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First published online August 17, 2007
Journal of Experimental Biology 210, 2969-2978 (2007)
Published by The Company of Biologists 2007
doi: 10.1242/jeb.005942
Analytical and numerical investigation of the flow past the lateral antennular flagellum of the crayfish Procambarus clarkii
1 Department of Biology, University of Virginia, Charlottesville, VA 22904,
USA
2 Department of Mechanical and Aerospace Engineering, University of
Virginia, Charlottesville, VA 22904, USA
* Author for correspondence (e-mail: jach{at}virginia.edu)
Accepted 11 June 2007
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Summary |
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 TOP Summary IntroductionMethods, Results and DiscussionReferences |
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Key words: crustacean, hydrodynamics, mathematical analysis
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Introduction |
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TOPSummaryIntroductionMethods, Results and DiscussionReferences |
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). In that study the authors observed a
hydrodynamic-dependent asymmetry in the spiking responses recorded from
single, bimodally sensitive local interneurons (Type I) in the crayfish
deutocerebrum. Specifically, they found that in an abruptly initiated flow of
water (or odorant) past the flagellum, consistently larger numbers of spikes
were recorded in response to the hydrodynamic stimulation when the flow
direction was proximal-to-distal. Among other things, in this communication we
show that the proximal-to-distal and the distal-to-proximal flows produced in
the flagellum-in-tube experiment correspond closely to the downward and upward
flicks, respectively, of a free-flicking flagellum. The putative mechanoreceptors of interest are beak-shaped sensilla distributed circumferentially and along the length of the flagellum. We show from calculations of the drag forces acting on them for free-flicking conditions that there are at least three sources of hydrodynamic asymmetry possibly related to the electrophysiological asymmetry: (i) the sense of the drag forces acting on medial and lateral mechanoreceptors changes in the same way for both with change in flick direction; (ii) during a downward (an upward) flick, a ventral (dorsal) mechanoreceptor experiences a larger drag force magnitude than a dorsal (ventral) mechanoreceptor; (iii) because of the difference in speeds between downward and upward flicks, the magnitudes of the drag forces acting on medial, lateral and ventral mechanoreceptors during a downward flick are about two times larger relative to the forces acting on medial, lateral and dorsal mechanoreceptors during an upward flick.
By necessity, the analyses presented here are based on simplified
geometrical models for the two flagellum flow configurations considered.
However, they are guided by kinematic observations and all the essential
physics is retained in order to ensure that: (i) the free-flicking flagellum
results are generally applicable and of broad interest; (ii) a comparison
between the free-flicking and flagellum-in-tube results is helpful for
interpreting the electrophysiological findings of Mellon and Humphrey
(Mellon and Humphrey, 2007). In
this sense especially, the analyses strongly complement the
electrophysiological work, since it would be very challenging and require
considerable effort and time to perform comparable velocity measurements for
the two flagellum flow configurations.
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Methods, Results and Discussion |
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TOPSummaryIntroductionMethods, Results and DiscussionReferences |
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(1) Kinematics of the flows approaching a downward and upward flicking flagellum
The geometry of an idealized flagellum shown flicking downwards in
quiescent water is sketched in Fig.
1 relative to an (X–Y) coordinate system
(frame of reference) fixed on the rotation pivot point. Our kinematic
observations show that a downward flicking flagellum deflects by an angle of
approximately 15° in 0.05 s, corresponding to an angular velocity of 5.24
rad s–1, and that the downward/upward flick velocity ratio is
1.66, approximately. The fluid drag forces acting on the deformable flagellum
determine its final equilibrium shape during the downward and upward flicks,
respectively. We find that these shapes are acquired very early in a stroke
for either flick direction, and that they are retained throughout the
remaining motion of the flick. During the downward flick the flagellum curves
by as much as 40°, approximately, along the distal third of its length
relative to the proximal two thirds, which remains essentially straight.
During the upward flick the flagellum is essentially straight along its entire
length.
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, relative to the
(x–y) coordinate system fixed at z, is
aligned along the y coordinate. Call 
the angle formed by the
x-coordinate and the tangent, tf, to the flagellum at
point z. Then, also from construction, the approaching velocity
components normal and tangent to the flagellum at point z are given
by Unf=Uyfcos and
Utf=Uyfsin, where
Uyf is the magnitude of
. The ratio of
these two velocity components is given by
Utf/Unf=tan. During a downward
flick the angle is very small along the proximal two thirds of the
flagellum but increases to values of the order 40° along the distal third,
towards the flagellum tip. Thus, for a downward flick we expect
Utf/Unf to be close to zero along the
proximal two thirds of the flagellum and to vary from 0 to 0.84,
approximately, along the distal third. During an upward flick the flagellum is
essentially straight so that Utf/Unf
is very close to zero along its entire length. The conclusions are that: (i)
the approaching velocity component tangent to a downward flicking flagellum is
always positive, directed from base to tip along its distal third, as sketched
in Fig. 1; (ii) along the
proximal two thirds of a downward flicking flagellum, the magnitude of the
velocity component normal to it significantly exceeds the tangential
component, whereas along the distal third the two velocity components
gradually become comparable in magnitude; (iii) for an upward flicking
flagellum the magnitude of the velocity component normal to it significantly
exceeds the tangential component everywhere along its length.
(2) Kinematics of the flow approaching the aesthetascs and putative mechanoreceptor sensilla on a downward flicking flagellum
Fig. 2 is a sketch of an
idealized aesthetasc or beak-shaped mechanoreceptor sensillum attached to a
point z on the ventral side of the curved, downward flicking
flagellum shown in Fig. 1.
Relative to an (x–y) coordinate system (as defined in
Fig. 1) fixed at the base of
the sensillum, the vector velocity of the water approaching it,
, can be
decomposed to define the velocity components normal and tangent to the
sensillum. These two quantities are given by
Uns=Uyfcos
and
Uts=Uyfsin, respectively, where
=–
and is as defined in
Fig. 1. We know for both the
aesthetascs and the putative mechanoreceptors that 
45°, and
that 0° along the proximal two thirds of the flagellum and
0°
40° along the distal third. It follows that
–45° along the proximal two thirds of the flagellum and
–45°–5° along the distal third. As a
consequence, the ratio of the two velocity components,
Uts/Uns=tan, is close to
–1 along the proximal two thirds of the flagellum and ranges from
–1 to –0.09, approximately, along the distal third. The
conclusions are that: (i) the approaching velocity component tangent to an
aesthetasc or a mechanoreceptor on the ventral side of a downward flicking
flagellum is always negative, directed from sensillum tip to base, contrary to
the sense of motion assumed for analysis in
Fig. 2; (ii) the magnitudes of
the two velocity components approaching these sensilla are comparable along
the proximal two thirds of the flagellum, whereas along the distal third the
normal velocity component gradually exceeds the tangential significantly.
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Medially and laterally located mechanoreceptors also have inclination
angles 45° with respect to the local flagellum surface.
However, because they are oriented in the direction of the local tangent to
the flagellum, tf, these sensilla experience the same approaching
velocity components as the flagellum itself, meaning that
Uns=Unf and
Uts=Utf for both downward and upward
flicking motions. Because of the recirculating nature of the flow that arises
in the wake of a downward (upward) flicking flagellum, an analysis of the
velocity components approaching mechanoreceptor sensilla on the dorsal
(ventral) side of the flagellum must await more detailed numerical
calculations, presented below.
(3) Dynamics of the flow around a downward or upward flicking flagellum
A flicking flagellum can be approximated as a curved, slender, cone-like
object of length Lf and of diameter df
at location z along its length such that
Lf/df
1 mm, dt0.25 mm,
and Lf=25 mm, from which it follows that the cone angle is
3.4°.
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dfUnf/µ,
where (=996 kg m–3) and µ
(=8.6x10–4 kg m–1
s–1) are the density and dynamic viscosity of water at
25°C, respectively. The normal component of velocity is given by
Unf=Uyfcos where, from
Fig. 1,
Uyf=
R(z), and 
5.24 rad
s–1 for a downward flick and 3.14 rad
s–1 for an upward flick. Along the proximal two thirds of a
downward flicking flagellum 0°, and along the remaining
distal one third we assume to a good approximation that increases
linearly with distance z, from 0° to 40°. In contrast, an
upward flicking flagellum is essentially straight. The resulting distributions
of Ref as a function of dimensionless flagellum length are
shown in Fig. 3. For both flick
directions Ref initially increases with
z/Lf. However, because df
decreases distally, Ref eventually maximizes at a value of
50 at z/Lf=0.66 for a downward flick and at a
value of 30, also at z/Lf=0.66, for an upward
flick. For larger values of z/Lf the value of
Ref decreases relatively quickly because both
df and Unf decrease distally.
In the length range 0z/Lf0.75, for
which 0°8°, the flow field around a downward flicking
flagellum is predominantly two dimensional (2D) and the above definition of
Ref applies. With reference to
Fig. 5, discussed further
below, relative to an (x, y, z) Cartesian coordinate system fixed to
the flagellum and with its z axis coinciding with the z axis
of the flagellum, this 2D flow is contained in a plane normal to the
z axis and consists of streamwise (Ux) and
transverse (Uy) velocity components driven by the
approaching normal velocity component, Unf, of the
flow1. To elucidate
this flow field, numerical calculations were performed for the transient,
developing, 2D motion of water past a circular cylinder ultimately attaining a
value of Ref=50. This flow condition closely corresponds
to the transient flow around a downward-flicking flagellum in the region
0.50z/Lf0.75 and exactly corresponds to
it for z/Lf=0.66. In addition, because of the
closeness of the Reynolds numbers, the flow field calculated for
Ref=50 provides a fair representation of that arising
around an upward-flicking flagellum for Ref=30
(Tritton, 1988). The
calculations were performed using the Flow and Heat Transfer Solver (FAHTSO)
code developed by Rosales et al. (Rosales
et al., 2000; Rosales et al.,
2001), extended to include the Immersed Boundary technique
(Pillapakkam et al., 2007). In
this approach, curved surfaces are approximated on a Cartesian grid to allow
the calculations to be performed in Cartesian coordinates. FAHTSO solves the
full forms of the Navier–Stokes equations for constant property flows.
Details regarding the code and its testing and applications are given in the
above three references.
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),
rather than solve for the flow field generated by a flagellum (cylinder)
accelerating through water, we solve for the flow field of water accelerating
past a fixed flagellum (cylinder). The 2D domain employed for the numerical
calculations was 20dfx20df units
and the cylinder center was located at (x=5df,
y=10df). Checks were conducted to ensure that the
locations of the four domain boundaries were sufficiently removed from the
cylinder so as not to adversely affect the calculated results. The boundary
conditions employed were: (i) zero velocity at the cylinder surface; (ii)
unidirectional, uniform flow with velocity
Uin=Uo
(Unf)2
aligned in the x-coordinate direction at the inlet plane
(x=0); (iii) developed flow at the outlet plane
(x=20df); (iv) impermeable bottom (y=0)
and top (y=20df) domain surfaces sliding at the
inlet flow velocity Uo. For the calculations presented
here the (x,y) grid consisted of (400x400) nodes and was
unevenly spaced, being more refined near the cylinder surface to resolve the
boundary layer forming on it. The dimensionless calculation time step was set
to 
t*=5.363x10–4, where
t*tUo/df,
and at each time step the entire flow field was calculated according to an
iterative scheme until it had converged within that time step.
The calculations were started with the water initially at rest. The flow at
the inlet plane as well as the top and bottom domain boundaries was then
accelerated according to an S-shaped curve given by:
 | (1) |
is dimensionless final time, and
Tp is an adjustable dimensionless constant that changes
the slope of the S-shaped curve. The curve used, plotted in
Fig. 4 and corresponding to
Tp=0.5 and
, shows that by
t*=4 the flow field has acquired its final steady state
velocity. Thus, in this calculation, by the time a flicking flagellum
translates two body diameters it has essentially acquired its final steady
state velocity, a condition closely matching our laboratory observations.
Fig. 4 also shows as a function
of time the dimensionless distance
(xf/df) traveled by the approaching
flow past a fixed flagellum or, equivalently, the distance traveled by a
flagellum moving through still water.
Values of df=5x10–4 m and
Uo=8.63x10–2 m s–1
were set numerically to give the final developed flow value of
Ref=50 of interest to this work using water as the fluid
medium. This value of df corresponds to the effective
diameter of a 25 mm flagellum two thirds along its length starting from the
base. However, note that, in dimensionless form, the calculated velocity
results correspond to any (df, Uo)
pair ultimately yielding Ref=50 and, because
t*tUo/df,
in the t* range explored other (df,
Uo) pairs will correspond to different physical times,
t.
As shown in Fig. 5, the
developing flow around a downward-flicking flagellum ultimately attaining a
value of Ref=50 is laminar and streamlined. In the plots
the streamlines are color-coded to indicate the local dimensionless magnitude
of velocity
.
Results are plotted for four dimensionless times t*=1, 2,
3 and 4, corresponding to physical times of
t=5.79x10–3 s,
1.16x10–2 s, 1.74x10–2 s and
2.32x10–2 s. The plots show the flow accelerating with
time, especially where it curves near to and around the flagellum surface. A
stagnation flow region arises immediately ahead of the flagellum, and a
recirculating wake region consisting of two counter-rotating vortices grows
with time immediately behind it. The flow in the vicinity of the stagnation
point is characterized by a boundary layer of thickness
/df=1.2Ref–1/2,
approximately, that increases in thickness with distance traveled around the
flagellum (Panton, 1996).
Inertial and viscous fluid forces are comparable in magnitude in the boundary
layer, which ceases to exist at the two points where the flow detaches from
the sides of the flagellum.
Figs 6 and 7 provide plots of the calculated profiles for the dimensionless Ux velocity component plotted along two coordinate directions. (Corresponding plots for the Uy velocity component, not provided here, show that, in comparison, it is negligibly small.) Fig. 6 provides the variation of Ux/Uo as a function of x/df for y/df=10, while Fig. 7 shows the variation of Ux/Uo as a function of y/df for x/df=5. The results shown are restricted to a subregion of the calculation domain corresponding to the flow of interest, immediately around the flagellum. Relative to the flagellum cylindrical cross-section shown in each figure, in Fig. 6 the flow approaches from the left (creating a stagnation region on the flagellum) and departs from the right (creating a recirculating region immediately behind the flagellum). In Fig. 7 the flow is directed vertically upwards along either side of the flagellum, at locations ±90° downstream of the stagnation point. Velocity profiles are plotted for four dimensionless times, t*. Also shown for reference in the figures are hypothetical ventral and dorsal (Fig. 6) and medial and lateral (Fig. 7) mechanoreceptor sensilla, of length 0.25xdf and oriented normal to the flagellum surface. Except for the profiles in the wake of the flagellum (Fig. 6), the gradients of all the other velocity profiles are observed to increase markedly with time near the flagellum surface. In contrast, in the wake region immediately behind the flagellum the Ux velocity component is always relatively small, initially being positive but eventually becoming negative as the wake forms and the flow in it reverses direction. For all times, the medial and lateral mechanoreceptors shown in Fig. 7 remain immersed in the boundary layer.
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To a very good approximation, this is the complex transient flow field to
which ventral, dorsal, medial and lateral mechanoreceptors are exposed to
along the length section 0.50z/Lf0.75
of a downward-flicking flagellum. However, because of the closeness of the
Reynolds numbers the same flow patterns arise for an upward-flicking
flagellum, with the sense of motions reversed and their magnitudes decreased,
and with the locations of the stagnation and wake flow regions interchanged.
As a consequence, fluid motion around a flagellum is such that medial and
lateral mechanoreceptor sensilla are torqued in the ventral-to-dorsal
direction during a downward flick and in the dorsal-to-ventral direction
during an upward flick. Also during a downward flick the ventral sensilla are
torqued towards the flagellum surface while the dorsal sensilla are minimally
affected. In contrast, during an upward flick it is the dorsal sensilla that
are torqued toward the flagellum surface while the ventral are minimally
affected. It has been tacitly assumed that the sensilla present on the
flagellum (both aesthetascs and putative hydrodynamic mechanoreceptors) do not
significantly alter the flow around the flagellum modeled as a cylinder. The
assumption is reasonable given the sparseness of the sensilla and their small
geometrical dimensions (diameter and length) relative to the flagellum
diameter.
(4) Dynamics of the flow around a flagellum in a tube
The FAHTSO code described above was also used to perform numerical
calculations of the flow of water at 25°C in a model of the
flagellum-in-tube experiment (Mellon and
Humphrey, 2007) using the inner dimensions of the experimental
tube, length Lt=25 mm and diameter
Dt=2.4 mm. Because of the flagellum's small cone angle
(3.4°, calculated above), and in order to emphasize the dynamics of the
flow in the flagellum length range
0z/Lf0.75, the flagellum was
approximated as a solid rod of length Lf=15 mm and
diameter df=0.5 mm and was concentrically located in the
tube. Two small openings at the ends of the tube and at right angles to it,
each of diameter d=1 mm, allowed setting up the proximal-to-distal
(P
D) and distal-to-proximal (DP) flow regimes investigated in the
experiments. A schematic of the flow configuration examined, with a definition
of the Cartesian coordinate system used for the calculations, is provided in
Fig. 8. [Note that the (x,
y, z) Cartesian coordinate system used for the flagellum-in-tube
calculations is defined so as to coincide with that used for the free-flicking
flagellum calculations and, thus, facilitate comparisons between the two sets
of results.]
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D flow,
or at opening B for the DP flow, of Uin=0.318 m
s–1 corresponding to a volumetric flow rate through the tube
of 
=15 cm3
min–1=2.5x10–7 m3
s–1. Because of the larger cross-section of the annular space
between the flagellum and the tube, this resulted in an average velocity of
Uann=5.78x10–2 m
s–1 in the annulus. The remaining boundary conditions imposed
were: (i) zero velocity at all solid surfaces; (ii) uniform velocity at the
outlet opening, equal in magnitude to that at the inlet. For the conditions
examined the flow everywhere was always laminar and had essentially developed
to its final state by t=0.025 s.
Profiles of the dimensionless Ux,
Uy and Uz velocity components in the
annular space are provided in Figs
9 and
10 for two typical axial
locations in the tube and valid for all times t
0.025 s. The
velocity components are non-dimensionalized using the average velocity in the
annulus, Uann, and the x and y
coordinates using the tube radius Dt/2. A clarification is
in order to better understand the discussion below concerning these results
and their comparison with the free-flicking flagellum calculations. With
reference to Fig. 8, note that
in these plots `S–N' denotes velocity component profiles plotted along
the south-to-north compass direction; that is, plotted as a function of
x for y=0. Similarly, `W–E' denotes velocity component
profiles plotted along the west-to-east compass direction; that is, plotted as
a function of y for x=0.
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D and DP flow regimes are
virtually the same in magnitude and distribution, but with directions
reversed, along the length of the annular space in the range
0.20z/Lt0.60, approximately. Based on
the flagellum length of 15 mm, these values correspond to
0.33z/Lf1, where the latter position
coincides with the tip of the flagellum. In this region the axial velocity,
Uz, is the only significant component of motion and has
the skewed parabolic shape characteristic of the developed flow in an
annulus.
In contrast, with reference to Fig.
10, between z/Lt=0 and 0.20,
corresponding to z/Lf=0 and
z/Lf=0.33, the third of the flagellum nearest to
its base, the two PD and DP flows differ in the following major
ways.
PD flow
The Uz component of motion is relatively large
everywhere around the flagellum and of sense such that it torques all
mechanoreceptor sensilla towards the flagellum surface. Along the medial and
lateral sides of the flagellum (W–E profiles), the
Ux component of motion is comparable to
Uz and of sense such that it torques medial and lateral
sensilla parallel to the flagellum surface and in the ventral-to-dorsal
direction. The Ux component of motion in the S–N
profiles is of sense such that it also torques dorsal and ventral sensilla
towards the flagellum surface while, in contrast, the Uy
component in the W–E profiles is of sense such that it torques medial
and lateral sensilla away from the flagellum surface. Thus, the
Ux component of motion reinforces the torque effects of
the Uz when acting on the dorsal and ventral sensilla, but
works to neutralize the torque effects of Uz when acting
on the medial and lateral sensilla.
DP flow
The Uz component of motion is comparable to that for
the PD flow everywhere around the flagellum, and of sense such that it
torques all sensilla away from the flagellum surface. Along the medial and
lateral sides of the flagellum (W–E profiles), the
Ux component of motion is comparable to the axial and of
sense such that it torques medial and lateral sensilla parallel to the
flagellum surface and in the dorsal-to-ventral direction. For this case, there
are no other significant components of motion affecting the mechanoreceptor
sensilla.
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z/Lf0.33 of the 15 mm flagellum
section in the tube, the PD flow closely mimics the motion past a
flagellum during a downward-directed flick and, in so far as the medial and
lateral mechanoreceptor sensilla are concerned, the DP flow closely
mimics the motion corresponding to an upward-directed flick. (An even closer
correspondence would be attained between the DP flow and an
upward-directed flick if the axial flow worked to torque the sensilla toward
the flagellum surface, as in the PD flow case, as opposed to away from
the flagellum surface.) For the flagellum length corresponding to
0.33z/Lf1, the PD and DP
flows in the annular space are strictly axially directed, equal in magnitude
and distribution, but of opposite sense. Along this section of the flagellum,
all sensilla experience the same torque magnitude but with sense dictated by
the axial flow direction. The main conclusions are that: (i) the PD flow
in the tube torques all mechanoreceptor sensilla towards the flagellum surface
and the medial and lateral in the ventral-to-dorsal direction; (ii) the
DP flow torques all sensilla away from the flagellum surface and the
medial and lateral in the dorsal-to-ventral direction. The first situation
corresponds closely to that expected for a downward-flicking flagellum, while
the second corresponds well, but to a lesser degree, to that expected for an
upward-flicking flagellum.
(5) Drag forces and torques acting on the hydrodynamic mechanoreceptor sensilla of free-flicking and tube flow flagella
The putative hydrodynamic mechanoreceptor sensilla of interest have
slightly curved beak-like shapes at their tips and, as a consequence, are also
referred to as `beaked' sensilla in the text. They are arrayed around the
circumference of each annular segment of the flagellum (dorsally, ventrally,
medially and laterally) and vary in length from 50 to 250 µm with diameters
ranging from 10 µm at the base to 2 µm at the tip. These sensilla point
distally along the flagellum, each being contained in a plane that passes
through the local flagellum axis, and they have orientation angles, ,
measured with respect to the tangent to the flagellum surface (see
Fig. 2), ranging from 35°
to 65°.
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=45°. For the purposes of an analysis based on
the fluid mechanics results presented in the sections above, in the case of a
flicking flagellum attention is restricted to representative sensilla located
in the range 0.50z/Lf0.75 of a 25 mm
flagellum of diameter very close to df=0.5 mm in this
range. In the case of the flagellum in a tube, we are primarily concerned with
representative sensilla in the range
0z/Lf0.33 of a flagellum 15 mm long
with diameter df=0.5 mm, for which the experiment closely
mimics downward and upward free-flicking flagellum conditions. The velocity distributions obtained for a free-flicking flagellum and for a flagellum in the tube flow configuration corresponding to the experiment, allow estimations of the drag forces and torques acting upon the hydrodynamic mechanoreceptors distributed circumferentially around a segment of the flagellum. Since certain characteristics of the mechanoreceptors such as their torsional restoring constants and damping constants are currently unknown, attention is paid to estimating the maximum drag forces and torques they can possibly experience. To this end, it is assumed that the sensilla are rigidly fixed (do not move relative to the flagellum) and are inflexible (do not bend). We know the latter assumption is correct from Atomic Force Microscope (AFM) experiments performed by others (H. C. Jennings and E. Berger, personal communication). They find that the application of a force, using an AFM tip at different locations along the length of a beaked-shape Procambarus antennular sensillum, deflects the entire sensillum without bending it. The former assumption allows us to obtain maximum estimates of the drag and torque, which are of the correct order of magnitude for a deflecting mechanoreceptor.
With reference to Fig. 2,
which defines the velocity component Uns normal to a
sensillum, it is easy to verify that the characteristic Reynolds number,
ReMR[
(dMRUns)/µ], for the flow
around a hydrodynamic mechanoreceptor, whether on a free-flicking flagellum or
the flagellum-in-tube experiment, is ReMR1. Because,
in addition,
LMR/dMR
) and adapted to
the present work, is given by:
 | (2) |
=0.577216 is Euler's constant. It follows that the total drag
force acting on a sensillum is given by:
 | (3) |
 | (4) |
is the distance along the length of the sensillum.
|
(dMR/2)2(dUns/dt)
contributes to the force per unit length acting on a mechanoreceptor. It is
easy to show that:
 | (5) |
0.1 for times
t*1 (t5.79x10–3
s). For times t*0.1
(t5.79x10–4 s) we find that
fAM/fMR10, but
fMR is so small at such early times as to render both
itself and fAM negligible. Thus in our analyses we neglect
the contribution of the added mass to the force and torque acting on a
mechanoreceptor.
In applying Eqn 3 and
Eqn 4, care must be taken to
evaluate Uns correctly along the length, , of the
sensillum. With reference to Fig.
11, for a medial or lateral sensillum it is easy to show that
Uns()=Ux(y), the
component of motion normal to the plane of the figure, where
y=sin. For a free-flicking flagellum
Ux(y) is obtained from the data used to generate
Fig. 7, and for the
flagellum-in-tube flow case Ux(y) is obtained
from the data used to generate the W–E profiles in
Fig. 10.
Fig. 12 shows the most general
situation for a ventral sensillum during a downward flick (or a dorsal one
during an upward flick). Now
Uns()=Ux(x)cos+Uz(x)sin,
where x=sin. In the z/Lf range of
interest here for a free-flicking flagellum
Uz(x)=0 so that
Uns()=Ux(x)cos and,
in this case, Ux(x) is obtained from the data
used to generate Fig. 6. In the
z/Lf range considered for the flagellum-in-tube
case, particularly the PD flow, both Ux(x)
and Uz(x) can contribute to
Uns() and their values are obtained from the data
used to generate the S–N profiles in
Fig. 10.
Fig. 13 shows the time variation of the calculated drag forces and torques acting on the medial, lateral and ventral mechanoreceptor sensilla of a downward-flicking flagellum ultimately attaining a final, developed flow Reynolds number of Ref=50. The plots reveal two distinct trends: (i) the total drag forces and torques for all these sensilla increase to asymptotic values associated with the developed flow condition attained by the flagellum by t*=4 (t=0.023 s); (ii) at any time, the drag forces and torques acting on the medial and lateral sensilla are 20–30 times larger than the corresponding forces/torques acting on the ventral sensilla.
Corresponding results for the drag forces and torques acting on the medial,
lateral and dorsal sensilla of an upward-flicking flagellum (not plotted in
Fig. 13) show exactly the same
trends but are half as large in numerical value because of the smaller
upward-flick velocity. That this should be the case is readily proven using
Eqn 2 to obtain the ratio of the
forces per unit length acting on a mechanoreceptor during an upward and
downward flick, respectively. The ratio is given by:
 | (6) |
it follows that
.
Table 1 summarizes and
compares the sensilla drag and torque results obtained for a downward and
upward free-flicking flagellum, and for the flagellum-in-tube configuration.
The values in the table for the flicking flagellum correspond to
t0.023 s (t*4), while those for the
flagellum-in-tube configuration correspond to t0.025 s. The
tabulated values reveal three trends. Relative to corresponding values
obtained for a downward and upward flicking flagellum, in the
flagellum-in-tube configuration: (i) ventral (dorsal) sensilla experience drag
forces and torques that are about 4.3 times larger (1.3 times smaller) for the
PD flow (for the DP flow); (ii) medial and lateral sensilla
experience drag forces and torques that are about 4.5 times smaller for the
PD flow and 4.7 times smaller for the DP flow; (iii) as for the
free-flicking flagellum, drag forces and torques are about 2 times larger for
the PD flow than for the DP flow.
|
The conclusions are that, for times t>0.025 s: (i) the
flagellum-in-tube experiment yields values of drag forces and torques acting
on the flagellum sensilla that are well within the range of the values
experienced by a free-flicking flagellum for both downward and upward flicking
conditions; (ii) the PD flow induces larger values of the drag forces
and torques acting on the sensilla than the DP flow, showing good
correspondence with the downward and upward flicks of a free-flicking
flagellum.
|
Acknowledgments |
|---|
|
Footnotes |
|---|
2 In the context of the present analysis, the inlet velocity
Uin is given the value Uo and
corresponds to Unf, the approaching velocity normal to the
flagellum.
|
References |
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TOPSummaryIntroductionMethods, Results and DiscussionReferences |
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Mellon, DeF. and Humphrey, J. A. C. (2007).
Directional asymmetry in responses of local interneurons in the crayfish
deutocerebrum to hydrodynamic stimulation of the lateral antennular flagellum:
experimental findings and theoretical predictions. J. Exp.
Biol. 210,2961
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Rosales, J. L., Ortega, A. and Humphrey, J. A. C. (2001). A numerical simulation of the convective heat transfer in confined channel flow past square cylinders: comparison of inline and offset tandem pairs. Int. J. Heat Mass Transfer 44,587 -603.[CrossRef]
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 D. Mellon Jr and J. A. C. Humphrey Directional asymmetry in responses of local interneurons in the crayfish deutocerebrum to hydrodynamic stimulation of the lateral antennular flagellum J. Exp. Biol., September 1, 2007; 210(17): 2961 - 2968. [Abstract] [Full Text] [PDF]
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