## SUMMARY

The aim of this work is to characterize the boundary layer over small appendages in insects in longitudinal and transverse oscillatory flows. The problem of immediate interest is the early warning system in crickets perceiving flying predators using air-flow-sensitive hairs on cerci, two long appendages at their rear. We studied both types of oscillatory flows around small cylinders using stroboscopic micro-particle image velocimetry as a function of flow velocity and frequency. Theoretical predictions are well fulfilled for both longitudinal and transverse flows. Transverse flow leads to higher velocities than longitudinal flow in the boundary layer over a large range of angles between flow and cylinder. The strong spatial heterogeneity of flow velocities around filiform-shaped appendages is a rich source of information for different flow-sensing animals. Our results suggest that crickets could perceive the direction of incoming danger by having air-flow-sensitive hairs positioned around their entire cerci. Implications for biomimetic flow-sensing MEMS are also presented.

## Introduction

Boundary flows have been studied in a biological context for several decades, including fish swimming (Anderson et al., 2001), 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.

## Materials and methods

### Air flow modelling around a cylinder

The signal emitted by a flying predator can be approximated by a spherical point source radiating an isotropic sinusoidal wave in all directions (Tautz and Markl, 1979) [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 *U*_{0} is the flow oscillation amplitude, and ω
is the angular frequency (rad s^{-1}), with ω=2 π*f,
f* being the frequency of flow oscillations (s^{-1}). A signal
impacting the cylinder from any angle can be decomposed into longitudinal
(*V*_{l}), radial (*V*_{r}) and circumferential
(*V* _{θ}) components
(Fig. 1A).

We first consider a flow oscillating parallel to a cylinder (longitudinal
flow). To avoid end effects, the length of the cylinder (*L*) is
supposed to be much larger than its diameter, *D*
(*L*>10*D*). Under these conditions, the fluid velocity at
time (*t*) can be expressed as a function of the distance to the
cylinder (*y*) as follows:
(2)

where β=(ω/2 ν_{air})^{½} is the
border effect factor (m^{-1}), ν_{air}=μ
_{air}/μ_{air} is the kinematic viscosity of the fluid
(m^{2} s^{-1}), μ_{air} is the dynamic viscosity of
the fluid (Ns m^{-2}), ρ_{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,φ, 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*_{θ}, 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 *X*_{i},
*Y*_{i}, *Z*_{i}, *C*_{i} and
*X*_{r}, *Y*_{r}, *Z*_{r},
*C*_{r} being their imaginary and real parts, respectively.
*H*_{n} 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), ϵ=[*i*(ω/ν)]^{½}, andθ
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 δ between
*V*_{r} and *V*_{θ}. This phase lag is
given by:
(7)σ
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*_{θ} 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, 5Ω; 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*(ω*t*)=*U*_{0}sin(ω*t*),
with amplitude *U*_{0}=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,ν
_{air}=1.56×10^{-5} m^{2} 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 2×2 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 λ=532 nm placed
under the camera. Measurements were conducted for velocities
*U*_{0}=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 *f*_{flow} with a frequency
(*f*_{acq}) slightly lower than a sub-multiple of the signal
frequency. This technique gives us the following pseudo sampling time interval
(*t*_{strob}):
(9)

where *T*_{acq} is the inverse of the PIV sampling
frequency, and *T*_{flow} 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 *t*_{strob}=0.28 ms, corresponding to a phase ofω
*t*_{strob}=0.2 rad=0.06π 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 φ corresponding to each sample point is obtained through inference,
not measurements. The known far-field velocity *U*_{∞}
and local velocity *U*_{0} are used with the relationshipφ
=arcsin(*U* _{∞}/*U*_{0}).

### 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, 32×32 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 32×32 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
*U*_{0}.

## Results

### Frequency

Fig. 2 shows the frequency
dependence of the velocity around a cylinder in transverse flow. It represents
the amplitude of the flow, i.e. the difference between the maximum velocity
fields [measured at ω*t*=(π/2)] and the minimum velocity
fields [measured at ω*t*=-(π/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 θ angles (90, 60, 45, 30 and 15°)
(Fig. 3), an angle θ 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 1.4*U*_{0} at 30 Hz to
1.6*U*_{0} at 180 Hz.

In longitudinal flow, we measured significant changes in fluid velocity at a short distance above the cylinder (<0.5 mm) as a function of flow oscillation frequency (Fig. 3, grey line). For longitudinal flow, the velocity profiles match nicely with those predicted by theory both in terms of amplitude and distance of the maximal velocity above the cylinder. Measurements very close (70 μm) to the surface of the cylinder are ∼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 π/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 θ 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).

The phase displacement with distance from the cylinder is represented in Fig. 6. The fit with the theory is good. Phase displacement goes from π/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.

## Discussion

### Measurements versus models

This PIV study is one of the most comprehensive regarding flow velocities
around cylinders of small diameters in oscillatory flows at low *Re*.
We could not use the approximation developed by Wang
(Wang, 1968) and later used by
Humphrey and Devarakonda [equation A.2.1
(Humphrey et al., 1993)] as
the conditions for its application [*Re*×*St*≫1 and
(*Re*/*St*)≪1] did not apply for the full range of values we
were interested in, particularly for the low frequencies
[*Re*×*St*=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 θ 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 θ 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 θ, 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 θ 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 θ angles, down to 37°.

What are the consequences in terms of hair sensitivity as measured by the maximal amplitude of hair displacement? Whatever its position around the cylinder, a long hair shows high displacement values at low frequency. Hair displacement amplitude sharply declines from 80 Hz and reaches a negligible value at frequencies above 500 Hz (Fig. 8A). By contrast, short hairs do not show such strong frequency-dependent behaviour over the values of interest (Fig. 8B). For the air velocity and frequency used in Figs 7 and 8, the influence of the transverse component is greater than that of the longitudinal component in the angle interval of [90°, 50°] for long hairs. This interval is further extended to [90°, 37°] for short hairs. Thus, short hairs are more sensitive than long hairs to the transverse component of flow from almost any direction. This result makes sense physically, as the largest differences between longitudinal and transverse flows are in the boundary layer, in which the small hairs are totally immersed. These considerations were tested at other frequencies (30 Hz and 180 Hz) and were found to be quite robust.

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.

- List of symbols
- d
- hair diameter
- D
- cylinder diameter
*f*_{acq}- PIV sampling frequency
*f*_{flow}- oscillating air flow frequency
*F*_{D}(*y,t*)- drag per hair unit length acting at height
*y* *F*_{VM}(*y,t*)- added mass force per hair unit length acting at height
*y H*_{n}- Hankel function of the first kind
- I
- moment of inertia
*I*_{h}- hair moment of inertia
*I*_{VM}- moment of inertia of the added mass stagnating around and moving with the hair
- L
- cylinder length
- L(t)
- angular momentum
*L*_{hair}- hair length
- N
- number of sampling points
- R
- damping factor including frictional terms at the hair base
*R*_{VM}- damping factor due to the friction between added mass of fluid moving with the hair and surrounding air
- S
- spring stiffness
- t
- time
*t*_{strob}- pseudo sampling time interval
*T*_{acq}- inverse of the PIV sampling frequency
*T*_{D}- drag torque
*T*_{flow}- inverse of the signal frequency
*T*_{R}- damping torque
*T*_{S}- restoring torque
*T*_{VM}- torque associated with the virtual mass of fluid
*U*_{∞}- far-field velocity
*U*_{0}- flow oscillation amplitude
*V*_{F}(*y,t*)- velocity acting on the hair
*V*_{l}- longitudinal component of a flow impacting the cylinder
*V*_{r}- radial component of a flow impacting the cylinder
*V*_{θ}- circumferential component of a flow impacting the cylinder
- X,Y,Z,C
- complex numbers
*X*_{i},*Y*_{i},*Z*_{i},*C*_{i}- imaginary parts of the complex numbers
*X,Y,Z,C* *X*_{r},*Y*_{r},*Z*_{r},*C*_{r}- real parts of the complex numbers
*X,Y,Z,C* - y
- distance to the cylinder
- α
- angular deflection of the hair with respect to its equilibrium orientation
- β
- border effect factor
- δ
- phase lag between
*V*_{r}and*V*_{θ} - γ
- Euler's constant
- λ
- laser wavelength
- μair
- dynamic viscosity of air
- νair
- kinematic viscosity of air
- ρair
- air density
- ρhair
- hair density
- θ
- angle between the cylinder and the flow
- ω
- angular frequency
- φ
- phase displacement

## Appendix

### Hair movement modelling

The physical behaviour of a single hair submitted to an air flow can be
modelled by the equation given by Shimozawa and Kanou (Shimozawa and Kanou,
1984a) and modified by Humphrey et al.
(Humphrey et al., 1993). 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 α is the angular deflection of the hair with respect to
its equilibrium orientation. The drag torque, *T*_{D}, arises
due to frictional drag acting along the hair shaft. The torque
*T*_{VM} is associated with the virtual mass of fluid, which at
any instant must be also accelerated along with the hair. The damping torque,
*T*_{R}, arises at the rotation point of the hair and results
from frictions between the hair base and the surrounding cuticle. The
restoring torque, *T*_{S}, 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 *I*_{h} is the hair moment of inertia:
(A3)

where *d* is hair diameter (m), *L*_{hair} is hair
length (m), and ρ_{hair} is hair density (kg m^{-3}).
*I*_{VM} 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 γ (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 ν_{air} is the air
kinematic viscosity (m^{2} 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, *L*_{hair} (m):
(A8)
(A9)

where *y* is the position along the hair (m), and
*F*_{D}(y,t) and *F*_{VM}(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 *F*_{D} and *F*_{VM}, 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
*V*_{F}(*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 (*R*_{VM}). Both
sources of torque always act so as to oppose hair motion. The total damping
torque, *T*_{R} (N m^{-1}), is given by:
(A12)

where *R* (Nm s^{-1} rad^{-1}), is a constant
damping factor which is allometrically related to *L*_{hair}
(μm) (Shimozawa et al.,
1998):
(13)
and *R*_{VM} is the damping factor due to the friction between
added mass of fluid moving with the hair and surrounding air:
(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 *L*_{hair} (μm)
(Shimozawa et al., 1998):
(A16)

## ACKNOWLEDGEMENTS

This work is part of the research conducted within the Cricket Inspired perCeption and Autonomous Decision Automata (CICADA) project (IST-2001-34718) and within the Customized Intelligent Life Inspired Arrays (CILIA) project (FP6-IST-016039). These projects are both funded by the European Community under the `Information Society Technologies-IST' Program, Future and Emergent Technologies (FET), Lifelike Perception Systems action.

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