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First published online October 18, 2006
Journal of Experimental Biology 209, 4398-4408 (2006)
Published by The Company of Biologists 2006
doi: 10.1242/jeb.02506
Air-flow sensitive hairs: boundary layers in oscillatory flows around arthropod appendages
1 Institut de Recherche sur la Biologie de l'Insecte-UMR CNRS 6035,
Faculté des Sciences et Techniques, Université François
Rabelais, Parc de Grandmont Avenue Monge, 37200 Tours, France
2 MESA+ Research Institute, Transducers Science and Technology group Faculty
of Electrical Engineering, University of Twente, PO Box 217, 7500 AE Enschede,
The Netherlands
* Author for correspondence (e-mail: jerome.casas{at}univ-tours.fr)
Accepted 22 August 2006
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Key words: viscous boundary layer, hair biomechanics, cercal system, cricket, flow sensing, sensor design
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), crustacean and moth olfaction
(Koehl et al., 2001;
Stacey, 2002) and fluid
sensing in arthropods in air and water
(Humphrey et al., 1993;
Barth et al., 1993;
Devarakonda et al., 1996;
Humphrey et al., 2003a;
Humphrey et al., 2003b).
Air-flow sensing using filiform hairs partially immersed in the boundary layer
around the body has been extensively studied in arthropods, especially in
spiders and crickets (Shimozawa and Kanou,
1984; Humphrey and
Devarakonda, 1993; Barth et
al., 1993). Filiform hairs are among the most sensitive biological
sensors in the animal kingdom (Shimozawa
et al., 2003). These outstanding sensitive structures are used to
detect approaching animals, in particular the oscillatory signals produced by
the wing beats of prey and predators
(Tautz and Markl, 1979;
Barth et al., 1993;
Gnatzy and Heusslein, 1986).
According to the body of work performed to date, a variation of hair length is
assumed to allow spiders and crickets to fractionate both the intensity and
frequency range of an air flow signal.
The biomechanics of hair movement in an oscillating fluid has stimulated
extensive research during the last few decades, and several models have been
developed (Fletcher, 1978;
Shimozawa and Kanou, 1984;
Humphrey and Devarakonda,
1993). In all these models, a hair is defined as an inverted
pendulum with a rigid shaft supported by a spring at its base. This mechanical
model can be described by four parameters
(Shimozawa et al., 2003;
Humphrey et al., 2003): the moment of inertia that represents the mass
distribution along the hair shaft; the spring stiffness which provides the
restoring torque towards the resting position; the torsional resistance within
the hair base; and the coupling resistance between the hair shaft and the air
(see Appendix).
An important assumption of these models concerns the nature of the
oscillating flow in the boundary layer over the cerci, as it greatly impacts
the movement of hairs of different lengths. The boundary layer has been
theoretically predicted and often experimentally confirmed on large structures
for both longitudinal and transverse oscillatory flows
(Raney et al., 1954;
Bertelsen et al., 1973;
Williamson, 1985;
Obasaju et al., 1988;
Justesen, 1991;
Tatsuno and Bearman, 1990).
These studies were, however, conducted for Reynolds and Strouhal numbers very
different from those of biological relevance, with the exception of the
studies by Holtsmark et al. (Holtsmark et
al., 1954) and Wang (Wang,
1968). While the longitudinal flow over arthropod appendages has
been thoroughly characterised by Barth et al.
(Barth et al., 1993), the
transverse flow has been much less studied. Barth et al. characterized the
boundary layer in longitudinal and transverse flows using laser Doppler
anemometry (LDA) over a spider leg (Barth
et al., 1993). However, only a few measurements were carried out
and they were limited by the 300 µm resolution of the LDA system, hence
missing out the region nearest to the substrate. Moreover, the influence of
the source orientation on the boundary layer was not investigated.
The aim of the present work is to characterize, in a systematic way in terms of phase and frequency, the boundary layer over small antenna-like appendages in crickets in an oscillatory flow for both longitudinal and transverse flows. Situated at the rear end of crickets, these appendages of conical shape, called cerci, are equipped with hundreds of filiform hairs of varying length. We use stroboscopic micro-particle image velocimetry (µPIV) to carry out measurements on artificial cerci placed in oscillatory flows of varying frequencies and orientations with respect to the source. Thus, our work deals with the flow around a cricket cercus, and not with the flow around a single hair. Our results are compared to existing models for longitudinal and transverse flows. We discuss the implications of our findings in terms of hair length and angular hair location around the cercus to extract as much information as possible. The perspectives are in terms of cricket perception of attacking predators and in terms of biomimetic Micro-Electric-Mechanical Systems (MEMS) flow sensors.
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) [see
Sueur et al. (Sueur et al., 2005) for finer distinctions in the near-field
region]. The far-field velocity of air particles is described by the equation:
 | (1) |
where U0 is the flow oscillation amplitude, and 
is the angular frequency (rad s-1), with =2 
f,
f being the frequency of flow oscillations (s-1). A signal
impacting the cylinder from any angle can be decomposed into longitudinal
(Vl), radial (Vr) and circumferential
(V 
) components
(Fig. 1A).
|
 | (2) |
where ß=(/2 
air)
is the
border effect factor (m-1), air=
µair/µair is the kinematic viscosity of the fluid
(m2 s-1), µair is the dynamic viscosity of
the fluid (Ns m-2), 
air is the density of the fluid
(kg m-3) and D is the diameter of the cylinder (m)
(Humphrey and Devarakonda,
1993).
Due to the viscosity effect, there is a phase displacement,
, with the
distance from the cylinder:
 | (3) |
Second, we consider the flow oscillating perpendicular to a cylinder,
theoretically described by Holtsmark et al.
(Holtsmark et al., 1954). We
compared our results to the first degree of this analytical approximation as
we are interested in a simple semi-analytical solution of the Navier Stokes
equation for modelling hair movement. This first-order solution corresponds to
the oscillatory part of the solution. The circumferential component of the
flow velocity, V, triggers hair defection. The
cylinder represents a cercus, not a hair on a cercus. This circumferential
component is:
 | (4) |
with
 | (5) |
X, Y, Z, C are complex numbers with Xi,
Yi, Zi, Ci and
Xr, Yr, Zr,
Cr being their imaginary and real parts, respectively.
Hn is the Hankel function of the first kind
(Abramowitz and Stegun, 1965),
y is the distance to the substrate (m), D is the diameter of
the cylinder (m), 
=[i(/)], and
is the angle between the cylinder and the flow (rad). The radial
component of the flow, which is parallel to the hair and has no influence on
its motion, is expressed as follows:
 | (6) |
Eqns 4 and 6 show the existence of a phase lag 
between
Vr and V. This phase lag is
given by:
 | (7) |
being given by:
 | (8) |
Therefore, the fluid far from the cylinder oscillates along a straight line
whereas the fluid near the cylinder oscillates in an elliptic fashion
(Holtsmark et al., 1954). We
solved numerically the phase advance of V with
distance from the cylinder.
Experimental set-up
We used cylinders (D=1 mm, L=40 mm) made of steel, of
identical base diameter but of a longer length than in real animals. This
enabled us to avoid boundary effect at both extremities. These cylinders were
stiff. They were held by a micromanipulator placed in a way to avoid
vibrations produced by the loudspeakers.
Flow measurements were performed using two-dimensional PIV. It is now well
established that PIV and LDA techniques are suitable for acoustical
measurement (Campbell et al.,
2000). Recently, these techniques have been used to characterize
the laminar acoustic viscous boundary layer and the acoustic streaming in tube
and wave guide (Castrejón-Pita et
al., 2006). Artificial cylinders were placed in a cylindrical
glass container (L=200 mm, D=100 mm), with two loudspeakers
(40 W, 5
; SP 45/4; Monacor, Brême, Germany) at both ends,
connected to a sinusoidal signal generator (2 MHz; TG 230; Thurlby-Thandar,
Huntingdon, Cambs, UK).
We checked for signal integrity at all frequencies in the centre of the
container using an LDA system (FlowLite 1D 65X90; Dantec Dynamics A/S,
Skovlunde, Denmark). At the central point,
U(t)=U0sin(t),
with amplitude U0=35 mm s-1. The loudspeaker
delivered good signals from 25 Hz upwards. Measurements were conducted at
25°C corresponding to an air kinematic viscosity,
air=1.56x10-5 m2 s-1.
The air inside the sealed glass box was seeded with 0.2 µm oil particles
(Di-Ethyl-Hexyl-Sebacat, 0.5 L; TPAS, Dresden, Germany) using an aerosol
generator (ATM 230; ACIL, Chatou, France). The laser of the PIV illuminated
the flow produced by the wave through the glass (NewWave Research Solo PIV 2,
532 nm, 30 mJ, Nd:YAG, dual pulsed; Dantec Dynamics A/S). The laser sheet
(width=17 mm, thickness at focus point=50 µm) was operated at low power (3
mJ at 532 nm) to minimize glare. A target area was then imaged onto the CCD
array of a digital camera (Photron FastCam X1280 PCI 4K) using a
stereomicroscope (LEICA M13, X10) that produced a 2x2 mm window around
the substrate. In order to measure high-frequency and low-amplitude
oscillatory movements, we set the CCD to capture a light pulse at 30 Hz in
separate image frames every 500 µs, with a laser impulsion length of
4±1 ns (Fig. 1B). To
reduce undesirable reflections, the cylinders were covered with fluorescent
paint, which re-emits the light of the laser at another wavelength. We used a
narrow band filter working at the laser wavelength 
=532 nm placed
under the camera. Measurements were conducted for velocities
U0=35 mm s-1, within the range of airborne
vibration amplitude generated 10 cm in front of a flying predatory wasp
(Tautz and Markl, 1979). Flow
frequencies ranged from 30 to 180 Hz. For a cylinder diameter of 1 mm, the
corresponding peak Reynolds number (Re) is 1.6 and the Strouhal
numbers (St) range from 2.7 to 16.2.
Stroboscopic measurements
We used the stroboscopic principle to sample high-frequency sinusoidal
signals (ranging from 30 Hz to 180 Hz) with a PIV system limited to 30 Hz
(Schram and Riethmuller, 2001). As explained in
Fig. 1B, this consists of
sampling a signal of period fflow with a frequency
(facq) slightly lower than a sub-multiple of the signal
frequency. This technique gives us the following pseudo sampling time interval
(tstrob):
 | (9) |
where Tacq is the inverse of the PIV sampling
frequency, and Tflow is the inverse of the signal
frequency. The number of sampling points, N, covering a full period
of the signal is given by:
 | (10) |
A numerical example is given in the legend of
Fig. 1B. In a second example,
let us assume we sample a 119 Hz signal at 30 Hz. The pseudo time interval is
then tstrob=0.28 ms, corresponding to a phase of
tstrob=0.2 rad=0.06 rad, giving N=33
sampling points per full period of signal. We repeated this measurement five
times and proceeded to a phase average on the basis of 50 and 150 pair images
in the first and second example, respectively. The estimation of the signal
phase corresponding to each sample point is obtained through inference,
not measurements. The known far-field velocity U
and local velocity U0 are used with the relationship
=arcsin(U /U0).
Data acquisition
Flow measurements for both longitudinal and transverse flows were conducted
for six frequencies (30, 60, 90, 120, 150 and 180 Hz). The two-dimensional
(2-D) velocity vector fields were derived from sub-sections of the target area
of the particle-seeded flow by measuring the movement of particles between two
light pulses. Images were divided into small subsections (width, 70 µm;
resolution, 32x32 pixels; covering rate, 50%) and cross-correlated with
each other using a flow map software (Flow Manager 4.4; Dantec Dynamics A/S).
The correlation produced a signal peak, identifying the common particle
displacement. An accurate measurement of the displacement (and thus of the
velocity) was achieved with sub-pixel interpolation.
We averaged five vector fields for each phase of the signal. For longitudinal flows, velocity profiles were obtained by averaging point measurements over 2 mm along the cylinder. For transverse flows, the velocity profile was extracted at five different angles (90, 60, 45, 30 and 15°) but not averaged along the length of the cylinder.
With subsection windows of 32x32 pixels, it is possible to obtain
valid measurements down to 0.1 pixels
(Mayinger and Feldman, 2001).
In our set-up, this corresponds to 0.3 mm s-1, equivalent to 1% of
U0.
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t=(/2)] and the minimum velocity
fields [measured at t=-(/2)]. The boundary layer thickness
around the cylinder is significantly reduced at high frequencies. From these
measurements, we extracted the velocity profiles for five different
circumferential angles (90, 60, 45, 30 and 15°)
(Fig. 3), an angle of
0° being directly upwind. As predicted by theory, the flow velocity
profiles vary as a function of the sine of this angle. At 30°, the
curvature effects of the cylinder produce a noticeable peak in flow velocity,
increasing at higher angles and levelling off at 90°. Maximal velocities
also increase with flow frequency, from 1.4U0 at 30 Hz to
1.6U0 at 180 Hz.
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50% higher than theoretical values. We do not
know if it is due to a PIV uncertainty or a failing of the theory. The high
light intensity in the focalisation part of the laser light sheet, which is
then very near to the substrate, may heat the surface, leading to an increase
of velocity (Brownian motion). The thickness of the boundary layer decreases
with increasing flow frequency, as predicted by the theory. The 90° and the 60° transverse flows produce higher velocities than the longitudinal one throughout the entire profile. This pattern is also observed for the 45° transverse flow but only within the first 1000 µm above the cylinder for a 30 Hz flow and within the first 700 µm for a 180 Hz flow. For all frequencies, the 30° and the 15° profiles are smaller than the longitudinal one, and the 45° profile is greater than the longitudinal one along the first 300 µm.
Phasing
Fig. 4 represents the
temporal evolution of the velocity field around a cylinder in a transverse
flow. The circulating zones typical of oscillatory flows are clearly visible.
Particles trapped in this region rotate in antiphase on both sides of the
cylinder (red arrows in Fig.
4). This is due to the interplay between their circumferential
components, which are moving in phase, and their radial components, which are
out of phase by /2. We extracted the velocity profiles of the
circumferential component of flow around the cylinder from the data
represented in Fig. 4 and
plotted them in Fig. 5. Hair
movement, which we did not measure, is induced by this component. The phase is
constant for each value of and is a function of y.
Fig. 5 also shows the evolution
of the velocity profile over a cylinder with time for a longitudinal flow. We
obtained a good agreement between theory and measurement except at very small
distances (<90 µm).
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/4 close to the surface to 0 rad at
the largest distances. There is therefore a phase advancement in the boundary
layer. While the observed changes in amplitude between transverse and
longitudinal flows are important, there is almost no phase lag between
them.
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) and later used by
Humphrey and Devarakonda [equation A.2.1
(Humphrey et al., 1993)] as
the conditions for its application [RexSt>>1 and
(Re/St)<<1] did not apply for the full range of values we
were interested in, particularly for the low frequencies
[RexSt=3 and (Re/St)=0.4 for
f=30 Hz]. By contrast, the Holtsmark et al. model
(Holtsmark et al., 1954) gives
meaningful predictions over the full range of parameters. An exploration of
the predictions of these different models indicates that the model of Wang,
and its refinement by Humphrey et al., is valid from 180 Hz down to 90 Hz, but
deteriorates at lower frequencies. As mentioned before, we compared the
transverse measurement to the first approximation solution. The streaming
motion to the hair movement was confirmed by visual examination of the film
taken with the PIV camera, but we neglected it as its amplitude was estimated
to be 2.5% of the amplitude of the far-field oscillatory flow
(Holtsmark et al., 1954;
Raney et al., 1954).
The match between model predictions for both longitudinal and transverse
flows and experimental data is very good, except for the smallest measured
distance from the cylinder. Reasons for this lack of fit may originate either
from a breakdown of the many approximations made in the models or from
experimental errors, as their weight is large at those small velocities. The
slightest misalignment of the thin laser sheet with the tiny cylinder can
indeed produce such a mismatch. Our results confirm the pioneer results
obtained by Barth et al. (Barth et al.,
1993) regarding the relative velocities in longitudinal and
transverse flows around spider legs. In particular, these authors observed
that velocities would be stronger in transverse flow than in longitudinal
flow. We observed higher velocities for transverse flow as long as the
angle is between 60° and 120°. As was previously predicted
(Humphrey et al., 1993), the
profiles are also quite different, transverse flows producing local velocities
up to 1.6 times stronger than the far-field values. By contrast, the maximal
amplification of the far-field value is only 1.1 for longitudinal flows.
Implications for air-flow sensing in animals and MEMS
The strong spatial heterogeneity of flow velocities around appendages in
transverse flow is a rich source of information for flow-sensing animals, in
particular for those using flow-sensing hairs. A single hair submitted to an
air flow from any angle will experience longitudinal and transverse forces
over its entire length. The relative importance of these air flow components
will be a function of circumferential location and hair length. We
distinguish between long (1500 µm) and short (300 µm) hairs in the
following discussion, as they are known to differ in best-tuned frequencies
and represent two extreme situations. We computed the drag torque (see
Appendix) in order to understand the relative influence of the longitudinal
and transverse components of any flow on hairs of different lengths
(Fig. 7), the virtual mass
torque being almost negligible. Long hairs experience boundary layer effects
only on their bottom portion and are submitted to the far-field velocity over
most of their remaining length. These far-field velocities decline with
decreasing angle , so that the transverse drag torque acting on a long
hair will also decline with decreasing angle and become inferior to the
longitudinal one for an angle of 57°. Short hairs experiencing
similar air flow are totally immersed in the boundary layer. The decline of
transverse flow velocity with the angle is lower in the boundary layer, so
that transverse drag torque will be higher than longitudinal drag torque over
a larger range of angles, down to 37°.
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Our results have important implications for hair canopy arrangement in arthropods using air flow to sense prey or predators. First and foremost, the spatially heterogeneous information provided by a transverse flow around a cylinder implies that hairs should be placed all around a cylinder, maximising the chances to perceive a source coming from any angle. Indeed, an isotropic distribution of hairs ensures that some hair will always be perpendicular to the flow, thereby experiencing the smallest possible boundary layer, and hence the largest possible displacement. For similar reasons, hairs of the same length vibrate with different amplitudes around a cylinder depending on their radial location. Arthropods may therefore perceive the direction of an incoming air flow in a differentiated manner using the radial distribution of the canopy, very much as their hair lengths act as a Fourier transform for frequency decomposition through different natural frequencies. This directionality filter may help crickets to identify the direction of the approaching source.
The rich content of information in the spatially heterogeneous flow around
a cylinder has escaped the attention of previous workers, who cited the great
computational advantages of considering a cylinder of low curvature as a plate
for the purpose of computation of hair movement. We now need to revisit both
the hair biomechanical models and the neurocomputational models of danger
perception in crickets on the basis of these results. Cricket air flow sensors
have recently been a source of inspiration to build artificial air flow
sensors (Dijkstra et al.,
2005). Design guidelines for building flow-sensing MEMS arrays
were also based on biomimetic ideas borrowed from the cricket's cerci. A
spatial arrangement of MEMS hairs with a large range of angles relative to
flow direction on a dedicated platform would increase the sensitivity of such
sensors by a large margin. Such design could represent a major advance to the
actual mounting, on a horizontal plate, of MEMS hairs restricted to measuring
longitudinal flows. This is however not a trivial task in MEMS
fabrication.
air
air
hair
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). Most
of this information is also contained in Magal et al.
(Magal et al., 2006). For a
rigid hair oscillating relative to a fixed axis of rotation, conservation of
angular momentum, L(t), states that the rate of change of
angular momentum is equal to the sum of torques acting on the hair:
 | (A1) |
where I is the moment of inertia of the hair relative to the axis
of rotation and is the angular deflection of the hair with respect to
its equilibrium orientation. The drag torque, TD, arises
due to frictional drag acting along the hair shaft. The torque
TVM is associated with the virtual mass of fluid, which at
any instant must be also accelerated along with the hair. The damping torque,
TR, arises at the rotation point of the hair and results
from frictions between the hair base and the surrounding cuticle. The
restoring torque, TS, is equivalent to spring stiffness,
expressing the elasticity of the socket membrane, and arises at the rotation
point of the hair. The velocity profiles are integrated in the first two
torques, which drive hair motion, whereas the last two torques are always
opposed to hair deflection.
Hair's moment of inertia
According to Humphrey et al. (Humphrey
et al., 1993) and Kumagai et al.
(Kumagai et al., 1998), the
total inertial moment of a filiform hair, I (Nm s-2
rad-1), is given by:
 | (A2) |
where Ih is the hair moment of inertia:
 | (A3) |
where d is hair diameter (m), Lhair is hair
length (m), and hair is hair density (kg m-3).
IVM represents the moment of inertia of the added mass of
the fluid stagnating around and moving with the hair of constant diameter and
is given by Humphrey et al. (Humphrey et
al., 1993) as:
 | (A4) |
Stokes (Stokes, 1851) shows
that for values of the dimensionless parameter:
 | (A5) |
such that s <<1:
 | (A6) |
with:
 | (A7) |
where (dimensionless) is Euler's constant, d is hair
diameter (m) at height y above the cercus, f is the
oscillating air flow frequency (Hz), and air is the air
kinematic viscosity (m2 s-1).
Fluid-induced drag and added mass torques
The fluid-induced instantaneous drag and added mass torques are obtained by
integrating the fluid-induced drag and added mass forces per unit length
acting along the total length of the hair, Lhair (m):
 | (A8) |
 | (A9) |
where y is the position along the hair (m), and
FD(y,t) and FVM(y,t) are the drag and
added mass forces per hair unit length, acting at height y,
respectively. Eqns A8 and A9 state that the total torque that acts to deflect
the hair from its resting position is given by the integrated sum of all
torques over the arm length of rotation y. Each of the torques is
generated on an infinitesimally thin disc of the hair shaft. Theoretical
expressions for FD and FVM, applicable
to a fluid oscillating perpendicular to a hair, have been derived by Stokes
(Stokes, 1851) and previously
used in filiform hair modelling studies
(Humphrey et al., 1993;
Shimozawa et al., 1998).
Drag and added mass forces per unit length
For a fluid oscillating perpendicular to a cylindrical hair segment, the
drag force acting on the cylinder at height y above the cercus
surface is:
 | (A10) |
where µ is the fluid dynamic viscosity (kg m-1
s-1), G is given by Eqn A6 and
VF(y,t) is the velocity acting on the hair. The
added mass force per unit length is given by:
 | (A11) |
where G is given by Eqn A6 and g is given by Eqn A7.
Damping torque
This torque results in part from the friction between the hair base and the
surrounding cuticle (Shimozawa and Kanou, 1984a;
Shimozawa et al., 1998). The
damping factor (Nm s-1 rad-1) includes frictional terms
at the hair base (R) as well as friction between added mass of fluid
moving with the hair and surrounding air (RVM). Both
sources of torque always act so as to oppose hair motion. The total damping
torque, TR (N m-1), is given by:
 | (A12) |
where R (Nm s-1 rad-1), is a constant
damping factor which is allometrically related to Lhair
(µm) (Shimozawa et al.,
1998):
 | (13) |
 | (A14) |
Restoring torque
The socket joint membrane acts as a spring, causing a restoring torque that
always acts to oppose hair motion:
 | (A15) |
where S (Nm rad-1) is the spring stiffness, which is
allometrically related to Lhair (µm)
(Shimozawa et al., 1998):
 | (A16) |
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Acknowledgments |
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References |
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Humphrey, J. A. C., Barth, F. G. and Voss, K. (2003a). The motion-sensing hairs of arthropods: using physics to understand sensory ecology and adaptive evolution. In Ecology of Sensing (ed. F. G. Barth and A. Schmid), pp.105 -115. Berlin, Heidelberg, New York: Springer-Verlag.
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