection. With low
acceleration (figure 2c), a water bulge at the bottom air–water interface is formed during an
initial deceleration (just after the collision with the spring). The bottom air–liquid interface
does not break, but instead, it is restored back to its original position. As the acceleration
increases (figure 2d), a water bulge cannot recoil back to its original position and starts
to flow out without any cavitation bubble (Pan et al. 2017). In that case, the acceleration
is denoted as the critical acceleration, acr . At even higher acceleration (figure 2e), a water
bulge is further stretched, and thus, a much larger amount of water can be ejected.
5. Expansion of an upper air cavity to resist water ejection
Figure 2(ce) shows how the upper interface of the water column (i.e. the lower interface
of the air cavity) gets lower during the deceleration of the artificial ear. Since the air cavity
above the water column is sealed by the closed top, the downwards displacement of the
upper interface indicates that the air cavity volume increases during impact. Therefore,
the air cavity decreases its internal pressure so that the lowered pressure pulls up the water
column and consequently resists the ejection of water. To estimate the magnitude of air
expansion, we balance an acceleration force, ρ(a+g)Ll(πR2), with the resisting force,
(p0pmin)πR2, where ρis the water density, Llis the length of the water column and R
is the inner radius of the tube. We assume that the effect of dynamic pressure is negligible
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Mechanics of removing water from the ear canal
t = 0 11 ms 15 ms 27 ms 42 ms t = 0 11 ms 16 ms 19 ms 28 ms
3 mm
Ear canal mold
3 mm
t = 0 t = 07 ms
g
6 ms 8 ms 12 ms 17 ms9 ms 15 ms 23 ms t = 0 6 ms 10 ms 13 ms 16 ms
3 mm 3 mm 3 mm
Lu
Ll
La
R
R
(c)
(a)(b)
(d)(e)
Figure 2. Temporal evolution of a water drop inside a polydimethylsiloxane (PDMS) ear replica when
(a)|a|=77 m s2and when (b)|a|=242 m s2. Here, the 30 % scaled-down-model of adult’s ear canal
is used and water is placed 15 and 25mm above the exit of ear canal. (ce) Image sequences of the water
movement inside a glass tube with an inner diameter of 2.4mm, where water is placed 5 and 10mm above the
open end of the tube. The acceleration is 108, 129 and 242ms2for panels (ce), respectively. The dashed lines
in the second image of panel (e) indicate a difference in the upper interface of the water column. This shows
that the volume of air above the upper air–water interface is expanding during the collision. Corresponding
movies 2–6 are included in the supplementary material.
since the ratio of acceleration to dynamic forces, ρ(a+g)Ll/[ρ(Lu/ti)2], is of the order
of 103. Here, p0represents the initial pressure and pmin represents the pressure when the
ear canal reaches its lowest position, p(t=Ti/2). By assuming the adiabatic expansion
of the air column and LuLawith Lubeing the vertical change of the position of
the upper interface and Labeing the initial length of the upper air column, the pressure
change, p0pmin , can be approximated as Γp0Lu/Laas the leading order term with Γ
being the adiabatic constant (Γ=1.4 for air). Therefore, the ratio of air cavity length to
liquid length becomes
Lu
La1
Γ
ρgLl
p01+a
g.(5.1)
According to the above equation, the scattered data of the vertical change in the air cavity
length in figure 3(c) are collapsed into a single line of (5.1) as shown in figure 3(d). The
system mentioned above is the small-scale counterpart of the dynamics of emptying a
bottle, where a similar coupling between the air compressibility and the liquid outflow is
also observed (Clanet & Searby 2004).
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S. Kim, A. Baskota, H. Kang and S. Jung
0 10203040
50
100
150
200
250
|a| (m s–2)
Lu/LaLu (mm)
|a| (m s–2)
|a| (m s–2)
Γ–1(ρgLl/p0) (1+|a|/g)
0 102030
0
100
200
300
400
Ll (mm)
La(mm)
0 100 200 300 400
0
1
2
3
4
1.2
1.2
1.2
1.2
1.7
1.7
1.7
1.7
2.3
2.3
2.3
2.3
2.3
2.8
2.8
2.8
2.8
Symbol R (mm) L (mm)
2.5
5.0
15.0
25.0
2.5
5.0
15.0
30.0
5.0
10.0
15.0
25.0
40.0
2.5
5.0
15.0
25.0
0.05 0.100
0.1
0.2
0.3
0.4
3
1
(a)
(b)
(c)
(d)
Figure 3. (a) Acceleration, |a|,versus the vertical length of water, Ll, with the tube having an inner diameter
of 2.4 mm where the vertical length of the air cavity, La, is fixed to be 15 mm. (b) Acceleration, |a|,versus the
vertical length of the air cavity, La, with the tube having an inner diameter of 4.6 mm where the length of the
water, Ll, is maintained to be 10 mm. Here, closed and open symbols represent the ejection and the non-ejection
of the water drop during the impact. (c) Changes in the vertical position of the upper air–liquid interface, Lu,
versus acceleration, |a|, where different tube diameters and different lengths of the water column are tested.
(d)Lu/Laplotted based on (5.1).
6. Rayleigh–Taylor instability
A fluid interface is stable when a fluid with a low density stays above another fluid with a
high density. However, our system of interest is reversed (a fluid with high density, i.e. a
water drop, placed above a fluid with low density, i.e. air) so that the Rayleigh–Taylor
instability (Rayleigh 1882;Taylor1950) would explain the water ejection dynamics.
Figure 2 shows that the water ejection begins by forming a bulge at the lower water–air
interface. Here, the classical Rayleigh–Taylor instability is modified to predict the critical
acceleration with an air cavity behind. The ejection of the water column is driven by an
inertial force but resisted by a surface force. The inertial force of the water column can
be scaled by ρR3a, where Rdenotes the radius of the bulge (see the inset of figure 2d).
Then, the resisting force can be estimated as γR. Mass conservation shows that the bulge
volume balances with the volume change in the air cavity, thereby estimating the radius of
bulge curvature as RL1/3
uR2/3. By balancing the driving and resisting forces together
with the above Rrelation, we simply get the following relation of critical acceleration for
the water removal:
acr γ
ρ3/5Γp0
ρ2/51
LlLaR22/5
.(6.1)
It is worth noting that we obtain a similar relation by calculating the dispersion relation of
a cylindrical interfacial wave (see Appendix B).
Figure 4(a) shows the critical acceleration in terms of the liquid volume. Based on (6.1),
we re-scale the x-axis and find that all critical acceleration values are collapsed into a
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Mechanics of removing water from the ear canal
Ejection
Non-ejection
2460
50
100
150
200
250
VL(µl)
|acr| (m s–2)
100 200 3000
100
200
300
Symbol Liquid
Water
Water
Water
Water
Water
IPA
IPA
Water
Water
Water 4.4
Effective radius
3.0
1.9
1.2
1.7
2.3
2.5
2.8
1.2
1.7
R (mm)
(×10–2)
(γ/ρ)3/5(Γp0/ρ)2/5(LlLaR2)–2/5(m s–2)
(a)(b)
Figure 4. (a) Experimentally measured critical acceleration, |acr |,versus the volume of the liquid inside an ear
canal, VL, where different liquids and tube radii are tested as listed in the right legend. Circles and rectangles
represent experimental results using glass tubes and PDMS replicas of a human ear canal, respectively. (b)|acr|
plotted based on (6.1).
single line, as shown in figure 4(b). Moreover, (6.1) shows that a water column hardly
gets removed, especially when a small amount of water is trapped just beneath the ear
drum and/or when the radius of the ear canal is small. It implies that the forceful motion
of head shaking can be dangerous for infants and children since the critical acceleration
value becomes large due to thesmallsizeoftheir ear canal. Once the acceleration exceeds
the critical value, water begins to flow out of the ear canal and the ejection volume appears
to be proportional to LuD2(see Appendix C).
7. Conclusions
In this study, we manufactured artificial ear canals similar to those of younger children and
adults and performed ear shaking experiments. Our experiments revealed the dynamics of
an artificial ear canal partially filled with water. The critical acceleration for the lodged
water to flow out was measured in terms of canal size, acceleration, and location and
volume of water. We developed a theory of modified Rayleigh–Taylor instability by
considering a resisting pressure arising from the expansion of the upper air cavity and
analytically predicted the critical acceleration. We showed that the critical acceleration
strongly depends on the size of the ear canal, the volume and location of the trapped
water. Our experiments were in good agreement with the theoretical predictions.
More importantly, our study implies that the critical acceleration, especially in young
children and infants, could grow to 14gdue to their narrow ear canals. Figure 5
illustrates the typical value of critical linear acceleration for brain injuries (concussion,
subconcussive impact and symptomatic impact) for juveniles and adults. Our estimated
critical acceleration, 14g, is not small compared to other critical accelerations for
brain injuries (|acr|≈14gfor concussion during rugby (King et al. 2016), |acr|≈16 g
for heading in soccer with subconcussive impacts (Naunheim et al. 2003), |acr|≈
18gfor subconcussive collision between football players (Rowson et al. 2009)and
|acr|≈4gfor low-speed vehicle collision with brain-pain symptoms McConnell et al.
1993). Subconcussive impacts, where someone is exposed to several repeated impacts
without immediate injury symptoms, can also lead to altered neurophysical impairments
(Breedlove et al. 2012; Talavage et al. 2014). Therefore, shaking the head aggressively
to remove water can have long-term effects on the brain similar to the brain injuries of
subconcussive impacts. In conclusion, head-jerking motions must be avoided especially
for infants and young children.
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S. Kim, A. Baskota, H. Kang and S. Jung
Infant
Adult
Sub-concussive impact
Concussion impact
Symptomatic impact
Hockey player collision
Football player collision
Elbow to head hit
Elbow to wrist hit
Soccer players collision
Player fell
Player hit the ground
Unintentional ball to the head
Soccer heading
Rugby player collision
Low-speed vehicle collision
100
101
102
Linear acceleration (g)
Figure 5. Comparison of linear acceleration of the head depending on various situations (McConnell et al.
1993;Varney&Varney1995; Naunheim et al. 2003; Pellman et al. 2003; Withnall et al. 2005; Guskiewicz
et al. 2007;Rowsonet al. 2009; Mihalik et al. 2010;Criscoet al. 2012; Daniel, Rowson & Duma 2012;Hanlon
&Bir2012; Daniel, Rowson & Duma 2014; Wilcox et al. 2015;Kinget al. 2016). There are three different types
of impact: concussion impact in red circle, sub-concussive impact in yellow square and symptomatic impact in
orange diamond. A grey band shows the range of critical acceleration of removing water out of the ear canal
from infants to adults.
As alternatives to shaking the head, well-known home remedies include blowing air into
the nose (Yale 2005; Williamson et al. 2015) and inserting a few drops of vinegar into the
ear (Nuttall & Cole 2004; Djalilian 2013). Here, the first air-blowing method can increase
the internal pressure of the middle ear by pressurizing air in the eustachian tube from the
nose. Then, the elasticity of the TM allows the membrane to buckle towards the external
ear (Gaihede, Liao & Gregersen 2007), which could increase the pressure of the air column
and finally lower the critical acceleration. The second remedy of using vinegar can reduce
the surface tension coefficient of the trapped water since the vinegar is miscible with water
and has a low surface tension. Similarly, using a hair dryer into the ear canal also lowers
surface tension by increasing the water temperature. Thus, other home remedies help to
remove water trapped inside human ear canals, which is well explained by our theory.
Supplementary movies. Supplementary movies are available at https://doi.org/10.1017/jfm.2023.309.
Acknowledgements. The authors thank Ms. K. Averett for her initial contribution to this study.
Funding. This work was partially supported by the National Science Foundation (grant no. CMMI-2042740
and no. CBET-2002714) and by the National Research Foundation of Korea (grant no. 2022R1F1A1076192 and
no. 2020R1A5A8018822) via PNU EPIC.
Declaration of interests. The authors report no conflict of interest.
Author ORCIDs.
Seungho Kim https://orcid.org/0000-0001-9164-444X;
Sunghwan Jung https://orcid.org/0000-0002-1420-7921.
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Mechanics of removing water from the ear canal
(a)(b)
Front
Rear
Ear canal
Right-hand side
Left-hand side
Figure 6. (a)Image showing a 3-D structure of the ear canal with a human skull. (b) CT-scanned right ear
canal, where the image is seen from the top of the head.
Appendix A. Three-dimensional and CT-scanned structures of ear canal
Figure 6(a)shows a 3-D structure of a human head with the ear canal (grey), and
figure 6(b) a zoomed top view of CT-scan image showing the internal structure of the
right ear canal. The ear canal gets narrower as it approaches the tympanic membrane.
The image source is described in §3, where the skull image is achieved via overlapping
multiple CT-scanned images.
Appendix B. Dispersion relation of a modified Rayleigh–Taylor instability
We consider the interfacial deformation on the bottom liquid–air surface, which has a
solution of the Bessel function in the radial (r) direction and sinusoidal functions in the
azimuthal (θ) direction. The axial velocity profile of the bottom liquid–air interface, uz,
can be approximated as
uzuzekz+st+imθJm(kr), (B1)
where zand trepresent a vertical coordinate and time, respectively. Here, ˜uz,k,sand m
correspond to the velocity magnitude, wavenumber, growth rate and an integer index (m=
0,1,2,3,...), respectively. For the Bessel function of the first kind, Jm(kr), we considered
the simplest wave mode (m=0) which is consistent with our experimental observations
of the interfacial deformation during the ejection of a trapped water drop (see figure 2d).
Then, the displaced volume by the expansion of the air column, πR2Lu, should be
equal to the volume of liquid bulge at the lower liquid–air interface,  uzdt(rdr)dθ.
By plugging (5.1)and(B1) into the volume conservation relation, we get
πR2ρ
Γp0
LlLa|a|=˜uz
1
sekz+st
t
02πR
0
J0(kr)rdr.(B2)
It can be then expressed as
sk2=Γp0
ρ
c
R2LlLa|a|,(B3)
where c∝˜uzR
0J0(kr)kr d(kr).
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S. Kim, A. Baskota, H. Kang and S. Jung
To obtain the explicit expression of sin terms of k, we first considered the axial
displacement of the lower liquid–air interface, ζ,as
ζ=˜
ζestJ0(kr). (B4)
Then, the linearized Euler equation together with Young–Laplace equation yields the
following relation:
ρ∂tuz=−z(γ∂
rrζ+ρaζ). (B5)
Here, we use a small slope approximation (dζ/dr1). Next, the kinematic boundary
condition at the lower liquid–air interface is further considered as
uz=tζ. (B6)
Using (B1), (B4), (B5)and (B6), we can obtain the dispersion relation as
s2=ak γ
ρk3.(B7)
This is known as the Rayleigh–Taylor instability. The maximum occurs when s2/∂k=0.
Then, we get the most unstable wavenumber, k=ρ|a|/(3γ), which finally results in
the following relation of the most unstable modal number, s:
s=2
3|a|k.(B8)
By plugging (B8)into(B3), we can finally get the relation of critical acceleration for
ejecting a water drop inside the ear canal as
a11/10
cr ∼˜u2/5
zγ
ρ1/2Γp0
ρ2/51
LlLaR22/5
.(B9)
One can see a similarity between (B9)and(6.1) despite different derivations. Equation
(B9) is from the dispersion relation using a perturbation method, whereas (6.1) is based
on the scaling argument. The small differences between the two models originate from
integrating the the first kind of Bessel function in (B2).
Appendix C. Ejection volume of dislodging water
Figure 7(a) shows the ejection volume of the dislodging water, Vejection,versus
acceleration, where the experimental value of Vejection is obtained via analysing images
before and after the first impact. Since the ejection of water occurs at the lower water–air
interface, we consider Vejection as
Vejection =Ti
0
uzdA.(C1)
By considering the continuity, uzscales as ˙
Lu, which leads to the following relation:
Vejection LuR2. Using (5.1), we can obtain the following relation as
Vejection ρ
Γp0LlLaR2|a|.(C2)
Based on (C2), we rescale the x-axis in figure 7(b) and find that the ejection
volume follows a single line. Deviations from the diagonal line may originate for the
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Mechanics of removing water from the ear canal
100 200 300 400 500
0
50
100
150
Vejection (mm3)
Vejection (m3)
0.2 0.4 0.6 0.8 1.0
(×10–8)
0
0.4
0.8
1.2
1.6 (×10–7)1.2
1.2
1.2
1.2
1.7
1.7
1.7
1.7
2.3
2.3
2.3
2.3
2.3
2.8
2.8
2.8
2.8
Symbol R (mm) L (mm)
2.5
5.0
15.0
25.0
2.5
5.0
15.0
30.0
5.0
10.0
15.0
25.0
40.0
2.5
5.0
15.0
25.0
|a| (m s–2)[ρ/(Γp0)]LlLaR2|a| (m3)
(a)(b)
Figure 7. (a) Experimentally measured ejection volume of dislodging water, Vejection,versus acceleration, |a|,
where different liquid lengths and tube radii are tested. (b)Vejection versus (C2).
following reasons. First, the scaling model above explains the initial instability, but does
not consider long-term behaviours. As shown in the third image in figure 2(e), a thin jet is
formed and drags more water out of the tube by inertia. Second, water droplets could stick
to the inner wall of the ear canal right after the first impact, as shown in the last image of
figure 2(d). Additionally, the water droplets adjacent to the lower water–air interface could
be absorbed into the bulk due to surface tension. All of these contribute to the uncertainties
of Vejection.
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