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Laser interferometric flow measurements in the lateral line organTsang, P

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Laser interferometric flow measurements in thelateral line organ

The investigations were supported by the Life Sciences Foundation (SLW), whichis subsidized by the Netherlands Organization for Scientific Research (NWO).

RIJKSUNIVERSITEIT GRONINGEN

Laser interferometric flow measurements in the lateralline organ

Proefschrift

ter verkrijging van het doctoraat in deWiskunde en Natuurwetenschappenaan de Rijksuniversiteit Groningen

op gezag van deRector Magnificus, dr. F. van der Woude,

in het openbaar te verdedigen opvrijdag 28 november 1997des namiddags te 4.15 uur

door

Peter Tjin Sjoe Kong Tsang

geboren op 5 mei 1970te Nieuw Nickerie, Suriname

Promotor: Prof. Dr. Ir. H. DuifhuisReferent: Dr. S. M. van Netten

ISBN 90-367-0840-0

Contents

Chapter 1 Introduction 1

Chapter 2 Temperature dependence of the viscosity of lateralline canal fluid 7

Chapter 3 Fluid flow profiles measured in the supraorbital lateralline canal of the ruffe 27

Chapter 4 Cupular influence on fluid flow in the supraorbitallateral line canal of the ruffe (Acerina cernua L.) 35

Chapter 5 A 2-dimensional model of the supraorbital lateral linecanal organ of the ruffe 57

Summary 75

Samenvatting 77

Acknowledgements 79

1 Introduction

Hearing research

Mammalian hearing is a complicated process which involves many steps. Anacoustic pressure wave propagating through air is received by the pinna andchannelled to the ear-drum by the external auditory canal. Hereafter, the stimulus istransferred from air to labyrinthine fluid. The medium mismatch is compensated bythe middle ear. After this, the stimulus reaches a snail-shell like spiral tube, calledthe cochlea. The moving structures and the hydrodynamics within the cochlea forman electromechanical structure which enables frequency-, spatial- and temporalanalysis. The transduction of the mechanical stimulus to nerve impulses isperformed by the mechano-sensory hair cells housed inside the organ of Corti.These electrical impulses are "heard" by the brain as sound.

Just from this simple overview alone, it is clear that understanding thecomplete hearing process involves knowledge of many disciplines. Hence researchinto hearing has been divided into many specialisations, each investigating aparticular part of the auditory organs in order to understand the specific functionsof each part. Present research involves both practical measurements and computermodelling. Measurements supply the necessary data for the development of thecochlea models. In return, the models are used to verify the measurements or togive an insight into what is important to measure. The knowledge obtained fromhearing research may lead to improvements in e.g. hearing aids and speechrecognition programs.

Besides the complexity of the mammalian cochlea structure, the organ isencased in thick protective temporal bone. This makes it difficult to conductmeasurements in the internal structures of the cochlea without disturbing thenatural conditions. The complexity of hearing motivates scientists to expand theirstudies to include birds, reptiles, amphibians and fish. In fact, when it comes tostudying the transductional and physical properties of the sensory hair cells it is fareasier to conduct the research on fish and amphibians. They possess a far simplerand more easily accessible sensory hair cell organ, the lateral line organ.

1

Lateral line sensory organ

The ancestors of all jawed vertebrates emerged in the warm waters of the earth'svast primordial sea around 500 million years ago (Litman, 1996). These animalsare believed to resemble certain members of a later group of fish, known as theplacoderms. These ancestors evolved to more advanced fishes, including thosewhich eventually crawled onto land and evolved further into amphibians, reptiles,birds and mammals.

The lateral line organ, which is unique to fish and amphibians, has beenknown since ancient times. The first ever (accurate) description of the lateral lineorgan appears to be given by Stenonis in 1664 (Walker, 1967). At that time, thelateral line organ was thought to be responsible for producing slime. It was notuntil the early nineteenth century before it was recognised to be a sensory organ(Jacobson, 1813; Walker, 1967), specifically for water motion (Knox, 1825;Walker, 1967). With improvements in measuring techniques, our knowledge of thisorgan has advanced immeasurably in terms of anatomy, physiology and lateral linemechanics.

The structure of the lateral line organ is relatively simple in comparison withthe mammalian cochlea. If we take for example the lateral line organ of the animalused for these studies, the ruffe (Acerina cernua L.) a common freshwater fish, itbasically consists of a number of neuromasts contained in a bony canal (see Fig. 1).The neuromast comprises of a cluster of sensory hair cells surrounded bysupporting cells and a cupula.

supraorbital canal

infraorbital canal

preopercular canal

nerve

vein epithelium

bony bridge

cupula

Figure 1: A diagram of the lateral line organs found on the ruffe's head. Part of the supraorbital

lateral line organ is also shown with the skin removed.

2

The stereocilia of the sensory hair cells are embedded in the matrix structure ofthe cupula (Kelly and van Netten, 1991). Thus, any mechanical stimulusexperienced by the cupula is directly coupled to the hair cells. Although thestructure of the lateral line organ is relatively simple, it is nevertheless extremelysensitive. For the ruffe, the threshold displacement sensitivity is in the order of 1nm at » 100 Hz (Kuiper, 1956).

The lateral line organ plays a major role in the detection and positioning forstriking at the prey (Janssen and Corcoran, 1993), avoiding predators, avoidingobstacles (Dijkgraaf, 1963) and for schooling (Partridge and Pitcher, 1980). Thisremarkable organ may be a factor in explaining the success fishes and amphibianshave in colonising a wide variety of habitats, ranging from clear to dark murkywater.

Labyrinthine fluid

Bárány's work (1907) on the endolymph, a fluid with suitable density and viscosity,led to the conclusion that it was responsible for the stimulation of the hair cells(Dohlman, 1967). A little more than half a century later, interest in the propertiesof the labyrinthine fluids and vestibular organs reached new heights. During theearly stages of the space race, there was a lot of interest from NASA in thevestibular organs with regard to the problems faced by astronauts during spaceexploration. The vestibular organs, which evolved to operate within thegravitational forces of earth, are exposed to unique gravitoinertial forcesencountered in the exploration of space. This made it necessary to understand thesensory information, which may differ quantitatively and qualitatively from what isexperienced on earth, in order to cope and adjust to the extra-terrestrial conditions.

Due to the important roles played by the labyrinthine fluids, there has beenmuch research into the physical properties of these fluids. Physical properties suchas viscosity, density, thermal coefficient of viscosity and coefficient of thermalexpansion were measured by Steer et al. (1967).

The stimulation of the sensory hair cells of the lateral line organ is analogousto that found in the vestibular system of mammals. In this case, the role of theendolymph is taken by the lateral line fluid which drives the cupula by acombination of viscous and inertial fluid forces (van Netten, 1991). Unlike themultitude of studies on endolymph and perilymph, research carried out on thelateral line canal fluid is very modest in comparison.

In the eighties and early nineties, Denton and Gray devised many novel

3

measuring techniques to conduct in vivo studies in the lateral line canal. Lateralline canal fluid flow (Denton and Gray, 1983, 1988, 1993) and the interactionswith different types of lateral line canal geometry were measured in physicalmodels (Denton and Gray, 1988). With aid of a model, the viscosity of the lateralline canal fluid was estimated to be 1.7 mPa s for a simple canal with pores(Denton and Gray, 1988). A higher value of 5.1 mPa s (van Netten, 1991) wascalculated for the lateral line canal fluid of the ruffe.

The flow profiles in the lateral line canal have long been assumed (Denton andGray, 1988; van Netten, 1991) to resemble the flow profiles found in a closed tube(see e.g. Sexl, 1930; Schlichting, 1960). Although the lateral line canal fluid flowwas measured by Denton and Gray (1988), it was limited to single points in spaceand no flow profiles could be deduced. The lack of measurements in this area oflateral line research needs to be filled to advance our understanding of this organ.

This research

The study presented in this thesis is a continuation of the research carried out onthe lateral line organ of the ruffe (Acerina cernua L.) at the Biophysics departmentof the University of Groningen. Much of the previous research has beenconcentrated on the morphology and the mechanics of the lateral line, from thecupula down to the level of the sensory hair cells; see e.g. Wubbels (1990), vanMaarseveen (1994), van Netten and Kroese (1987), and van Netten (1991).Surprisingly, very little detailed research has been conducted on the lateral linecanal fluid which surrounds the neuromasts. The importance of this fluid has longbeen recognised for the role it plays in driving the cupula. However, research inthis field has been limited due to the fact that specialised measurement techniquesare required to carry out a comprehensive study on the physical and flow propertiesof this fluid in vivo.

This thesis consist of several parts, all dealing with different aspects of thelateral line fluid. The first part (chapter 2) deals with measuring the viscosity of asmall volume (0.04 ml) of lateral line canal fluid as a function of temperature. Forthis purpose a novel viscometer was developed. This viscometer has the capabilityto measure the viscosity of biological fluids in vivo or in vitro over the full,physiologically relevant temperature range. It consists of a laser interferometerclosely related to the one described by van Netten (1988), which was used to trackthe motion of a driven pendulum submerged in the sample fluid. The viscosity ofthe sample fluid can thus be worked out from the resonance characteristicsexhibited by the oscillating pendulum.

4

The second part (chapter 3) focuses on the flow characteristics of the lateralline canal fluid. In this chapter, the flow measurements in a lateral line canal withthe cupula removed are described. The removal of the cupula is necessary toinvestigate the influence of the lateral line canal on the stimulus received by thecupula. As with the viscosity measurements, a specialised measurement techniquewas developed. A micro sense probe, constructed from a tapered glass fibre with a Æ50 mm sphere cemented on, is used for tracking the fluid flow. This micro senseprobe has the advantage that it can be placed nearly anywhere in 3-D space withinthe lateral line canal and allows velocity profiles to be made with a high spatialresolution (≈ 50 mm) and a vibrational velocity sensitivity down to the mm/s range.

In chapter 4, longitudinal and radial profiles measured with the cupula in placeare presented. Included in this chapter is an extended version of the hydrodynamicmodel for cupular motion (van Netten, 1991) to describe the flow caudal to thecupula.

In conjunction with the flow measurements, the flow in the lateral line canal isalso computed with a finite element package, SEPRAN (SEPRA-analysis, Delft, theNetherlands). The models' formulation and the computed results are presented inchapter 5. The limited 3-D capabilities of SEPRAN restrict the lateral line models(with and without cupula) to 2-D. The models mimic the conditions as encounteredin the measurements described in chapters 3 and 4. These models are in fact firstattempts at modelling the influence of a partially open canal on the flow of thelateral line canal fluid in the region of the cupula, with the aim of gaining someinsight into the flow in areas which are inaccessible to the micro sense probe.

References

Denton, E. J. and Gray, J. A. B (1983). Mechanical factors in the excitation ofclupeid lateral lines. Proc. R. Soc. Lond. Biol. Sci., 218; 1-26.

Denton, E. J., and Gray, J. A. B. (1988). Mechanical factors in the excitation ofthe lateral lines of fishes. In Sensory Biology of Aquatic Animals, edited byJ. Atema, R. R. Fay, A. N. Popper, and W. N. Tavolga (Springer, NewYork), pp. 595-617.

Denton, E. J. and Gray, J. A. B (1993). Stimulation of the acoustico-lateralissystem of clupeid fish by external sources and their own movements. Phil.Trans. R. Soc Lond. B 341, 113-127.

Dijkgraaf, S. (1963). The functioning and significance of the lateral line organs.Biol. Rev. 38, 51-105.

Dohlman, G. F. (1967). Secretion and absorption of the endolymph. NASA SP-

5

152, pp 101-123.Jacobson, L. (1813). Extrait d'un memoire sur une organe de sens dans les raies

et les squales. Nouv. Bul. Sci. Soc. Philomotique Paris 3, 332.Janssen, J. and Corcoran, J. (1993). Lateral line stimuli can override vision to

determine sunfish strike trajectory. J. Exp. Biol. 176, 299-305.Kelly, J. P., and Netten, S. M. van (1991). Topography and mechanics of the

cupula in the fish lateral line. I. Variation of cupular structure andcomposition in three dimensions. J. Morphol. 207, 23-36.

Knox, R. (1825). On the theory of the 6th sense in fishes. Edinburgh J. Sci. 2,12.

Kuiper, J. W. (1956). The microphonic effect of the lateral line organ. Ph.D.thesis, Rijksuniversiteit Groningen.

Litman, G (1996) Sharks and the origins of vertebrate immunity. ScientificAmerican, pp.47-51.

Maarseveen, J. Th. P. W. van (1994). Mechanofysiologisch onderzoek aan dezijlijn van de pos. Ph.D. thesis, Rijksuniversiteit Groningen.

Netten, S. M. van and Kroese, A. B. A. (1987) Laser interferometricmeasurements on the dynamic behaviour of the cupula in the fish lateralline. Hearing Res. 29, 55-61.

Netten, S. M. van (1988). Laser interferometer microscope for the measurementof nanometer vibrational displacements of a light-scattering microscopicobject. J. Acoust. Soc. Am. 83, 1667-1674.

Netten, S. M. van (1991). Hydrodynamics of the excitation of the cupula in thefish canal lateral line. J. Acoust. Soc. Am. 89, 310-319.

Partridge, B. L. and Pitcher, T. J. (1980). The sensory basis of fish schools;relative roles of lateral line and vision. J. Comp. Physiol. A 135, 315-25.

Schlichting, H. (1960). Boundary-layer theory (McGraw-Hill PublishingCompany, New York).

Sexl, T. (1930). Über den von E.G. Richardson entdeckten "Annulareffekt". Z.Phys. 61, 349-362.

Steer, R. W. (1967). Physical properties of labyrinthine fluids and quantificationof the phenomenon of caloric stimulation. NASA SP-152, pp 409-420.

Walker, T. J. (1967). History, histological methods, and details of the structureof the lateral line of the walleye surfperch in Lateral line detectors (P. Cahn,ed.), Indiana university press, pp. 13-25.

Wubbels, R. J. (1990). Afferent activity in the supra-orbital canal of the rufflateral line. Ph.D. thesis, Rijksuniversiteit Groningen.

6

2 Temperature dependence of the viscosityof lateral line canal fluid

Introduction

Lateral line organs such as the ones found on the head of the ruffe are used by theanimal to detect water motion around it (Dijkgraaf, 1963). The supraorbital lateralline organ of the ruffe consists of bony canals with a number of neuromasts locatedunder bony bridges. Neuromasts are the sensory elements of the lateral line organand are constructed from dome shaped extracellular matrices (Kelly and vanNetten, 1991), cupulae, which have bundles of sensory hair cells attached to theirbases. The cupulae are stimulated by a combination of viscous and inertial fluidforces (van Netten, 1987), conveyed to them by the surrounding lateral line canalfluid. The lateral line canal fluid is in mechanical contact with the outside watervia the flexible skin windows which cover the canal (van Netten and vanMaarseveen, 1994).

Through these mechanical connections, energy from the outside water motionis utilised to drive the cupulae. The resultant motion of the hair bundles istransduced into receptor potentials by the hair cells. Part of the stimulus energy islost through viscous damping from the lateral line canal fluid. To gain a betterinsight into the extent of energy dissipated, the viscosity of the lateral line canalfluid needs to be known.

Previous studies of the frequency response of the cupula in the ruffe hasindirectly yielded a viscosity value of 5.1 mPa s for the lateral line canal fluid at atemperature of approximately 18 °C (van Netten and Kroese, 1987; van Netten,1991). This was achieved by applying a hydrodynamic model to measurements ofthe frequency response of the cupula. The viscosity was calculated from the best fitthrough the results.

In the present paper, direct measurements are described and the effect oftemperature on viscosity is also investigated. There are many types of viscometerswhich can be used for measuring fluid viscosity. The most well-known are based onoscillating-body, capillary-flow and falling-ball techniques (Kestin, 1988; Fung,1981). One of the simplest type of viscometers is the oscillating-body viscometer.This uses a probe which is axially symmetrical and is designed to oscillatetorsionally in the sample fluid. By measuring the torque and angular frequency, theviscosity can be determined.

7

A type of viscometer that is applicable to the measurement of viscosity in avolume of about 0.1 ml sample fluid, is the oscillating magnetic microrheometer(Lutz et al., 1973). With this technique a small iron sphere is driven by a magneticfield in a small volume of sample fluid and its displacement is measured. Using amathematical model the viscosity of the fluid can be obtained. The fact that 0.1 mlis needed makes it unsuitable for measuring lateral line canal fluid viscosity,because only ≈ 0.04 ml can be obtained from one fish.

An alternative use of an oscillating-body is employed by the techniquedescribed in this paper (see Fig. 1). Instead of measuring the angular frequency andtorque of a probe performing torsional oscillations, the resonance characteristicsexhibited by an oscillating sphere performing swinging oscillations in a sample

fluid are utilised. The Q dB3 value of the resonance peak depends upon the sample

fluid's viscosity. The Q dB3 value is given by the ratio f f fh l0 / −b g where f0 is the

resonance frequency, fl and fh are the frequencies at which the response is 3 dB

lower than the maximum with f f fl h

< <0

.

Prior to the determination of the sample fluid's viscosity, a graph ofexperimentally measured Q-factors versus viscosity is made from a series ofglycerol/water mixtures with known viscosities. The viscosity range made up by theglycerol/water mixtures is chosen to cover the expected sample fluid's viscosity.Then the measured Q-factor of the sample fluid is used to find its correspondingviscosity from the graph.

The method of measuring viscosity is based on a comparison method andtherefore no specific hydrodynamic theory of forces on the probe and relatedfrequency response is necessary. The accuracy is improved, however, by fitting amodel through the measured frequency responses. In this way, the resonancefrequency of the oscillating sphere measured in a fluid can be determined moreprecisely. This in turn makes the process of transforming Q-factors to viscositymore accurate. Therefore a hydrodynamic model of the fluid forces on theoscillating sphere is described in the theory section.

Theory

Oscillating sphere

The oscillating sphere probe employed by the viscometer used is shown in theenlarged circle of Fig. 1. It consists of a piezo element driving a wire shaft with a

8

bending stiffness constant S. Attached to the tip of the wire shaft is a small sphere.This is submerged in the fluid of which the viscosity has to be measured. Evokedoscillatory movements of the piezo element are directly coupled to the wire shaft.As the sphere, on the other end of the wire shaft, is impeded by the fluid andbecause the wire is elastic, its displacement is not equal to that of the driver. Thedifference in displacement is determined by the hydrodynamic properties of thefluid, amongst others the viscosity.

A large part of the related theory is based on the work by Stokes (1851) on themotion of pendulums in viscous fluids and on a hydrodynamic model derived byvan Netten (1991) for viscously driven spheres.

Prior to describing the motion of the sphere, a few assumptions must be made.The fluid is incompressible and Newtonian. A fluid is considered to beincompressible for a periodic flow when v « c and τ » l/c, where v is the velocity ofthe fluid, c is the speed of sound in the fluid (1500 m/s), where l and τ denote thedistance and time over which the fluid flow noticeably changes (Landau andLifshitz, 1987). These conditions for an incompressible fluid are satisfied in thepresent case, because the velocity of the sphere at the resonance frequency is ≈ 0.02m/s. Also, the shortest vibration period τ is 1.25 ms and the length over which thefluid flow noticeably changes is ≈ 1 cm, hence τ » l/c.

To linearise the fluid flow equations, the radius, a, of the sphere has to be far

greater than its displacement amplitude, i.e. a »X0 . For the present measurements,

a = 0.5 mm and X0 = 10 µm and thus linearisation is justified.

Consider a wire shaft with a displacement of u t U ei ta f = 0ω driving a sphere in

a viscous, incompressible Newtonian fluid which is at rest at infinity. If the sphere

oscillates with a frequency w and a displacement x t X ei ta f = 0ω , then the total force

exerted by the wire shaft on the sphere is given by:

F t u t x t S Mx t Fwire fluida f a f a fb g a f= − = +&& , (1)

for a » X0 and the wire shaft length L » X0 and with Ffluid (Stokes, 1851; Landauand Lifshitz, 1987) given by:

Fa a

x aa

xfluid = +FH

IK

+ +FH

IK

61

2

96 1

2π µδω δ

π µδ

&& & , (2)

where m is the dynamic viscosity of the fluid, δ is the boundary layer thickness,which is given by:

9

δ µωρ

= 2, (3)

with r being the density of the sample fluid.

Solving for x(t) yields an expression for the displacement amplitude, X0 of thesphere in complex notation:

XP U

Pff

i ff

iff

c

ct t t

00

2 3 219

21

1

2

=

− +FHG

IKJF

HG

I

KJ +

− F

HG

I

KJ +

F

HG

I

KJ

σρ

a f (4)

with

fa

t = µπρ2 2 , (5)

PSa

c = ρπµ6 2 , (6)

and s is the density of the sphere. The parameter ft is equal to the transition

frequency at which the inertial fluid force component starts to dominate over theviscous component for a periodic flow. The Reynolds number ReAC for a periodic

flow is equal to (f/ ft ). Pc is a dimensionless parameter which plays a major role in

governing the shape of frequency response along with s/r, whereas Uo and ft

serve as scaling factors.

According to Eq. 4, resonance occurs when Pc»1. A rough estimation of Pc forthe sense probe used can be made using Eq. 6. The sense probe has a stiffness, S, of ≈ 40 N/m, s = 7000 kg/m3, a radius a = 0.5 mm and m is expected to be between 1

to 10 mPa s. This means that Pc ranges from 104 to 106. As Pc»1 it means that an

approximation of the resonance frequency f0 can be obtained by setting the twomost significant real terms of the denominator of Eq. 4 to zero:

f f Pt c0 9 1 2= +/ ( / )σ ρ . (7)

10

Furthermore, it also means that the shape of the frequency response around theresonance frequency is determined by the first two terms of the denominator of Eq.4. The Q-factor may be approximated by:

QMS

R= , (8)

where M is the mass of the sphere, S is the stiffness of the wire and R is thedissipative fluid force per unit velocity. Although R is frequency dependent, the Q-factor is determined by the damping around the resonance frequency. As the

resonance frequency is ≈ 500 Hz and Pc » 1, R » 6 2π µ δa / (see Eq. 2), because the

factor a/δ » 20 ≥ 1. Substituting M and R into Eq. 8 gives Q as a function of µ at f

= f0 and for Pc » 1:

Qa S

a=

43

3 2

3

20

π σ

π ρµω. (9)

Methods

Laser Doppler interferometer

The displacement of objects moving in the submicron region can be accuratelymeasured using laser interferometry (see Lading, 1971, or Drain, 1980). The setupshown in Fig. 1 is based on the laser interferometer as described by van Netten(1988).

Polarised light emitted from the He-Ne laser (Spectra Physics Stabilite 124B) isfirst rotated by a λ/2 retardation plate to ensure that the laser light is of the correctpolarisation for maximum transmission through subsequent polarisedbeamsplitters. The beam then passes through a pair of lenses L1 and L2, whichform a telescope used for correcting the Gaussian beam/lens interactions (Hanson,1973). Following the telescope, the beam is then split by the beamsplitter BS1 andthe two resulting beams then pass through a pair of Bragg-cells (Isomet 1205c-2);one of which is driven by a R.F. source of 80 MHz and the other by 79.6 MHz.Thus, there is an optical frequency difference of 400 kHz between the two emergingbeams. These are subsequently directed into the prisms P1 and P2 via two mirrors.

11

laserbeam

aira

piezoelement

attenuator

M

BS1BC1 BC2 L1

L2

BS2BS3

MM

M MM

MMO

photodetector

Peltier element

brass trough

f=200 mm

occular

P1 P2

l 4/l 2/

l 2/

S-Pol

P-Pol

S-Pol

Cir-Pol

samplefluid

x(t)

u(t)

wire shaft

Figure 1: Diagram of the laser interferometer viscometer setup. L1 (f=80 mm) and L2 (f=200

mm) are lenses, BS1 BS2 BS3 are beamsplitters, BC1 (80 MHz) and BC2 (79.6MHz) areBragg-cells, M are mirrors, λ/4 and λ/2 are retardation plates, P1 and P2 are prisms, Cir-Pol, P-

Pol and S-Pol indicates polarisation, MO is a microscope objective lens.

After being reflected by a series of mirrors and beamsplitter BS2, the twobeams are focused onto the surface of the oscillating sphere by a microscopeobjective lens (Zeiss Apochromat 40 mm), thus forming a fringe pattern. Thesphere's surface scatters the laser light in all directions as it moves through thefringe pattern. Some of the Doppler shifted light is scattered back in the directionof the objective lens and is collected. It is then focused onto a photodiode via BS2.

12

The laser light intensity is conserved as much as possible by the effective use ofpolarisation. Light falling on the beamsplitter BS2 is P-polarised and passesdirectly through. P-polarised light is then converted to circularly polarised light bythe λ/4 retardation plate as it travels towards the microscope objective lens.Backscattered circularly polarised light returning via the optical axis is converted toS-polarised light by the same λ/4 plate and is now reflected by BS2. This S-polarised light can then be redirected via BS3 towards the ocular or to thephotodiode by adjusting the λ/2 plate located in front of BS3.

The sensitivity of the interferometer can be adjusted by varying the distancebetween the two beams, simply by moving the two prisms P1 and P2 in tandem,backwards or forwards. This has the effect of increasing or decreasing the fringespacing of the fringe pattern as the angle (α) between the beams and the optical

axis at the focus point is varied. The fringe spacing λ λαfringe n

v

f= =

2 sin ∆ is

governed by the angle α, where n is the refractive index and λ the laserwavelength. The velocity of the moving sphere v is then given by the modulationfrequency ∆f multiplied by the fringe spacing.

The photodiode signal is amplified and frequency demodulated by ademodulator (Polytec VIB-VDEC) to give a voltage which is proportional to thevelocity of the sphere. The signal from the demodulator is sampled and averaged bya spectrum analyser (2034, Brüel & Kjær).

Oscillating sphere

A steel sphere (∅ 1 mm) taken from a Parker ball-point cartridge (Parker, finepoint) is used as the sense probe. A hole of 0.3 mm in diameter was drilled throughthe steel sphere and a short length of tungsten wire (9.3 mm in length, ∅ 0.3 mm)was glued in with epoxy resin. This unit was then attached to a piezo element(Physik Instrumente) which was driven by a pseudo-random noise signal of 3-800Hz generated by the spectrum analyser.

13

Calibration fluids

In the present measurement technique, the Q-factor of a measured resonance curveof a sample fluid is compared to that of a series of calibration fluids with knownviscosities. This means that the calibration fluids must have a similar density withrespect to the sample fluid for the resonance frequency to be in the same region (seeEqs. 7 and 9).

The density of the supraorbital lateral line canal fluid is very close to that ofwater (1010 kg/m3, Jielof et al., 1952) and this led to the choice of using differentglycerol/water mixtures. Glycerol-water mixtures have densities close to that of thelateral line canal fluid and, furthermore, viscosity data on glycerol/water mixturesare readily available.

Glycerol/water weight data were taken from the Handbook of Chemistry andPhysics (1979) and converted and plotted as a series of points to give a graph ofviscosity versus glycerol/water volume ratio (see Fig. 2) (ρglycerol =1260 kg/m3).

0.0 0.2 0.4 0.6 0.8 1.0

0

2

4

6

8

10 data points fit using Eq. 10

visc

osity

(m

Pa

s)

glycerol-water volume ratio

Figure 2: Viscosity as a function of glycerol/water volume ratio at 20 °C. The solid line is a fit

made with Eq. 10 through the data taken from the Handbook of Chemistry and Physics (1979).

14

By use of a curve fitting routine an empirical relationship between viscosity andglycerol/water volume ratio was obtained as:

µ = − + −28 2 118 3 117 30 263. . ..x e x , (10)

where µ is the viscosity in mPa s and x is the glycerol-water volume ratio.Using the graph as depicted in Fig. 2, a series of calibration fluids was

produced, starting with a viscosity of 1 mPa s and ending with the final mixture of10 mPa s with increments of 1 mPa s per mixture. This viscosity range was used forpilot experiments to roughly determine the range that covered the sample fluid'sviscosity. After this, a viscosity range with finer steps was used for increasedaccuracy. For these experiments a viscosity range from 1 to 3 mPa s withincrements of 0.25 mPa s was used. For every mix required, the precise volumeratio needed was calculated and with the use of a 1 ml hypodermic syringe precisevolumes of glycerol and water were mixed.

A series of frequency response measurements on glycerol/water mixtures werecarried out at 20 °C, because the standard data in the literature are quoted for 20 °C.The resultant resonance curves were then curve fitted to obtain the Q-factor foreach solution (see Figs. 3 and 4). This was used for the comparison with the Q-factors of resonance curves belonging to the lateral line canal fluid measured attemperatures ranging from 4 to 20 °C. This particular temperature range waschosen, because it is close to the temperatures of the ruffe's habitat.

For each new temperature setting, a period of approximately 2 minutes wasallowed to elapse before new recordings were made. This gave the sample fluidtime to adjust and stabilise to the new temperature setting of the Peltier cooledbrass trough.

Preparation

The fishes, Acerina cernua L. used in these experiments were anaesthetised by I.P.injection with Saffan (Pitman-Moore) (Oswald, 1978). For the present experiments8 fishes were used.

Approximately 15 minutes after applying the anaesthetic, the lateral line canalfluid was extracted from the supraorbital canal using a 1 ml hypodermic syringe. Afish has on average approximately 0.04 ml of lateral line fluid in the supraorbitalcanal.

Prior to the extraction of the lateral line canal fluid, the fish were always takenout of the water tank to avoid mixing the lateral line canal fluid with water externalto the fish.

15

The contents of the syringe was then carefully emptied into a Peltier cooledbrass trough. The oscillating sphere was then lowered into the fluid until it wassubmerged by 0.1 mm similar to the procedure used for the measurement of thecalibration fluid viscosity. All the air bubbles were removed and the depth of thefluid in relation to the oscillating sphere was kept as constant as possible by closeinspection throughout the duration of the experiment.

Results

Typical examples of frequency responses of the oscillating sphere measured insome of the glycerol/water fluids and the lateral line canal fluid are shown in Fig.3. At 20 °C, the fluid with the lowest viscosity (1 mPa s) has the highest peak andthe fluid with the highest viscosity (3 mPa s) has the lowest peak. It is clear to see

that the peaks do not share the same resonance frequency, f0. There is a frequency

shift of 4 Hz from approximately 467 Hz for µ = 1 mPa s to 463 Hz for µ = 3 mPa s.

450 455 460 465 470 475 480 485

-64

-60

-56

-52

-48

-44

1 mPa s (water) 1.5 mPa s 2 mPa s 2.5 mPa s 3 mPa s sample fit

am

plit

ude

(dB

)

frequency (Hz)

Figure 3: Frequency responses of the oscillating sphere measured at a depth of 0.1 mm for a

range of calibration fluids and a sample fluid. All the frequency responses were measured at 20 °C. The solid line shows a fit of the water measurements with a second order filter model.

16

The resonance peak for lateral line canal fluid has a resonance frequency f0 =

466 Hz, which is very slightly lower than that of water (µ = 1 mPa s).The Q-factors of frequency responses measured in a complete set of

glycerol/water mixtures are plotted in Fig. 4 as a function of viscosity (m). The Q-factor of the lateral line fluid converted to a measure of viscosity is indicated by thedotted lines.

0.5 1.0 1.5 2.0 2.5 3.0

25

30

35

40

45

50

measurements fit with Eq. 9

Q-f

act

or

viscosity (mPa s)

Figure 4: Q-factor against viscosity for the different calibration mixtures. The continuous line is afit through the measurement points using equation 9. The values for the fit are σ = 7000 kg/m3, S

= 32 N/m, a = 5×10-4, f0 = 467 Hz and ρ = 1010 kg/m3. The point of intersection of the dotted

lines indicates the Q-factor measured in the lateral line canal fluid at 20 °C and the corresponding

viscosity.

Fitting the data points with equation 9 yields a function which is used forconverting the Q-factor of the frequency response measured in the lateral line canalfluid to the corresponding viscosity at various temperatures. These actualparameters from the fit closely match the parameters based on the physicaldimensions of the oscillating sphere.

Out of the total of 8 specimens used, results taken from 4 fishes are plotted inFig. 5 as a function temperature. Included in this figure are data points for water

17

taken from the Handbook of Chemistry and Physics (1979) and from actualmeasurements on water using the viscometer.

0 5 10 15 20

1.0

1.2

1.4

1.6

1.8

2.0 fish 1 fish 2 fish 3 fish 4 water (HCP) water (measured)

visc

osi

ty (

mP

a s

)

temperature (°C)

Figure 5: Results for 4 different fishes and for water are plotted as a function of temperature.

The continuous curve with small black dots represents the viscosity of water as taken from the

Handbook of Chemistry and Physics (1979).

It is clear that temperature has a significant effect on the viscosity of the lateralline canal fluid. At a temperature of 4 °C, the viscosity is approximately 1.90 mPa s,and it is therefore approximately 0.30 mPa s higher than that of water. At 20 °C thelateral line canal fluid viscosity decreases to a value of around 1.26 mPa s and thisis 0.26 mPa s higher than that of water. This upward shift of approximately 0.30mPa s in viscosity as compared to that of water holds for the entire temperaturerange for which the measurements were conducted.

The reliability of the measurement technique can be gained by themeasurements with water from 5 to 20 °C. A comparison of these results tostandard water viscosity data gives a precise indication of the accuracy of theglycerol/water mixtures and the overall performance of the technique.

The error in the measurement of the temperature is 0.25 °C due to the Peltierelement reacting to the feedback signals sent back by the temperature sensor

18

embedded into the brass trough. The maximum error that can occur in thedetermination of the viscosity is predominantly dependent on the error that canoccur in the measurement of the Q-factor. The maximum error in the measurementof the Q-factor due to the fluid level not being held constant is approximately 5%.This translates to an error of 0.08 mPa s at m = 1.20 mPa s and 0.20 mPa s for m =1.90 mPa s. The error increases with viscosity because at high viscosity a relativelysmaller span in the Q-factor covers a larger viscosity range.

Discussion

The relative simplicity of the measurement technique presented makes it possible tomeasure viscosity of small sample volumes both quickly and reliably. Temperatureeffects can also be easily investigated by using a built-in Peltier element.

It is clear from Fig. 3 that there is a systematic shift downwards in theresonance frequency with increasing viscosity for the different glycerol/water

mixtures. This reflects the dependence of f0 on ρ which changes with the mixtures,see Eq. 7. The difference in resonance frequency between the 1 mPa s solution to

that of the 3 mPa s solution is less than 1% (4 Hz) of f0. This 1% shift in f0 onlyalters the Q-factor by 0.4%, whereas the difference in the densities of theglycerol/water mixtures used contribute to less than 2% difference in Q-factor. Theviscosity is by far the most dominant parameter in determining the Q-factor,therefore no corrections were needed.

It is obvious from the results that the measured viscosity is much lower thanthe previous value of 5.10 mPa s obtained by van Netten and Kroese (1987) for atemperature of approximately 18 °C. For the natural temperature range of the ruffe(4-20 °C) the highest viscosity measured was 1.90 mPa s (4 °C) and 1.20 mPa s forthe lowest (20 °C). Therefore, the difference between the results must be due to anadditional factor other than temperature. The difference may be due to van Netten'smodel not incorporating the canal walls. This leads to an effectively higherviscosity value when fitted through the data of the cupular frequency response (e.g.van Hengel, 1996). A viscosity of 1.2 mPa s gives the best fit to measured data ofcupular motion (van Netten and Kroese, 1987) when the effects of the canal wallsare incorporated into the model. This explanation is further supported by the factthat the flow profile in the lateral line canal is frequency dependent and that it isseverely influenced by the boundary layer of both the bony bridge and canal (Tsangand van Netten, 1997). Therefore, hydrodynamic models of the lateral line organ

19

should not neglect the influence of the integral structure of the bony bridge andcanal walls when they are utilised for determining the viscosity.

Table I. Viscosity of labyrinthine fluids

Author(s) Animal Fluid type

(Endolymph/

Perilymph)

Viscosity

(mPa s)

Tempera-

ture °CTime elapsed

(hours)

ten Kate &

Kuiper (1970)

Pike E

E

1.20 ± 0.08

1.30 ± 0.08

23

22.6

Directly

60

ten Doesschate

(1914)

Cod

Cod (dead)

Haddock

Plaice

E

E

E

E

1.297

1.364

1.220

1.195

0

0

0

0

-

-

-

-

Rauch (1959) Human E

P

1.03-1.05

1.02-1.03

27

27

-

-

Schnieder &

Schindler

(1964)

Guinea Pig P

P

P

E

0.84-0.87

1.15

1.45

1.00

20

20

20

Directly

2

2-24

Directly

Steer (1967) Human

Human

Cat

E

P

P

0.852 ±2%

0.802 ±2%

0.78 ±2%

35

35

35

Directly and

after 1-7 days

The temperature dependence of lateral line canal fluid viscosity is similar tothat of water. The viscosity is, however, approximately 0.3 mPa s greater than thevalue of water for the temperature range of 4-20 °C. Such a dependence ontemperature was also predicted by previous work on the viscosity of pikeendolymph (ten Kate & Kuiper, 1970). Ten Kate & Kuiper only measured at onetemperature (23 °C) but concluded that the viscosity as a function of temperaturecan be described by:

µ =(2.31±0.16)/(1+0.036T+1.85×10-4T2) (11)

where µ is the viscosity and T is the temperature in °C. This function is based onthe equation for water viscosity as a function of temperature (Tietjens, 1960).Equation 11 was then used to compare the pike endolymph viscosity to otherendolymph viscosity data of other species measured at different temperaturesettings. Due to the lack of endolymph viscosity data measured in a single speciesover a range of temperature, data from other species had to be used.

20

Fig. 6 is an overview of the endolymph viscosity data from Table I for differentspecies, lateral line canal fluid of the ruffe and water. Included in this graph is aplot of the viscosity function for water (Tietjens, 1960), a fit of the pike endolymphdata with Eq.11 and a fit of the lateral line canal fluid data with Eq. 11 with thenominator as a free parameter.

0 5 10 15 20 25 30 35

0.5

1.0

1.5

2.0

2.5 ruffe water Eq. 11 ruffe fit Tietjens human Eq. 11 pike human cod (dead) cod haddock plaice pike (dead)

visc

osity

(m

Pa

s)

temperature (°C)

Figure 6: A plot of the endolymph viscosity data from Table I for different species, lateral line

canal fluid of the ruffe and water. Two human endolymph data points are indicated by the star

symbol (Steer, 1967) and the triangle symbol (Rauch, 1959). Included in this graph is a plot of

the viscosity function for water (Tietjens, 1960), a fit of the pike endolymph data with Eq. 11 anda fit of the ruffe lateral line canal fluid data with µ = 2.18/(1+0.036T+1.85×10-4T2).

Fitting Eq. 11 through the viscosity data measured in the ruffe yields anominator of 2.18 instead of 2.31 used for the pike endolymph. This is within theerror margin of the nominator of equation 11. This fit to the lateral line data passesclosely through most of the data points for human, pike, plaice, cod and haddockendolymph. Only the data measured in dead animals do not fit closely. Thissuggests that the endolymph belonging to different species all have similar viscositycharacteristics in relation to temperature. Naturally, different species function atdifferent temperatures and hence the viscosity of the endolymph is dependent uponthis.

21

The relationship between viscosity and temperature is often neglected in thefield of macro mechanical cochlea modelling. Frequently endolymph andperilymph are assumed to have the viscosity of water (1 mPa s). A prediction ofwhat the endolymph in mammals could be, can be obtained by extrapolating the fitthrough the lateral line canal fluid to 37 °C. The viscosity of endolymph at atemperature of 37 °C is expected to be µ = 0.84 mPa s. The fit also gives a viscosityof 0.87 mPa s at 35 °C, which differs by less than 3% from the measured humanendolymph viscosity.

There is clear evidence that time (postmortem) has an effect on the viscosity.Measurements in fluids taken from animals which have been dead for some timeshow a noticeable increase in viscosity. This increase can be as much as ≈ 30 %over a period of 2 hours, as can be seen from guinea pig measurements (Schnieder& Schindler, 1964). A further increase to 66 % was observed in the period between2-24 hours. Furthermore, ten Kate & Kuiper (1970) observed an increase from 1.2to 1.3 mPa s for the pike endolymph after the pike's head has been in therefrigerator for 60 hours and ten Doesschate (1914) also reported an increase forcod endolymph from 1.279 to 1.364 mPa s (time after death is not known). OnlySteer (1967) reported that no observable change occurred in the viscosity of humanendolymph after a day and for up to a week. Increase in viscosity was not observedin the measurements of the lateral line canal fluid because the experiments tookless than 45 minutes to complete. This agrees with ten Kate's & Kuiper's (1970)observation that experiments conducted within 60 minutes after death showed nonoticeable change in viscosity.

One other point of concern is the effect on the lateral line fluid by lowering thetemperature and then increasing it. The first measurements were made at 20 °Cand thereafter the temperature was lowered to 4°C. Further measurements werethen made with the temperature being increased in small steps until 20°C wasreached. Over a duration of around 45 minutes, no significant differences was seenbetween the first measurement at 20 °C and the last measurement at the sametemperature. Therefore, varying the temperature does not seem to produceirreversible effects on the viscosity.

Conclusion

The method employed by our viscometer for measuring the viscosity of a fluid witha volume of 0.04 ml or less has proved to be simple and yet able to deliverreproducible results. The system can be refined further to enhance its sensitivityand accuracy by using smaller spheres and wire shafts with different bending

22

stiffness constants. An additional advantage of this measurement technique is thatit can also be used for in vivo measurements. In vivo measurements were notconducted in the ruffe due to the difficulties of avoiding the contamination of thelateral line fluid with the water around the fish. Moreover, the temperature couldnot be varied with ease.

We can further conclude that the viscosity of the lateral line canal fluid istemperature dependent and ranges from 1.2 ± 0.1 mPa s at 20 °C to 1.9 ± 0.2 mPa sat 4 °C (physiological temperature range of the ruffe). The lateral line canal fluidthus has viscosity characteristics (as a function of temperature) which are verysimilar to those of endolymph from other species.

Throughout the duration (45 minutes) of the experiments, there were nodetectable changes in the viscosity of the lateral fluid due to deterioration caused bytime or by varying the temperature from 20 °C to 4°C and then back again.Although the viscosity did not change over the 45 minutes of measurement time,the viscosity is, however, expected to increase if the measurement period would belonger than 45 minutes.

Finally, the results of this study imply that hydrodynamic models of the lateralline organ must include the effects of the canal walls and bony bridge.

Acknowledgements

We would like to thank J. Land, E. Zevenberg and J. van Maarseveen for their helpwith the building of our apparatus and electronics. In addition we are grateful toProf. Dr. H. Duifhuis for his helpful discussions and comments on improving thismanuscript.

The investigations were supported by the Life Sciences Foundation (SLW),which is subsidized by the Netherlands Organization for Scientific Research(NWO).

References

Dijkgraaf, S. (1963). The functioning and significance of the lateral line organs.Biol. Rev. 38, 51-105.

Doesschate, G. ten (1914). Onderzoekingen gedaan in het FysiologischLaboratorium der Utrechtse Hogeschool 5e reeks XIV. De eigenschappenvan de endolymph in Beenvisschen.

23

Drain L. E. (1980). The laser Doppler technique. Wiley, New York.Fung, Y. C. (1981). Biomechanics mechanical properties of living tissues.

Springer-Verlag, New York.Handbook of chemistry and physics, 60th edition. (1979). R. C. Weast and M. J.

Astle (eds.). The Chemical Rubber Company.Hanson, S. (1973). Broadening of the measured frequency spectrum in a

differential laser anemometer due to interference plane gradients. J. Phys.D: Appl. Phys. 6, pp. 164-171.

Hengel, P. J. W. van (1996). Emissions from cochlear modelling. Thesis,University of Groningen.

Jielof, R., Spoor, A., and de Vries, H. (1952). The microphonic activity of thelateral line, J. Physiol. 116, pp. 137-157.

Kate, J. H. ten and Kuiper, J. W. (1970). The viscosity of the pike's endolymph.J. Exp. Biol. 53, pp. 495-500.

Kelly, J. P., and Netten, S. M. van (1991). Topography and mechanics of thecupula in the fish lateral line. I. Variation of cupular structure andcomposition in three dimensions. J. Morphol. 207, 23-36.

Kestin, J., W. A.Wakeham, (1988). Transport properties of fluids. Cindas dataseries on material properties volume I-1

Lading, L (1971). Differential heterodyning technique. Appl. Opt. 10 pp. 1943-9Landau, L. D., Lifshitz, E. M. (1987). Fluid Mechanics (Pergamon, Oxford),

2nd ed.Lutz, R. J., Litt, M., and Chakrin, L. W. (1973). Rheology of biological systems,

Gabelnick, H. L., and Litt, M. (eds.). Charles C Thomas, Springfield, Ill.,pp. 119-157.

Netten, S. M. van (1988). Laser interferometer microscope for the measurementof nanometer vibrational displacements of a light-scattering microscopeobject. J. Acoust. Soc. Am 83, pp 1667-1674.

Netten, S. M. van (1991). Hydrodynamics of the excitation of the cupula in thefish canal lateral line. J. Acoust. Soc. Am. 89, pp. 310-319.

Netten, S. M. van and Kroese, A. B. A. (1987). Laser interferometricmeasurements on the dynamic behaviour of the cupula in the fish lateralline. Hearing Res. 29, 55-61.

Netten, S. M. van and Maarseveen, J. Th. P. W. van (1994).Mechanophysiological properties of the supraorbital lateral line canal inruffe (Acerina cernua L.). Proc. R. Soc. Lond. B 256, 239-246.

Oswald, R. L. (1978). Injection anaesthesia for experimental studies in fish.Comp. Biochem. Physiol. 60c, pp. 19-26.

Rauch, S. (1959). La biochimie de l'endolymphe et de la périlymphe. C. R. Soc.Franç. oto-rhino-laryng. 56, 238.

24

Schnieder, E. A. and Schindler, K. (1964). In Biochemie des Hörorgans. S.Rauch.

Steer, R. W. (1967). Physical properties of labyrinthine fluids and quantificationof the phenomenon of caloric stimulation. NASA SP-152, pp 409-420.

Stokes, G. G (1851). On the effect of the internal friction of fluids on the motionof pendulums. Trans. Camb. Phil. Soc. 9, pp. 6-106.

Tietjens, O. (1960). Strömungslehre BD I. Springer Verlag.Tsang, P. T. S. K. and Netten, S. M. van (1997). Fluid flow profiles measured in

the supraorbital lateral line canal of the ruff. Diversity in AuditoryMechanics. World Scientific, Singapore.

25

3 Fluid flow profiles measured in the supraorbitallateral line canal of the ruffe

Introduction

Fishes detect water motion by means of a lateral line organ. One type of lateral lineorgan is located in recessed canals on the head of the fish. Water motion outsidethe fish is passed on via skin membranes (van Netten and van Maarseveen, 1994)to canal fluid which in turn drives a number of dome shaped structures, calledcupulae. The cupular motion is sensed by stereocilia of hair cells attached to thebase of the cupula (Kroese and van Netten, 1989). The lateral line organ is easilyaccessible and thus offers the opportunity for the study of a sensitive hair cell organin a relatively undisturbed natural condition.

Cupular motion in response to an oscillating stimulus sphere placed in thecanal has been measured (van Netten and Kroese, 1987), but little data existregarding the fluid flow within the canal. So far, the analyses of lateral linemechanics have neglected the influence of the canal wall assuming the flow drivingthe cupula to be spatially uniform (van Netten, 1991). Measurements of fluidmotion in the lateral line canal of sprats using seeding particles have beenconducted (Denton and Gray, 1983), but these were limited to a single point withinthe canal. Free floating seeding particles suffer from Brownian motion andconvection, making them unsuitable for monitoring canal fluid flow atphysiological stimulus levels. This led to the development of a sense probe thatfollows the fluid motion, with a measuring volume fixed in place and sensitiveenough to detect flow velocities down to the order of 1 µm/s.

To further our understanding of the stimulus transduction in the peripherallateral line organ, the influence of the canal wall on the fluid stimulus was studied.

Methods

The ruffes, Acerina cernua (L), used in these experiments were anaesthetised byI.P. injection of Saffan (25 mg/kg) (Pitman Moore) (Oswald, 1978) and placed in asmall tank where they were fixed in position by head and body clamps.

27

l /2

l

l/4

f=200 mm

waveplatewaveplate

imaging system

attenuator

/2 waveplate

f = 80 mm80 MHz

79.6 MHz

Bragg-cells

f = 120 mm

photodetector

Ruffe ( L.)Acerina cernua

laser beams

stimulus

sense probe

lateral line canal

bony bridge

Figure 1: The laser interferometer used for the velocity measurements is very similar to the one

described by van Netten (van Netten, 1988). The two laser beams emerge from the Bragg-cells

with an optical frequency difference of 400 kHz between them. They are directed via a series ofmirrors through a polarising beamsplitter and l/4 waveplate before finally being brought to focus

upon the surface of the sense probe. The sense probe, driven by the fluid flow, Doppler-shifts the

laser light and some of the scattered light travels back along the optical axis and is focused onto

the photodetector. An image of the sense probe can be redirected to the imaging system byadjusting the l/2 waveplate.

The cephalic lateral line canals consist of skin covered cartilaginous canals,approximately 1 mm deep and 1.8 mm wide, with bony bridges covering the

28

cupulae.For the experiments, a small section of the skin above the supraorbital canal

was removed to expose cupula II (following the notation of Jakubowski(Jakubowski, 1963) and the region just behind cupula I. The experiments describedhere were performed in empty lateral line canals from which the epithelium andcupula II were removed, while the bony bridge above cupula II was left intact.

To measure the fluid flow at a selected position within the canal, a sense probeconsisting of a small sphere (Æ 50 mm) attached to a highly flexible taperedborosilicate glass fibre was used. The glass fibre is around 2.5 mm long and has adiameter of about 4 mm (see Fig. 1). The sense probe is mounted on a x,y,z-positioner and can be positioned at any location in the fluid.

The positions of the canal walls and the bony bridge were determined byscanning the laser beams from one side to the other. The sense probe was thenpositioned in the midpoint of the canal at approximately 0.3 mm caudal from thebony bridge. A small stimulus sphere (Æ 0.67 mm) driven by a piezoelectricstimulator was then placed on the other side of the bony bridge just behind cupula Iin order to produce a sinusoidally changing fluid flow in the canal. The magnitudeof the stimulus sphere’s velocity was kept at 3.14 mm/s over the entire frequencyrange at which we measured. The velocity of the sense probe moving in response tothe evoked fluid motion was measured with a heterodyne laser interferometer atdifferent depths in the canal.

The sense probe was calibrated by measuring its response to a large (Æ 4 mm)stimulus ball vibrating sinusoidally in free field, placed a few mm away from it.The response was found to be frequency independent as expected from theory (vanNetten, 1991). In addition, these calibration measurements confirmed that, at thefrequencies used, the sense probe’s motion is equal to that of the calculated fluidmotion to within a few percent (see Fig. 2).

The fish was artificially respired with tap water throughout the duration of theexperiment. The flow of water was stopped during the measurements to avoidunwanted vibrations caused by the flow of water through the gills. The condition ofthe fish was checked periodically by looking at the blood flowing through a bloodvessel inside the eye. The temperature of the surroundings (15 °C) was keptconstant throughout the duration of the experiment. This slowed the formation ofair bubbles in the fish tank and reduced the production of fish slime in the canal.

29

1.0 1.5 2.0 2.5 3.0

0

20

40

60

80

100

120

measurements theory

am

plitu

de

(µm

/s)

distance from the edge of the stimulus ball (mm)

Figure 2: A plot of the free field water flow velocity as a function of horizontal distance away from

the edge of the stimulus ball, as measured with the sense probe. The stimulus ball has a radius of2 mm and is vibrating at 10 Hz with an amplitude of 6.75 µm. The solid line is the velocity

calculated from theory (van Netten, 1991) and shows that the sense probe's motion is reliably

following the fluid flow.

Results

First, measurements were conducted in free field at the same positions with regardto the stimulus sphere and sense probe as to be used in the canal experiments, toisolate the mechanical effects of the lateral line canal. The amplitude and phase ofthe fluid flow velocity in free field are shown in Fig. 3, they have a fairly flatprofile for all frequencies, as expected from calculations on the fluid field caused bya vibrating sphere in a large fluid volume (van Netten, 1991) for the geometry used.

30

0 200 400 600 800

0

5

10

15

20

25

30

am

plitu

de (

µm/s

)

vertical distance (µm)

10 Hz 30 Hz 70 Hz

0 200 400 600 800

-30

-20

-10

0

10

20

30

phas

e (d

eg)

vertical distance (µm)

Figure 3: A plot of the amplitude and phase of the velocity as a function of vertical distance from

the bottom edge of the stimulus sphere as measured with the sense probe in free field at 2.5 mm

away from the stimulus sphere for different stimulus frequencies. The data from 20, 40, 60 Hz

were not plotted, since they do not significantly differ from the data plotted above.

These characteristics are to be compared with Fig. 4, which show typicalmeasurements in the lateral line canal at various depths and frequencies. For all thefishes investigated, the flow pattern measured in the empty lateral line canal isdependent on the frequency of the stimulus. The smallest separation between thesense probe and the canal floor is approximately 8 mm (allowing room formovement) and together with the radius of the sense probe (25 mm), it means thatmeasurements started at a distance of 33 mm. Clearly, the profiles found in thelateral line canal differ significantly from those measured in free field (Fig. 3),demonstrating that the canal plays an important role in shaping the actual stimulusto the cupula.

0 200 400 600 800

0

5

10

15

20

25

30

am

plitu

de (

µm

/s)

distance from the canal floor (µm)0 200 400 600 800

-30

-20

-10

0

10

20

30 10 Hz 20 Hz 30 Hz 40 Hz 60 Hz 70 Hz

phas

e (d

eg)

distance from the canal floor (µm)

Figure 4: A plot of the amplitude and phase of the velocities measured at different depths in the

empty canal for a range of stimulus frequencies. The distance to the edge of the bony bridge is 0.3

mm.

31

The maximum of the profile in the canal at 10 Hz is found at approximately400 mm above the canal floor. As the stimulus frequency increases, the maximumgradually shifts to a value of about 150 mm at the higher frequencies used. For stillhigher frequencies (results not shown), the profiles do not change significantlyfrom this.

The phase shown in Fig. 4 is very different from that of Fig. 3. There is aphase lead near the canal wall, but as the distance away from the floor is increasedit changes to a phase lag. From 600 mm onwards the phase is heading towards zero,as the edge of the canal is approached (nearing free field). This was found for allthe frequencies at which we measured. An increase in stimulus frequency has theeffect of flattening the phase found at the mid region of the canal.

Discussion

The experimental results clearly indicate the influence of the lateral line canal'sboundary layer. In contrast to measurements made in free field (Fig. 3), the profilesmeasured in the lateral line canal (Fig. 4) not only show a dependence on verticaldistance but also depend on stimulus frequency.

To further characterise the effect of the lateral line canal wall on the fluid flow,the experimental results can be compared with two extreme models that maysimulate the canal wall. One model consists of fluid flowing past an (infinite) plate,while the other model considered is fluid flow inside an (infinitely) long tube withradius r. Clearly, the behaviour of lateral line canal fluid is expected to behave inan intermediate fashion, since the lateral line canal consists of grooves recessed inbone and is, at least in a hydrodynamic sense, partially open because of theexistence of the skin covered openings in the bone. Moreover, duringmeasurements the skin covering these openings was removed.

The fluid velocities in both models can be solved in analytical terms(Schlichting, 1987) for (spatially) constant pressure gradients, and both can becharacterised by an a.c. boundary layer thickness defined by d=(2µ/ωρ)1/2, whichgives a measure of the distance over which the fluid is significantly affected byeither the plate or the tube’s wall. Here, m is the fluid viscosity (1 mPa s for water), w is the stimulus frequency, and r is the fluid density (1000 kg/m3). This yields for δ approximately 180 mm at 10 Hz and 70 mm at 70 Hz.

The a.c. fluid flow past a plate increases monotonically from zero at the plate’ssurface, to the free field value far from the plate. This transition occurs within acharacteristic distance equal to d. This behaviour is unlike the experimental results.

32

The profiles measured show maxima over the canal cross-section, similar to thebehaviour of fluid in tubes. The measured shift of the maximum towards the canalfloor, as frequency increases, follows the pattern in a long tube in which at lowfrequencies the profile is parabolic with a maximum in the centre. At highfrequencies the profile has a ring-like shape with maxima not in the centre but at afrequency dependent distance from the wall given by (Schlichting, 1987; Sexl,1930; Womersley, 1955): Dmax = 2.28⋅d. Substituting the high stimulus frequencyused (70 Hz) results in 150 µm, which is comparable to the position of the maximafound in the lateral line canal at this frequency. Thus, although the canal was openduring the measurements, the profiles measured compare favourably with that of along closed tube.

Conclusions

The novel method we employ for in vivo flow measurement is sensitive enough todetect velocities down to 1 µm/s at a spatial resolution of 50 µm.

The flow profiles obtained for an empty lateral line canal clearly demonstratethat the cupula receives a frequency dependent stimulus, which deviates from a flatprofile. The fluid flow in the canal is comparable to that in a long tube. Thestimulus to the cupula, therefore, is controlled by the boundary layer of the canal.Experiments are now underway to investigate the flow profile and boundary layergenerated by the integral structure of canal wall and cupula.

An interesting consequence of the position of the profile maxima found in thelateral line canal may be that to detect frequencies below 100 Hz, the canal cupulaeshould extend to at least 150 µm from the canal floor into the fluid to benefit fromthe maxima in flow velocity. This condition is certainly met in the case of the ruffewhere the cupulae have an average height of approximately 600 µm.

Acknowledgements

We are very grateful to Prof. H. Duifhuis, P. W. J van Hengel, C. J. Kros, M. P. M.G. van den Raadt and J. E. C Wiersinga-Post for their comments on an earlierversion of this manuscript. The investigations were supported by the Life SciencesFoundation (SLW) which is subsidized by the Netherlands Organisation forScientific Research (NWO).

33

References

Denton, E. J. and Gray, J. A. B. (1983). Mechanical factors in the excitation ofclupeid lateral lines. Proc. R. Soc. Lond. Biol. Sci., 218: 1-26.

Jakubowski, M. (1963). Cutaneous sense organs of fishes. I. The lateral-lineorgans in the stone-perch (Acerina cernua L.), Acta Biol. Cracoveniensia,Zool. 6, 59-82.

Kroese, A. B. A. and Netten, S. M. van (1989). Hair cell transduction. In: Themechanosensory lateral line (Neurobiology and evolution) (ed. by S.Coombs, P. Görner and H. Münz), pp. 265-284. Springer-Verlag.

Netten, S. M. van and Kroese, A. B. A. (1987). Laser interferometricmeasurements on the dynamic behaviour of the cupula in the fish lateralline. Hearing Res. 29, 55-61.

Netten, S. M. van (1988). Laser interferometer microscope for the measurementof nanometer vibrational displacements of a light-scattering microscopicobject. J. Acoust. Soc. Am. 83, 1667-1674.

Netten, S. M. van (1991). Hydrodynamics of the excitation of the cupula in thefish canal lateral line. J. Acoust. Soc. Am. 89, 310-319.

Netten, S. M. van and Maarseveen, J. Th. P. W. van (1994).Mechanophysiological properties of the supraorbital lateral line canal inruffe (Acerina cernua L.) Proc. R. Soc. Lond. B 256, 239-246.

Oswald, R. L. (1978). Injection anaesthesia for experimental studies in fish.Comp. Biochem. Physiol. 60C, 19-26.

Schlichting, H. (1987). Boundary-layer theory (McGraw-Hill PublishingCompany, New York).

Sexl, T. (1930) Über den von E.G. Richardson entdeckten "Annulareffekt".Z.Phys. 61, 349-362.

Womersley, J. R., (1955). Method for the calculation of velocity, rate of flow andviscous drag in arteries when the pressure gradient is known, J. Physiol.127, 553-563.

34

4 Cupular influence on fluid flow in the supraorbitallateral line canal of the ruffe (Acerina cernua L .)

Introduction

Fishes and amphibians have a lateral line system for the detection of water motionwithin the close vicinity of the animal (Dijkgraaf, 1963). The cephalic part of thelateral line system is situated on the head and in certain species of fish it consists ofrecessed bony canals with a series of protective bony plates arching over the canal,called bony bridges. Situated under the bony bridges are the sensory units, orneuromasts. A neuromast consists of sensory hair cells grouped together with theirstereocilia anchored in a dome shaped extracellular matrix (Kelly and van Netten,1991), called cupula.

Lateral line canal organs share a common working principle. Lateral line canalfluid couples the water motion outside of the fish to the cupula, thus driving thesensory hair cells, which code the mechanical information passed on to them intoaction potentials which are sent to the brain (Dijkgraaf, 1963; Kroese and vanNetten, 1989). To understand the process from water disturbance to thetransduction of the stimulus by the hair cells, the stimulus that is conveyed to thecupula must be known. Specifically, the interaction of the boundary layer close tothe cupula and that in the vicinity of the canal wall is of interest.

Morphological differences in cephalic lateral line organs can be found betweenthe many species of fish (Coombs et al., 1988). These variations include differencesin canal size, ranging from approximately 0.1 mm up to 7 mm (Denton and Gray,1988: Münz, 1989), canal shape and bony bridge covering. In addition to this, theshape of a neuromast can vary greatly between species (Pumphrey, 1950). It isexpected that these variations in the lateral line organs lead to differences in theflow of lateral line fluid and the stimulation of the neuromast (Denton and Gray,1983; 1988; Coombs and Montgomery, 1992; van Netten, 1991; Wiersinga-Postand van Netten, in preparation).

It has been proposed that the velocity profiles within the lateral line canal aresimilar to the flow profiles in pipes and thus are controlled by the boundary layersof the wall (Denton and Gray, 1988; van Netten, 1991). In circular pipes, anoscillatory flow profile is characterised by the (linear) Reynolds number for

periodic flow, R rAC = 2ωρ µ/ (Batchelor, 1967; Schlichting, 1987), where r is the

radius of the pipe, ω the angular frequency of the oscillatory flow, ρ the fluid

35

density and m the fluid viscosity. At low frequencies (RAC «1), at which the flow isdetermined by viscous fluid forces, the flow has a parabolic profile across the

diameter of the pipe. At high frequencies (RAC »1), the flow profile takes on anannular form (Sexl, 1930; Womersley, 1955).

It has been experimentally demonstrated (Tsang and van Netten, 1997) thatthere are indeed many similarities between oscillatory flow profiles in thesupraorbital lateral line canal of the ruffe (Acerina cernua L.) and those in circularpipes. In those experiments, the cupula was removed to investigate the influence ofthe canal and bony bridge on the flow in the canal. The flow in the vicinity of thebony bridge was found to be comparable to the flow in a pipe, in showing a gradualchange from a parabolic like profile to an annular profile with increasingfrequency. The frequency characteristics of the flow were clearly found to becontrolled by the canal wall, similar to the flow behaviour found in pipes.

As expected, there are also differences between the measured flow in the lateralline canal and the flow in a pipe. These differences were most pronounced at thetop of the canal, away from the bony bridge region, where under intact conditionsno overlying bone is present and only skin covers the canal. It has been suggestedthat these boneless regions of the canal wall, termed hydrodynamic windows,permit the transmission of fluid motion outside the fish into canal fluid motion (vanNetten and van Maarseveen, 1994). Obviously, the absence of rigid bone in thecanal wall causes the velocity profile close to the top of the canal to differ from theflow profiles found near the wall of a completely closed pipe. In addition, in theexperimental situation used for those flow measurements, the skin covering thewindows was removed, to facilitate the insertion of a sense probe. From previouswork (van Netten and van Maarseveen, 1994) it is known that the skin overlyingthe hydrodynamic windows is very compliant and that removal is not likely toaffect the velocity distribution dramatically. The results obtained under theexperimental conditions may therefore be expected to give a reasonablerepresentation of the normal flow profiles.

As a continuation of the flow measurements of the lateral line fluid in the(empty) supraorbital lateral line organ of the ruffe (Acerina cernua L.), this chapterdeals with the situation in which the cupula is present. This provides informationon the interaction between the boundary layers controlled by both the cupula andthe canal wall. For instance, it shows to what extent the stimulus received by acupula is affected by the presence of neighbouring cupulae. It thus givesinformation on whether an array of cupulae in the lateral line canal work in adependent mode, controlled by the hydrodynamics of the canal and other sensoryunits, or merely as individual fluid flow detectors.

36

Methods

Preparation

Thirty-nine ruffes (Acerina cernua L.) were used for studying radial andlongitudinal canal fluid flow profiles in the supraorbital lateral line canal. The ruffehas been the subject of various studies in which the mechanosensitivity of thelateral line system has been investigated in relation to its peripheral hydrodynamicsand hair cell mechanics (e.g. van Netten, 1997). Fish were anaesthetised with anintraperitoneal injection of Saffan (Pitman Moore, 25 mg/kg body weight) andplaced in a fish tank with tap water, where they were artificially respired and heldrigidly in place with head and body clamps.

The right supraorbital canal was opened by removing a small piece of skincovering the canal in between cupula I and III (numbering follows Jakubowski,1963), see Fig. 1, to enable the placement of a stimulus sphere and the sense probefor detecting fluid velocity.

Figure 1: A simplified

diagram of the supra-

orbital lateral line canals

of the ruffe showing the

locations of the cupulae

(according to

Jakubowski, 1963). For

these experiments, the

flow profiles are all

made in the region

caudal to cupula II as

shown by the dotted

line. The bony bridges

are not shown.

During a measurement the artificial respiration to the ruffe was stopped (≈ 2minutes) to reduce unwanted vibrations. It was turned on in between the

measurements (≈ 4 minutes). To avoid the water from degassing and to reducefluctuations in the lateral line canal fluid viscosity (Tsang et al., chapter 2), the

37

temperature of the surroundings was kept constant at 15°C. Excess slime aroundthe head was removed, preventing the slime from flowing into the lateral line canalduring the measurements, which otherwise would engulf the sense probe.

Mechanical stimulation

A small piezo-electrically driven stimulus sphere (Æ 0.68 mm), placed at about 3.5mm rostral from the caudal edge of bony bridge II was used to produce oscillatoryfluid flow in the lateral line canal. The stimulus sphere was moved with a constantvelocity amplitude (22 mm/s for radial profiles and 30 mm/s for longitudinalprofiles) in the direction of the longitudinal axis of the canal with frequenciesranging from 10 to 500 Hz. The effect of the boundary layer of the sphere wascorrected for in the measured velocity profiles.

Sense probe for fluid flow measurement in the lateral line canal

Measuring fluid flow with a spatial resolution down to the order of tens of micronsrequires a sense probe of comparable dimensions. Additional requirements of thesense probe include sufficient sensitivity to oscillating flow, down to the order of 1 mm/s in the relevant physiological frequency range of the ruffe. For in vivomeasurements further restrictions are imposed by the relatively small dimensionsand varying geometry of the lateral line canal.

These conditions can be met by using a measurement technique speciallyadapted to this problem, as described by Tsang and van Netten (1997). Thistechnique makes use of a sense probe which consists of a micro-pendulumsuspended from a very flexible glass fibre shank (see Fig. 2). This arrangement isdesigned to simulate a seeding particle that maintains a stable equilibrium position,and yet is allowed to move as freely as possible with the fluid. It also offers thepossibility to measure at selected points within the lateral line canal.

38

glass shank

resin

resin sphere

glass fibre

Figure 2: Diagram of the

sense probe used for

measuring the fluid flow

inside the supraorbital lateral

line canal. The resin spherediameter (∅ ≈ 50 µm), glass

fibre thickness (∅ ≈ 3 µm)

and length (l ≈ 2 mm) are not

drawn to scale.

To determine the motion of the small sphere at the tip of the sense probe, alaser interferometer was used (see Fig. 3). The operation of the laser interferometeris described in more detail in chapter 2. Basically, the fluid flow is followed by thesense probe’s sphere which scatters the laser light as it moves through the (moving)fringe pattern (Æ 20 mm) formed by the two laser beams at their point of focus. The

laser beams

stimulus

nerve

microscope objective

lateral line canal

ruffe ( L.)Acerina cernua

glass shank

resin sphere

bonybridge

cupula

x

y

z

Figure 3: The two laser beams having an optical frequency difference of 400 kHz are brought to

focus upon the surface of the sense probe. The sense probe can be moved in tandem with the

microscope objective lens in the vertical direction (y-axis) with a micrometer. Velocity profiling in

the horizontal plane direction is carried out by moving the fish in the x-z direction. For a more

detailed description of the laser interferometer used for the velocity measurements, see chapter 2.

sense probe was attached to an x, y, z-manipulator fixed to the support of theobjective lens. Using the manipulator, the surface of the sense probe’s sphere waspositioned exactly in the fringe pattern. This ensured continuous back-scattering of

39

the laser light at any depth of focus. The scattered Doppler-shifted light wasdetected and converted to an instantaneous velocity signal using a modifiedfrequency demodulator (Polytec).

Response velocity signals were low-pass filtered using an 8 pole Butterworthfilter (Frequency Devices) set at 8x the stimulus frequency before being digitisedwith a 16 bit A/D converter (Ariel, DSP-16) at a sampling frequency of 64 timesthe stimulus frequency. Responses consisting of 32 periods were usually averaged15 times, from which amplitude and phase of the harmonic component of the fluidvelocity at the stimulus frequency were calculated based on a FFT.

Radial and longitudinal mapping of fluid flow

The sense probe and the stimulus sphere were lowered into position as depicted inFig. 4. Care was taken not to come into contact with the cupula or canal wall toavoid damaging the sense probe and to avoid picking up particles.

canal floorepithilium

radial

sense probe

longitudinal

hair cells

cupula II

bony bridge

stimulus sphere

skin skin

Figure 4: Diagram of the supraorbital lateral line canal of the ruffe and experimental set-up. The

lateral canal fluid is driven by the stimulus sphere and oscillates in the longitudinal direction. Radial

velocity profiles consist of measurements of the longitudinal velocity of the lateral line fluid flow at

different depths above the canal floor. Longitudinal velocity profiles are conducted as a function of

distance away from the cupula at a height of 0.2 mm above the canal floor.

Radial profiles were obtained by moving the sense probe up and down withrespect to the fish and measuring the longitudinal velocity of the lateral line fluid.Measurements started at a distance of 75 mm above the canal floor and wererepeated with steps of 0.1 mm to the top of the canal at about 0.8 mm above thecanal floor.

Longitudinal velocity profiles were produced by moving the fish in thehorizontal plane and started at approximately 0.3 mm from the cupula and were

40

repeated with a step size of 0.05 mm within the close vicinity of the cupula and 0.1mm further away up to a distance of 1.3 mm. At each measurement position afrequency response was determined usually using 20 stimulus frequencies rangingbetween 10 and 100 Hz or 70 and 500 Hz.

Radial and longitudinal velocity profile measurements were conducted inseparate fishes for each set of profiles. Due to the time needed for mapping thegeometry of the canal and carrying out each set of measurements it was not possibleto obtain a complete set of radial and longitudinal velocity profiles in the same fishbefore the canal was filled with slime.

Sense probe fabrication

The sense probe consists of a glass fibre made from the tapered tip of amicropipette pulled from a borosilicate glass capillary with an outer diameter of 1.5mm and inner diameter of 0.86 mm using a micropipette puller (Flaming BrownP97). Micropipettes were produced to have tips which taper down from ≈ 20 µm to ≈ 3 µm over a distance of 20 mm. The last 2 mm of the tapered tip (Æ < 5 µm),which is extremely flexible, is used for the sense probe.

Measuring the flow with the glass fibre alone would give a weighted averagevelocity of the flow experienced along the total length of the glass fibre. In order tolocalise the measurement region to the tip, a small sphere (Æ 50 mm) is added tothe glass fibre’s tip. This sphere is formed by dipping the glass fibre’s tip into adrop of Epoxy resin (Bison). The size of the resin sphere that is formed at the glasstip is determined by the length of the tip that is inside the resin, which was chosento be approximately 50 mm. Once the resin sphere has been hardened the other endof the glass fibre is glued onto a supporting glass shank.

Sense probe calibration

An ideal probe should move exactly with the flow to be measured, thus having avelocity sensitivity of unity over the frequency range of interest. The probe'sfrequency response, however, is high-pass filtered, as a consequence of itshydrodynamic interaction with the fluid. At low frequencies the motion of theprobe is attenuated as the probe is viscously driven. Then, the viscous fluid forceson the probe mainly balance the elastic bending force, resulting in the probe havinga velocity proportional to the frequency of the oscillatory flow. At relative high

41

frequencies the probe will move in unison with the fluid, since it has a densitywhich is similar to that of the fluid and is predominantly driven by inertial fluidforces. An estimate of the cut-off frequency of the high pass behaviour can beobtained from an expression derived for the motion of an elastically suspendedspherical pendulum driven by an oscillatory viscous fluid flow. The 3 dB cut-off

frequency (fco) of such a pendulum is proportional to the stiffness S of the glass

fibre according to: fS

aco ≈

12 2π µ (van Netten, 1991), where a is the radius of the

sphere and m the dynamic viscosity of the driving fluid. It can be seen that fco canbe lowered by decreasing S or increasing a. In practice, a is kept small to maintainthe required spatial resolution (≈ 25 mm) while a minimum value of the stiffness, ofthe order of 10-4 N/m, is imposed by the construction of the glass fibre but also byavoiding detection of thermal noise (see below). This yields cut-off frequencies inthe range between 10 and 100 Hz in water, which covers the lowest frequenciesused in this study. In practice, it means that for each individual sense probe acalibration of its frequency response is needed.

The calibration of the sense probe’s frequency response involves measuring thesense probe's velocity while submerged in a tank of water, in response to a stimulussphere (Æ 0.68 mm) oscillating at various frequencies with a constant velocity (22mm/s), placed at a distance of approximately 2.5 mm from the probe. All velocityprofiles of the lateral line canal are presented including a correction for thefrequency characteristics of the sense probe used.

A typical example of a calibration measurement is shown in Fig. 5. The pointsshow the amplitude and phase of the velocity of the sense probe to a constantvelocity flow and confirm the expected high-pass behaviour of the probe. The solidlines depict the calculated frequency response (van Netten, 1991) of the sense probebased on a stiffness S = 3.6 10-4 N/m and an effective radius a = 33 mm (µ = 1

mPa s) and density ρ = 1000 kg/m3). The related cut-off frequency (≈ 90 Hz) is

close to the lowest frequency used in this study (70 Hz). It is clear that correction ofthe measured data is especially required with respect to the phase at lowfrequencies.

Noise of the sense probe

A physical restriction of the sense probe’s sensitivity to detect low fluid flowvelocities is related to the probe’s low stiffness and the thermal motion of the fluid.

42

An estimate of the probe's rms displacement, x, due to Brownian motion can be

obtained using x kTS= , where k is Boltzmann’s constant and T is the absolute

temperature. At room temperature this yields x ≈ 3.5 nm. Using the fluctuation-dissipation theorem and assuming a friction coefficient equal to 6πaµ, the relatedpower density of the sense probe’s velocity can be estimated to be about 10-15

(m/s)2/Hz (logarithmically symmetrically) distributed around 250 Hz with a fullfrequency width at half maximum power of about 700 Hz. This yields an rmsvelocity of the order of 1 mm/s. This figure is close to the equivalent (rms) noise

velocity (≈ 1 mm/s) of the frequency demodulator used and is also similar to thepractically found threshold value of the measuring system.

10 1000

30

60

90500

500

pha

se (

deg

)

frequency (Hz)

10 10010

100

measurements theory

am

plit

ude

(µm

/s)

Figure 5: Measured

frequency response of a

sense probe (circles) in

response to an oscillating

sphere producing a

constant fluid velocity. A

fit to the measured data

of an equation describing

an elastically coupled

spherical pendulum (van

Netten, 1991) is depicted

by the solid line.

Parameters used: glassfibre’s stiffness S = 3.6

10-4 N/m; sense probe

radius a = 33 mm; fluid

viscosity µ = 1 mPa s;

fluid density ρ = 1000

kg/m3 (Pc = 0.64; ft =

142 Hz). The related cut-off frequency (fco = Pc

ft) is 91 Hz.

43

Results

Radial flow profile

Radial velocity profiles as a function of height above the canal floor and at severalfrequencies are shown in Fig. 6 and are representative for the measurement setobtained (N=25). The approximate positions of the stimulus, bony bridge, cupulaand projection of measurement points with respect to each other in the particularfish used for the measurement are shown in Fig. 7.

0 200 400 600 8000

30

60

90

pha

se (

deg)

distance from the canal floor (µm)

0 200 400 600 8000

20

40

60

80 70 Hz 115 Hz 160 Hz 200 Hz 300 Hz

am

plitu

de (

µm/s

)

Figure 6: Amplitude

and phase of the

radial flow velocity

and phase profiles

measured in the

lateral line canal at adistance of »0.3 mm

away from the edge

of the cupula for a

range of stimulus

frequencies.

44

Close to the canal floor, the flow can be seen to be highly frequency dependent.At low frequencies (70 and 115 Hz), the flow is significantly less than at higherfrequencies (> 200 Hz). At a stimulus frequency of 200 Hz there is a flowmaximum at a height of approximately 150 µm above the canal floor. Thefrequency at which a maximum occurred was variable among the fish used andranged from 180 to 300 Hz. Towards the top (800 µm) of the canal, nearing theopen water outside of the fish (free field), all of the profiles converge to a velocityamplitude of approximately 15 to 20 µm/s.

Figure 7: Diagram of the lateral

line canal for the ruffe used for the

radial profile measurements close

to the cupula. The distances

between the stimulus (S), bony

bridge (BB), cupula II (C) and the

position of the projected measure-ment points (×) are also shown.

The phase profiles show that the flow leads the stimulus at all frequencies,most significantly at low frequencies. At a frequency of 70 Hz there is a phase leadof 40° for the lower section of the canal. The phase lead increases rapidly withincreasing stimulus frequency above 70 Hz. The largest phase leads, up to 90°,were usually observed at intermediate frequencies. At the highest frequencymeasured (300 Hz) the phase lead is small and the lateral line canal fluid thusflows almost in phase with the stimulus. Moving to positions near the top of thecanal (heading towards free field) causes the phase lead to be confined within therange of zero to 20° at all frequencies, showing that at those positions the flow isalmost in phase with the stimulus. It can thus be concluded that close to the canalfloor the flow deviates most severely from the stimulating fluid, most significantlyat the lowest frequencies, while at the top of the canal the fluid flow is in phasewith the stimulus fluid flow.

Frequency response of fluid flow close to the cupula

Fig. 8 shows more detailed information on the frequency response of fluid velocityclose to the cupula and the canal floor. Measurements were made at a distance of0.3 mm away from the cupular edge, at a height of 0.2 mm above the canal floor intwo different fishes. The flow profiles have a plateau of fairly constant velocity

45

between 70 to 150 Hz. Results at frequencies below 70 Hz were found to be similarto those found at 70 Hz. Amplitudes reach maxima at around 250 Hz. At higherfrequencies the velocity amplitude slowly decreases, approaching another constantvelocity plateau. There is a significant phase lead at frequencies ranging from 100to 250 Hz. At higher frequencies (300 Hz) the phase lead becomes small, indicatingthat the flow is nearly in phase with the stimulus.

To directly illustrate the influence of the cupula on the flow in the lateral linecanal, the flow measured in an empty lateral line canal, at approximately the samedistance from the stimulus sphere as in the case with the cupula present (Fig. 8), isshown in Fig. 9. The velocity measured is approximately constant (50 mm/s) overthe frequency range of 70 to 500 Hz. The flow is in phase with the stimulus from70-200 Hz; from 200-500 Hz there is a slight phase lead of up to 10°. The constantamplitude and (almost) zero phase are expected in free field, but are also in linewith the equations of the Sexl/Womersley profile (Sexl, 1930; Womersley, 1955) ina closed pipe with the dimensions of the canal (Æ ≈ 1 mm). The cut-off frequencyabove which the fluid flow in a pipe is in phase with the evoking stimulus flow is

given by fd

co = 22

µπρ

, with fluid viscosity µ, fluid density ρ and pipe diameter d,

which yields about 1 Hz, if applied to the lateral line canal. Therefore, in an emptycanal no frequency selectivity is expected within the measurement frequency range.

The comparison with Fig. 8 shows that, especially at low frequencies, the fluidflow in the canal is significantly attenuated by the presence of the cupula. These arethe frequencies at which the cupula is known to be only excited slightly and movesless than the evoking fluid flow. The attenuation of the flow at low frequencies cantherefore be attributed to the blocking effect of the cupula.

To further investigate the blocking effect of the cupula on the fluid flow, asimplified model was used to describe the frequency responses measured. In themodel it is assumed that the flow is the sum of the imposed stimulating fluid flowand the disturbance of a sphere driven by it (see Discussion). The sphere isassumed to mimic the cupula having a frequency dependent velocity as calculatedwith a model of cupular excitation (van Netten, 1991). The effects of the walls onthe flow are neglected in the model.

46

50 70 100 200 300 500 700

0

30

60

90

ph

ase

(de

g)

frequency (Hz)

50 70 100 200 300 500 700

10

100

20

am

plit

ud

e (

µm/s

)

fish 1 fish 2 model

50

50 70 100 200 300 500 700

10

100

50

am

plit

ud

e (

µm/s

)

50 70 100 200 300 500 700

0

30

60

90

20

ph

ase

(de

g)

frequency (Hz)

Figure 8: Amplitude and phase of fluid

velocity measured in the supraorbital lateral

line canals of two ruffes at a distance of

approximately 0.3 mm away from the edge of

the cupula at a height 0.2 mm above the canal

floor. Fits to the measured data made with the

model described in the text are depicted by the

solid lines. Parameters used for the data

depicted with the open circles: cupula slidingstiffness S = 0.163 N/m, canal fluid viscosity µ= 6.66 mPa s, cupula radius a = 2.04´10-4 m

and the distance to the cupula centre r = 2.42´

10-4 m, resulting in a Pc = 39.79 and ft = 25.46

Hz. Parameters used for the data depicted

with the filled circles: cupula sliding stiffness S= 0.163 N/m, canal fluid viscosity µ = 11.62

mPa s, cupula radius a = 1.67´10-4 m and the

distance to the cupula centre r = 2.15´10-4 m,

resulting in a Pc = 10.69 and ft = 66.25 Hz.

Figure 9: Amplitude and phase of fluid velocity

measured in the supraorbital lateral line canal

without cupula at a height of 0.2 mm from the

canal floor and at a comparable distance from

the stimulus sphere as the case presented inFig. 8.

47

The solid lines in Fig. 8 show the results of the model calculations. Fitting theequation of the fluid velocity to the amplitude and phase data sets involvesessentially 3 free parameters. If the elastic coupling of the cupula to the canal flooris fixed at 0.16 N/m, according to previous results from independent measurementson cupular mechanics (van Netten, 1991; Wiersinga-Post and van Netten, 1998),the free parameters are: the cupular radius a, the canal fluid viscosity m and thedistance between measurement position and the centre of the cupula r. The modelgives an accurate description of the two data sets depicted with a cupular radius aequal to 0.2 and 0.17 mm, with canal fluid viscosity m equal to 7 and 12 mPa s andthe distance to the cupula, r, set to 0.24 and 0.22 mm, for respectively the data setrepresented with open and closed symbols in Fig. 8. The variation in theparameters found are likely to be related to geometrical variations in thedimensions of the lateral line organ. The discrepancy between the values found for

r and the actual distance at which was measured (≈ 0.9 mm) suggest that the canalhas a significant confining effect on the flow measured close to the cupula (seeDiscussion).

Longitudinal flow profile

Longitudinal profiles were measured at a height of 0.2 mm above the canal floor(N=14) and all gave basically the same results. A typical example is shown in Fig.10. In this particular fish the approximate positions of the stimulus sphere, thebony bridge, cupula and range of measurement points with respect to each other areshown in Fig. 11.

Since at high frequencies (300 Hz) the cupula is not attenuating the flow, thelongitudinal profile measured at this frequency indicates the decay of the stimulusalong the length of the canal, if no cupulae were present. For comparison, a curvehas been added that shows the theoretically expected reduction of the flowamplitude of a dipole (vibrating stimulus sphere) in free field with respect to thedistance to its centre, rstim, which is proportional to (rstim)-3. Its similarity confirmsthat neither the cupula nor the canal wall has a significant effect on the fluid flowalong the canal at this frequency. It also predicts that with this particular geometry,cupula III, which is approximately positioned at 3 mm from cupula II, would detectabout 10% of the stimulus detected by cupula II.

48

0.0 0.5 1.00

30

60

90

pha

se (

deg

)

distance from cupular edge (mm)

0.0 0.5 1.00

10

20

30

40 70 Hz 115 Hz 160 Hz 200 Hz 300 Hz

~(rstim

)-3

am

plit

ud

e (µ

m/s

)

Figure 10: Amplitude and

phase of longitudinal flow

velocity profiles measured

at a height of 0.2 mm

above the lateral line canal

floor at several frequen-

cies. The theoretical flow

velocity as a function of

distance from the stimulus

ball in free field is also

included in the plot and is

depicted by the dotted line.

The edge of the cupula is

located at x = 0 and the

caudal edge of the bony

bridge is at approximately x

= -0.9 mm.

Figure 11: A diagram of the lateral

line canal for the ruffe used for the

longitudinal profile measurements

close to the cupula. The distances

between the stimulus (S), bony

bridge (BB), cupula II (C) and the

range of the measurement points is

indicated by the thick dotted line.

At the lower frequencies it can be seen that the influence of the cupula, via itsblocking action on the flow, is present along the whole trajectory measured,

49

although its amplitude also reduces with distance. This is in accordance with thephase, which remains relatively close to zero for the high and low frequencyplateaus measured close to the cupula but approaches to phase leads up to about 40°at frequencies which evoke a phase lead deviating significantly from zero atpositions close to the cupula (see also Fig. 8, frequency response close to cupula).The reduction along the canal length of the disturbance of the flow caused by thecupula (at low frequencies) shows that cupula III is only slightly affected by thepresence of cupula II.

Frequency response close and further away from the cupula

100-15

0

15

30

45 70 200 300 500

50070 300200

ph

ase

(de

g)

frequency (Hz)

1005

10 0.3 mm 1.3 mm

50

am

plit

ude

(µm

/s)

Figure 12: Amplitude and

phase of flow velocity

measured at distances of

0.3 mm and 1.3 mm away

from the edge of cupula II.

All the measurements

were conducted at a height

of 0.2 mm above the canal

floor.

More detailed frequency characteristics of the flow close (0.3 mm from edge of thecupula) and further away (1.3 mm) are compared in Fig. 12. The difference in thelow and high frequency amplitude plateaus, reminiscent of the presence of the

50

cupula, as described for locations close to the cupula in Fig. 8 [Frequency responseclose to cupula], is also present at larger distances, although the difference betweenthe plateaus reduces with distance. The difference in amplitude is merely a constantreduction at all frequencies. The phase response is nearly the same at the twodistances shown, except from some slight deviations at low frequencies. In linewith the notion from the longitudinal profiles it can be concluded that a(diminishing) fraction of the disturbance caused by cupula II is propagated alongthe canal.

Discussion

The results presented show that the flow measuring method used, consisting of aflexible sense probe in combination with laser interferometry, enable the in vivomeasurement of fluid flow in the lateral line canals of fish down to the order of 1µm/s, with a spatial resolution of the order of tens of micrometers. This method hasbeen utilised to map the fluid velocity along two principal directions in thesupraorbital lateral line canal of ruffe.

Several observations have been made that provide information on the effect ofthe presence of a cupula and the canal wall on the flow of canal fluid. Directexperimental comparison between two experimental situations, in which fluid flowwith and without a cupula was measured, show that the cupula influences the flowsignificantly in the lateral line canal at distances of the order of a cupular diameter(see Figs. 8, 10 and 12). This influence is most significant at frequencies which arebelow the resonance frequency of the cupula (≈130 Hz; Wiersinga-Post and vanNetten, 1998). The cupula is only slightly excited at those frequencies and thusblocks the flow of canal fluid. At frequencies above the resonance frequency,stiffness forces of the elastic coupling via the hair cell bundles to the canal floor arenegligible, resulting in a neutrally buoyant behaviour in which a cupulamechanically behaves like the canal fluid. This means that at these frequencies thecupula is mechanically transparent to flow in the canal.

From modelling the cupula as a sphere driven by a viscous fluid, an expressionwas derived that describes the resulting fluid flow along the direction of vibration(longitudinal canal axis). This model is based on the frequency response of the

velocity of a cupula, Vcup, in response to a spatially constant fluid stimulus with

velocity Vstim:

51

Vi f f

if f f f

P i f fi

f f f fVcup

t t t

c t t t

stim=+ − −

+ + − −

b ga f

b g b g

b ga f

b g b g

1

2

13

1

2

13

3 2 2

3 2 2

/

/,

with fa

t = µπρ2 2 and P

Sac = ρ

πµ6 2 , (1)

where S is the elastic coupling of the cupula to the canal floor, and a is the cupularradius, while m and r are fluid viscosity and density (van Netten, 1991). Theresulting fluid flow at a distance r from the centre of the sphere is then given by thesum of the spatially constant stimulus fluid flow and the disturbance relative to thisstimulus flow as caused by the presence of the sphere of this field:

V r Va

rV Vstim cup stima f d i= + −

3

3 . (2)

The second term, associated with the disturbance, is based on the potential flowalong the direction of vibration of a sphere vibrating with a small amplitude (e.g.Harris and van Bergeijk, 1962). This is valid because measurements wereperformed outside the boundary layer of the cupula at the frequencies used.Equation 2 adequately describes the measured data close to the cupula, as shown inFig. 8. The resulting fit parameters, however, deviate from the real known physicalvalues. The values found from fitting Eq. 2 to the data yields for the viscosity, 7-12mPa s, which is 5 to 10 times higher than the experimentally determined value (1.3mPa s at 15 °C, see chapter 2). Also, the range found from fitting for the distancefrom the centre of the cupula to the point of measurement (0.17 - 0.2 mm) is abouta factor 5 lower than the real distance (0.9 mm) (see Fig. 11). This discrepancy forboth viscosity and distance can be explained by the effect the canal wall has on thefluid flow. It is known from low Reynolds number hydrodynamics that the dragforce exerted by a viscous fluid flowing with velocity V past a rigid sphere in a

circular container, as given by Stokes' law, Fdrag = 6pamV, has to be corrected by a

factor larger than 1, depending on the distance between sphere and container wall(Happel and Brenner, 1983). This factor can be interpreted as an increase of theviscosity and amounts to about 10 if the canal wall is located 0.3 mm from thesphere’s surface, which is a realistic estimate for the distance between cupularsurface and canal wall. Also, the smaller distance found while fitting for r whenusing a free field model reflects the confining action of the canal on the flow. Itseems therefore likely that the influence of the cupula on the flow caudal to it, as

52

measured at low frequencies, is enhanced by the presence of the canal wall. Theeffect of the canal wall on the flow further away from the cupula and bony bridge,which is the region where the canal is not covered with bone, is less. In fact, in thisregion and at high frequencies, the decay with distance is similar to the dependency(r-3) found in free field. At these locations the filtering effect of the cupula(blocking of low frequencies) is still present but substantially reduced. In thisrespect, the presence of hydrodynamic windows between cupulae (van Netten andvan Maarseveen, 1994) make the neuromasts in the ruffe’s supraorbital lateral linecanal an array of relatively independent motion sensors. Placed in a closed tubewith the same dimensions, the neuromasts would all detect exactly the same flowsince the fluid velocity in a tube is essentially forced back into the Sexl/Womersleyprofile within a distance of a cupular diameter (Meeuwissen, 1994).

The frequency responses of fluid flow, also those measured close to the cupula,as presented in Fig. 12, differ significantly from the frequency responses asmeasured for cupula II (Wiersinga-Post and van Netten, 1998). At low frequencies,a plateau in velocity amplitude was always found, contrary to the velocity of thecupula which approaches zero. At high frequencies, both cupula and flow willattain the same frequency dependent velocity, since then the cupula is mechanicallythe same as the fluid. The difference between the velocity plateaus of the fluid

reached at the two frequency limits can be seen from Eq. 2 to equal Vstim(1-a3/r3)and thus depends only on cupular size and the distance measured at.

Conclusions

It can be concluded that, at frequencies below the resonance frequency of thecupula, the radial and longitudinal velocity profiles are predominantly determinedby the influence of cupula. At frequencies above the cupular resonance frequency,the cupular influence diminishes and the flow profiles resemble those found in anempty lateral line canal.

The decay of the flow velocity as a function of distance along the longitudinalcanal axis suggests that cupula II and cupula III operate relatively independentlyfrom each other, which seems to be related to the presence of the hydrodynamicwindows.

53

References

Batchelor, G. K. (1967). An introduction to fluid dynamics. CambridgeUniversity Press, Cambridge.

Coombs, S., Janssen, J. and Webb, J. F. (1988). Diversity of lateral line systems:evolutionary and functional considerations. In: Sensory Biology of AquaticAnimals, edited by J. Atema, R. R. Fay, A. N. Popper and W. N. Tavolga(Springer, New York), pp. 553-593.

Coombs, S. and Montgomery, J. (1992). Fibers innervating different parts of thelateral line system of an Antarctic Notothenioid, Trematomus bernacchii,have similar frequency responses, despite large variation in the peripheralmorphology. Brain Behav. Evol. 1992: 40: 217-233.

Denton, E. J. and Gray, J. A. B (1983). Mechanical factors in the excitation ofclupeid lateral lines. Proc. R. Soc. Lond. Biol. Sci., 218: 1-26.

Denton, E. J., and Gray, J. A. B. (1988). Mechanical factors in the excitation ofthe lateral lines of fishes, in Sensory Biology of Aquatic Animals, edited byJ. Atema, R. R. Fay, A. N. Popper, and W. N. Tavolga (Springer, NewYork), pp. 595-617.

Dijkgraaf, S. (1963). The functioning and significance of the lateral line organs.Biol. Rev. 38, 51-105.

Happel, J and Brenner, H. (1983). Low Reynolds number hydrodynamics,(Martinus Nijhoff Publishers, The Hague).

Harris, G. G. and Bergeijk, W. A. van (1962). Evidence that the lateral lineorgan responds to near field displacements of sound sources in water. J.Acoust. Soc. Am 34, 1831-1841.

Jakubowski, M. (1963). Cutaneous sense organs of fishes. I. The lateral-lineorgans in the stone-perch (Acerina cernua L.), Acta Biol. Cracoveniensia,Zool. 6, 59-82.

Kelly, J. P., and Netten, S. M. van (1991). Topography and mechanics of thecupula in the fish lateral line. I. Variation of cupular structure andcomposition in three dimensions. J. Morphol. 207, 23-36.

Kroese, A. B. A. and Netten, S. M. van (1989). Hair cell transduction Themechanosensory lateral line (Neurobiology and evolution) (ed. by S.Coombs, P. Görner and H. Münz), pp. 265-284 Springer-Verlag.

Münz, H. (1989). Functional organization of the lateral line periphery. In: TheMechanosensory Lateral Line, edited by S. Coombs, P. Görner, and H.Münz (Springer, New York), pp. 285-297.

Netten, S. M. van and Kroese, A. B. A. (1987). Laser interferometricmeasurements on the dynamic behaviour of the cupula in the fish lateral

54

line. Hearing Res. 29, 55-61.Netten, S. M. van (1988). Laser interferometer microscope for the measurement

of nanometer vibrational displacements of a light-scattering microscopicobject. J. Acoust. Soc. Am. 83, 1667-1674.

Netten, S. M. van (1991). Hydrodynamics of the excitation of the cupula in thefish canal lateral line. J. Acoust. Soc. Am. 89, 310-319.

Netten, S. M. van and Maarseveen, J. Th. P. W. van (1994).Mechanophysiological properties of the supraorbital lateral line canal inruffe (Acerina cernua L.) Proc. R. Soc. Lond. B 256, 239-246.

Oswald, R. L. (1978). Injection anaesthesia for experimental studies in fish.Comp. Biochem. Physiol. 60C, 19-26.

Pumphrey, R. J. (1950). Hearing. Symp. Soc. Exp. Biol. 4, 3-18.Sexl, T. (1930). Über den von E.G. Richardson entdeckten "Annulareffekt".

Z.Phys. 61, 349-362.Schlichting, H. (1987). Boundary-layer theory (McGraw-Hill Publishing

Company, New York).Wiersinga-Post, E. C. and van Netten, S. M. (1998). Relation between functional

mechanosensitivity and morphology of neuromasts in the lateral line of theruffe. In preparation.

Womersley, J. R., (1955). Method for the calculation of velocity, rate of flow andviscous drag in arteries when the pressure gradient is known, J. Physiol.127, 553-563.

55

5 A 2-dimensional model of the supraorbitallateral line canal organ of the ruffe

Introduction

The lateral line organ is unique to fish and amphibians. This organ is sensitive towater motion and is used by fish for avoiding predators, schooling (Partridge andPitcher, 1980), detecting and positioning for striking at prey (Janssen andCorcoran, 1993). An example of such an organ is the cephalic lateral line organfound on the head of the ruffe, Acerina cernua L. It consists of cartilaginous canalswith hydrodynamic windows located on the top of the canals (van Netten and vanMaarseveen, 1994). The bony structures between the openings are called bonybridges and arch over dome shaped extracellular matrices, called cupulae (Kellyand van Netten, 1991). Hair bundles of sensory hair cells protrude into the base ofthe cupula. These sensory hair cells transduce the cupular motion to electricalresponses.

Much of the mechanics concerning the functioning of the lateral line organ hasbeen extensively investigated, also in relation to the understanding of the detectionprocess down to the level of the sensory hair cells (see e.g. van Netten, 1997). Forthis to be possible it is essential that the stimulus driving the cupula, in otherwords, the fluid flow in the lateral line canal, is known exactly.

In vivo flow measurements in a relatively intact lateral line canal have beenhampered by the physical restrictions imposed by the tiny dimensions of the canal(of the order of 1 mm). Flow imaging techniques utilising seeding particlesilluminated by a laser sheet cannot be easily used. Although the options for in vivoflow measurements are limited, some measurements in the lateral line canals ofsprats have been conducted by Denton and Gray (1985). They used a travellingmicroscope to measure the displacements of seeding particles in the lateral linecanal stimulated by an oscillating sphere.

More recently, the flow in the lateral line canal of the ruffe was studied (Tsangand van Netten, 1997) using a micro-sense probe tracked with a laserinterferometer. This method allowed the flow in the lateral line canal to be mappedin the radial and longitudinal direction along the canal with a spatial resolution of50 µm. This technique, however, was limited to measurements up to the edge of thebony bridge. The flow in the region underneath the bony bridge could not bemeasured.

57

Measurements were conducted in the supraorbital lateral line canal of the ruffeeither with the cupula left intact, or removed from the canal. In this manner it waspossible to distinguish the influence of the bony bridge and that of the cupula. Asinusoidally moving glass stimulus sphere (Æ 0.68 mm) was placed in thesupraorbital lateral line canal in the region behind cupula I (Jakubowski, 1963) andjust in front of the bony bridge covering cupula II. The flow velocity profiles inboth radial and longitudinal directions were measured at a starting position of » 0.3mm longitudinally away from the edge of the bony bridge for the case with thecupula removed. The longitudinal velocity was measured at a constant height of 0.2mm above the canal floor. For the case where the cupula was left intact, theprocedure was exactly the same, apart from the starting position which was » 0.3mm longitudinally away from the edge of the cupula instead of the bony bridge. Onaverage the cupula protrudes approximately 0.6 mm beyond the edge of the bonybridge.

In the present paper a model of the lateral line canal based on the conditions ofthe measurements is developed to compute the flow in the lateral line canal. Inorder to compare the computations with the measurements, features such as thebony bridge and the opening at the top of the canal were incorporated into themodel.

Methods

For the computation of the flow in the lateral line canal the finite element packageSEPRAN (developed by SEPRA-analysis, Delft, the Netherlands) was used. Thecomputational domain is divided into a large number of triangular elements (seeFig. 1). In this package the pressure is computed in the centre points of theelements and the velocity components in the centres of the sides. The flowequations describing the motion of an incompressible fluid are used:

∇⋅ =v 0 (1)

ρ µ&v v v p v+ ⋅∇ = −∇ + ∇b gc h 2 (2)

in which v is the fluid velocity, ρ is the fluid density (1010 kg/m3; Jielof, 1952), pis the pressure and µ is the viscosity of the lateral line canal fluid, which wasmeasured to be 1.3 mPa s (chapter 2).

These equations are solved using a so-called pressure correction method. Thismeans that the continuity equation (1) is perturbed to:

58

εp v+ ∇⋅ = 0 (3)This perturbed equation can be used to eliminate the pressure from the momentumbalance (Eq. 2). After discretization the resulting system of the equations becomes:

Mu Su N u u L M LuTp& + + − =−b g 1

01

ε(4)

in which L denotes the numerical gradient operator, M is the mass-matrix, S the

stress-matrix, Mp is the pressure mass matrix and u is used instead of v to

indicate the difference between actual velocity and numerical approximation. N(u )is a discretization of the nonlinear convective terms. The nonlinearity in thisoperator is treated with a Newton iteration process. If the chosen ε is small (10-6)then the numerical errors introduced by the extra εp term do not outweigh theadvantage of achieving a coupling of the equations.

For the time integration an implicit Euler scheme was chosen, leading to:

Mu u

tSu N u u L M Lu

n nn n n T

pn

++ + − +− + + − =

11 1 1 11

0∆

c hε

(5)

(for more details see van Hengel, 1996).The motion of the cupula is coupled to the motion of the fluid. This is achieved

by computing the stress tensor:

σ µ= − + ∇ + ∇pI u uTb ge j (6)

and integrating this variable over the boundary of the cupula (assumed to be semi-

circular). This yields a force (Ffluid ) acting on the cupula. Since the cupula is

assumed to move only in the longitudinal direction, the longitudinal component

Ffluid x, of this force drives the cupula, which has mass m and is linked to its resting

position by a stiffness s. The equation of motion of the cupula thus becomes

F m sfluid x,&&= +ξ ξ (7)

in which ξ is the displacement of the cupula. The cupula is assumed to contain nointernal damping. The damping of its motion is assumed to be caused only by theviscous fluid drag (van Netten,1991).

For the time integration of the motion of the cupula a two-step Runge-Kutta

method was chosen. Using y1 = ξ and y2 = &ξ, equation (7) can be rewritten to:

59

&

&

,

y

y sm

y

yFfluix x

m

1

2

1

2

0 1

0

0L

NM

O

QP = −

L

NM

O

QPL

NM

O

QP +

L

NM

O

QP (8)

converting Eq. (7) into a system of coupled first order differential equations. Invector notation:

&y Ay b= + (9)

The computed motion of the cupula is coupled back to the SEPRAN part forthe fluid motion, by prescribing the velocity of the cupula as fluid velocity on theboundary representing the cupula. Because of the small amplitude of the cupulardisplacements caused by the stimulus amplitudes used in these computations, thedisplacement of the cupula is negligible compared to its size. This implies that forthe fluid computations the geometry can be taken fixed, prescribing the cupularvelocity on a boundary located at the average position of the cupula.

The order of computation in one time step thus becomes:Step 1: Computation of the fluid motion at time n∆t using the implicit Eulerscheme. The boundary conditions are given by the stimulus applied on the instream

boundary and the motion of the cupula, given by yn.

Step 2: y y t Ay bestn n n n+ = + +1 ∆ with the fluid force in bn derived from the stress

tensor σ computed from the fluid motion resulting from step 1.Step 3: Estimation of the fluid motion at time (n+1)∆t by implicit Euler integration,

using the estimated motion of the cupula yestn+1 for the boundary condition.

Step 4: y yt

Ay b Ay bn n n nestn

estn+ + += + + + +1 1 1

2

∆ with the fluid force in best

n+1

derived from the result of step 3.Previous work on such a simulation of the motion of a cupula in the

supraorbital lateral line canal using SEPRAN was performed by Meeuwissen(internal publication) and van Hengel (van Hengel, 1996). In both cases arotationally symmetric 3-D geometry was used, representing a sphere placed insidea cylinder.

In the simulations to be described here, the geometry was reduced to twodimensions. The results therefore attempt to describe the fluid flow in a verticalplane along the centreline of the canal. This approach was chosen instead of therotationally symmetric geometry, because in this manner the removal of (part of)the top of the canal was made possible. A complete 3 dimensional treatment of the

60

system could not be performed within reasonable limits of computation time andmemory use.

Figure 1: Mesh A is a 2-D mesh approximating the lateral line canal organ with cupula II

removed. Mesh B is identical to mesh A in dimensions and in boundary conditions, but in this

case cupula II is added. For the flow computations with and without cupula, the inflow comes

from the left hand side.

The dimensions for the computational domain were taken from averagedmeasurements of the ruffe in the region around cupula II. The height of the cupulais taken to be 0.55 mm and only half of it is covered by the bony bridge, unlikecupula III which lies under the bony bridge and is completely covered. The heightof the canal is 0.9 mm and the bony bridge is 0.1 mm thick. The mesh is made upof two chambers. One forms the lateral line canal and the other represents fluidoutside the canal encompassing a total height of 2.0 mm. The total length of thecanal is 7 mm, the edge of the bony bridge is located 4.3 mm from the inflow. Theinflow fluid profile at position x = 0 is flat and develops into a Sexl/Womersley(Sexl, 1930; Womersley, 1955) profile over a distance of » 2 mm. The inlet sectionof the model was extended to 4.3 mm in order to avoid influence of the numerical

61

treatment of the inflow on the results at the location of the cupula and/or the edgeof the bony bridge.

The resonance frequencies of cupula II and III have been experimentallydetermined and range from 115 to 130 Hz (van Netten and Kroese, 1987;Wiersinga-Post and van Netten 1998). For the present model, the mass andstiffness was chosen such that the resonance frequency of the cupula is 115 Hz

On the inflow boundary and the cupula, the velocity was prescribed (thevertical component for both was set to 0). On the bottom and top walls of the canal

a no-slip condition was imposed (u = 0 ). The boundary conditions imposed on all

other boundaries are so-called stress free conditions, meaning σ = 0. This conditiontheoretically holds for fluid infinitely far removed from any flow disturbance suchas the stimulus, the cupula and the edge of the bony bridge. This implies that theassumption was made that all these boundaries were sufficiently far removed fromthe region of interest where the velocity profiles were computed.

Results

Empty lateral line canal

Within the lateral line canal both the cupula and the bony bridge influence the flowof the lateral line canal fluid. Removing the cupula from the computational meshresults in computed flow profiles which contain the influence of the bony bridgealone. The flow profiles computed in an empty canal are shown in this section.

Radial flow velocity profile

Flow velocity measurements in the lateral line canal of the ruffe (Tsang and vanNetten, 1997) have shown that in an empty canal the flow profiles in the radialdirection change significantly for stimulus frequencies in the range of 10-70 Hz(see Fig. 2).

62

0.0 0.2 0.4 0.6 0.8

-30

-20

-10

0

10

20

pha

se (

deg)

distance above the canal floor (mm)

0.0 0.2 0.4 0.6 0.8

0

10

20

30

10 Hz 20 Hz 30 Hz 40 Hz 60 Hz 70 Hz

am

plitu

de

(µm

/s)

0.0 0.2 0.4 0.6 0.8

0

10

20

30

10 Hz 20 Hz 30 Hz 40 Hz 60 Hz 70 Hz

am

plitu

de

(a

.u.)

0.0 0.2 0.4 0.6 0.8

-30

-20

-10

0

10

20

pha

se (

deg)

distance above the canal floor (mm)

Figure 2: A plot of the radial velocity amplitude

and phase profiles measured at different

depths in the empty lateral line canal for a

range of stimulus frequencies. The distance to

the edge of the bony bridge is 0.3 mm. (from

Tsang and van Netten, 1997).

Figure 3: A plot of the radial flow velocity

amplitude and phase profiles computed at a

position of 0.3 mm away from the bony bridge

for a range of stimulus frequencies.

Similar changes can be seen in the computed flow velocity profiles as shown inFig. 3. At a stimulus frequency of 10 Hz, the computed velocity profile isapproximately parabolic. The point at which the flow is at a maximum is found at aheight of ≈ 0.4 mm above the canal floor. Increasing the stimulus frequency causesthe position of the flow maximum to shift closer towards the canal floor, at 70 Hzthe flow maximum is at ≈ 0.15 mm above the canal floor. The flow profilegradually changes from a parabolic to an asymmetric shape. A difference betweenthe measured and computed flow profiles is found in the mid-region of the canal.The computed amplitudes are relatively flat in the mid-region and only start to dipsharply at a height of > 0.7 mm above the canal floor. In the measurements there isa sharper but constant decline from ≈0.3 mm above the canal floor onwards. Forboth measurements and computations, the flow profiles resemble theSexl/Womersley (Sexl, 1930; Womersley, 1955) profiles found in a closed tube.

63

The computed phase profiles show similarities in absolute values to those ofthe measurements. There are however some discrepancies in the shape of the phaseprofile as well. The computed phase clearly shows a steady flattening over the mid-region of the canal with increasing stimulus frequency. The measurements show asmall amount of flattening over the mid-region of the canal, but this is far lesspronounced than in the computed results. In addition the measured phase profilesexhibit a phase shift of 10-20° in the phase profile when compared to the computedphases for stimulus frequencies of 20-70 Hz.

Longitudinal flow velocity profile

Measured longitudinal flow profiles were recorded at a height of 0.2 mm above thelateral line canal floor, (Fig. 4). The measurements were conducted at this heightabove the canal floor because it was within the region where the flow velocitywould be at its maximum. For the longitudinal flow measurements, a higherstimulus frequency range was used than that of the radial flow measurements inorder to have a higher velocity. The profiles for the different inflow frequencies areall very similar.

The corresponding computed flow velocity and phase profiles in thelongitudinal direction are shown in Fig. 5. These flow profiles at the differentstimulus frequencies are all similar, all exhibiting the same decay characteristics asa function of distance. There is however a slight downward shift in the velocityprofile with the increase in stimulus frequency. This is related to the flowmaximum in the radial direction shifting closer towards the canal floor.

The measured phase profiles in Fig. 4 show that the flow is nearly in phasewith the inflow, close to the bony bridge. Moving further down the canal causes thephase to increase and lead the inflow. The computed phase profile shown in Fig. 5is similar to the measurements, although the phase increase with distance issomewhat larger in the measurements.

64

0.0 0.5 1.0 1.5 2.0 2.5

-20

-10

0

10

20

pha

se (

de

g)

distance from the edge of the bony bridge (mm)

0.0 0.5 1.0 1.5 2.0 2.5

0

5

10

15

70 Hz 115 Hz 160 Hz 200 Hz 300 Hz

am

plit

ude

(a

.u.)

0.0 0.5 1.0 1.5 2.0 2.5

10

20

30

40

50

60

70

80

70 Hz 160 Hz 300 Hz

am

plit

ude

(µm

/s)

0.0 0.5 1.0 1.5 2.0 2.5

-20

-10

0

10

20

pha

se (

de

g)

distance from the edge of the bony bridge (mm)

Figure 4: A plot of the longitudinal flow

velocity amplitude and phase profiles at a

height of 0.2 mm above the canal floor at

inflow frequencies of 70, 160 and 300 Hz. The

results for other frequencies are not plotted,

because they are similar to these curves. The

edge of the bony bridge is located at 0 mm

along the canal.

Figure 5: A plot of the longitudinal flow

velocity amplitude and phase profiles at a

height of 0.2 mm above the canal floor for a

range of frequencies. The edge of the bony

bridge is located at a distance of 0 mm along

the canal.

Lateral line canal with cupula

Computations with the inclusion of the cupula yield results which differsignificantly from the case with an empty lateral line canal. The fact that the cupulais often found to protrude beyond the edge of the bony bridge forces themeasurements to be conducted at a different position as compared to the case withthe empty lateral line canal. The starting position at which the flow profiles in theradial and longitudinal directions were measured was 0.3 mm away from the edge

65

of the cupula. This means that the measurements were conducted approximately0.9 mm away from the caudal edge of the bony bridge II.

Radial flow velocity profile

The measured radial flow profiles (see chapter 4) plotted in Fig. 6 show adependency on the frequency of the stimulus. A different stimulus frequency rangewas used compared to that of the measurements in the empty lateral line canal. Itwas found that the most interesting changes in the flow profiles occurred atstimulus frequencies above 70 Hz. At 70 Hz the flow profile is relatively flat, butthe profile changes considerably when the stimulus frequency is increased beyond115 Hz. The flow velocity at these frequencies is greatest near the bottom of thecanal and steadily decreases with increasing height above the canal. At frequencies160, 200 and 300 Hz the flow profiles all share a similar form, with the flowreaching maximum velocity at 200 and 300 Hz.

The computed radial velocity profiles plotted in Fig. 7 share somecharacteristics with those of the measurements. At a stimulus frequency of 70 Hz,much of the flow is attenuated by the cupula, from the canal floor up to a height of » 0.4 mm. The flow increases slightly from a height of 0.4 mm to the top of thecanal. Increasing the stimulus frequency to 115 Hz shows an increase in thevelocity quite similar to the measurements. Just as with the measurements, theregion where the velocity decays rapidly is located at the bottom half of the canal.Increasing the stimulus frequency further produces flow profiles with similarshapes, but the height of the velocity peak decreases, which is opposite to what isobserved in the measurements. The reduction of the velocity at stimulus frequencies> 115 Hz is spread over the total height of the canal, as is also seen in themeasurements.

The plots of the measured phase profiles show the flow leading the stimulus.Most of the large phase changes occur at the lower section of the canal atfrequencies < 300 Hz. For 300 Hz, the phase profile is nearly flat. The largestphase change is observed at a frequency of 115 Hz and it amounts to approximately80°, whereas at the other frequencies changes from 5-50° are observed. Thecomputations show large phase changes at stimulus frequencies of 70 and 115 Hz.At 70 Hz, the phase close to the canal floor (» 0.05 mm) leads the stimulus by120°. This quickly decreases towards zero within 0.5 mm from the canal floor. At astimulus frequency of 115 Hz the computations show a lead in phase over thestimulus with a distinctive hump in the profile in the midsection of the canal. The

66

differences between the measured and computed velocity profiles are predominantlyreflected in the phase.

0.0 0.2 0.4 0.6 0.8

0

30

60

90

70 Hz 115 Hz 160 Hz 200 Hz 300 Hz

pha

se (

deg)

distance above the canal floor (mm)

0.0 0.2 0.4 0.6 0.8

0

20

40

60

80

am

plit

ude

(µm

/s)

0.0 0.2 0.4 0.6 0.8

-50

0

50

100

distance above the canal floor (mm)

pha

se (

deg)

0.0 0.2 0.4 0.6 0.8

0

10

20

30

40 70 Hz 115 Hz 160 Hz 200 Hz 300 Hz

am

plit

ude

(a.u

.)

Figure 6: Radial flow velocity amplitude and

phase profiles measured in the lateral linecanal at a distance of » 0.3 mm away from the

edge of the cupula for a range of stimulus

frequencies.

Figure 7: Computed radial flow velocity

amplitude and phase profiles at a distance of

0.3 mm away from the edge of the cupula.

Longitudinal flow velocity profile

Measured longitudinal velocity profiles (see chapter 4), as plotted in Fig. 8 alsoshow how the flow is influenced by the presence of the cupula. At stimulusfrequencies of 70 and 115 Hz, the flow profiles are predominantly flat,demonstrating that most of the flow is attenuated by the cupula. From 160 Hzupwards the flow velocity steadily increases with stimulus frequency. The decay ofthe velocity as a function of distance is quite rapid, i.e. the velocity for a stimulus of300 Hz decreases by around 60% over a distance of 1.2 mm.

67

0.0 0.5 1.0 1.5

0

30

60

90

120

150

phas

e (d

eg)

distance from the cupular edge (mm)

0.0 0.5 1.0 1.5

0

20

40

60

80

100

120 70 Hz 115 Hz 160 Hz 200 Hz 300 Hz

ampl

itude

(µm

/s)

0.0 0.5 1.0 1.5 2.0 2.5

0

10

20

30

40

ampl

itude

(a.

u.)

70 Hz 115 Hz 160 Hz 200 Hz 300 Hz

0.0 0.5 1.0 1.5 2.0 2.5

0

30

60

90

120

150

phas

e (d

eg)

distance from the cupular edge (mm)

Figure 8: A plot of the measured longitudinal

flow velocity amplitude and phase profiles at a

height of 0.2 mm above the lateral line canal

floor. The x-axis has been rescaled so that the

edge of the cupula is located at x = 0 mm.

Figure 9: A plot of the longitudinal flow velocity

amplitude and phase profiles at a height of 0.2

mm above the lateral line canal floor. The x-axis

has been rescaled so that the edge of thecupula is located at x = 0 mm.

The measured phase profiles clearly show that most of the phase changes occurnear the cupula. For stimulus frequencies of 70 Hz and 300 Hz the phase leads thestimulus by about 20-30° and within a distance of 1 mm away from the edge of thecupula the phase lead has reduced to approximately 0°. At the other stimulusfrequencies the phase lead differs between each frequency. Close to the cupularedge the phase lead at 115 Hz is »60°, at 160 Hz » 90° and at 200 Hz » 80°. For allof these phase profiles the phase reduces by about 40° over a distance of 1 mmaway from the cupular edge.

The computed longitudinal flow profiles plotted in Fig. 9 also show the flowprofiles to be very much dependent on the stimulus frequency. At 70 Hz the flowprofile is relatively flat as was seen in the measurements. One noticeable differencebetween measurements and computation is found at the resonance frequency of the

68

cupula (115 Hz). The computed longitudinal flow velocity is higher than at theother stimulus frequencies, contrary to the measurements which shows a maximumamplitude at 300 Hz. Increasing the stimulus beyond 115 Hz causes the flowprofiles to steadily shift downwards.

The computed phase profiles also show some significant differences from themeasurements. As with the measurements most of the rapid phase changes occurclose to the cupula. One striking feature is that the phase profile for a stimulusfrequency of 70 Hz shows a significant change in the phase from approximately140° to 15° over » 0.7 mm. At 115 Hz the phase drops from around 90° to 60° overa distance of 1 mm. At stimulus frequencies >115 Hz the phase profiles are flat andget closer and closer towards zero with increasing stimulus frequencies.

Discussion

Limitations of the finite element package SEPRAN in combination with limitationsof computer memory and computing time led to the computations of the flow in thelateral line canal organ to be reduced to a 2 dimensional model. The process ofreducing a 3 dimensional structure to 2 dimensions inevitably leads to the omissionof some of the canal features. The differences between the computed results and themeasurements indicate several shortcomings of the model. The differences with themeasurements in the case where the cupula is left intact indicate that the realsystem is too complex to be calculated with a 2-D model. The flow around thecupula is apparently, profoundly 3-D. The geometry of the lateral line canal seemsrelatively simple, but its shape changes as a function of distance along the fish'shead. Such shape changes cannot easily be accounted for in a 2-D mesh. In spite ofits shortcomings, the 2-D model provides insight into the flow behaviour in the reallateral line, especially in the empty canal. Furthermore, it may form the basis forfuture developments of 3-D lateral line models.

Empty lateral line canal

The radial velocity profiles computed for the empty lateral line canal are bothposition and frequency dependent. In comparison with the measurements, manysimilarities can be seen. In both cases, the position of the flow maxima can bedescribed by the a.c. boundary layer thickness in an infinitely long tube with a

radius r, δmax = 2.28(2µ/ωr)1/2 (Tsang and van Netten, 1997); (Schlichting, 1987)

69

where δmax is the distance from the canal floor, µ is the viscosity of the lateral line

canal fluid (1.3 mPa s), ρ is the lateral line canal fluid density (1010 kg/m3) and ωis the angular stimulus frequency. The largest variations occur at the lowerfrequencies; above 70 Hz the difference in the velocity profiles becomes smaller.Even though the top of the canal is open, the flow profile very close to the bonybridge (at the position where the cupula is located) is similar to the flow in a tube.The computed phase profiles show some similarities in shape with those of themeasurements. The measurements, however, show a phase difference of 10-20° atthe inflow frequencies ranging from 20 to 70 Hz when compared to thecomputations. Such shift in the phase cannot be caused by the compressibility ofthe fluid, which was not taken into account in the SEPRAN computations. Themaximum phase shift that can occur due to compressibility is about 0.001° using φ= ωL/c, where ω is the stimulus frequency, c is the speed of sound in water (1500m/s) and L is the distance to the stimulus sphere. This is far too small to havecaused the phase difference between the simulations and measurements. Thissuggests that the measured phase is perhaps influenced by the side walls of thelateral line canal, which the 2-D model cannot take into account. In order to furtherinvestigate this difference, the phase profile is calculated for both the flow in a 2-D(Meeuwissen, 1994) and 3-D, infinitely long, closed tube (Sexl, 1930; Womersley,1955) (see Fig. 10).

0.0 0.1 0.2 0.3 0.4

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-10

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3-D

pha

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)

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-10

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10

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30

40

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2-D 10 Hz 20 Hz 30 Hz 40 Hz 60 Hz 70 Hz

pha

se (

deg

)

Figure 10: Plots of the flow velocity phase profiles in a tube with a radius of 0.45 mm calculated

with the theory of Sexl/Womersley (Sexl, 1930; Womersley, 1955; Meeuwissen, 1994) for a 2

dimensional case (top) and a 3 dimensional case (bottom) for a series of inflow frequencies.

70

It is clear that there is a difference in the calculated phase profiles between 2-Dand 3-D. At 10 Hz, there is a slight difference at the edge of the tube, but overallthere is very little difference between the two cases. At stimulus frequencies from20-70 Hz, there is a shift of 5-9°. This suggests that some of the differencesbetween measurements and computations are due to the 2-D model not being ableto take the side walls of the canal into account.

10

100

543

SEPRAN (scaled) 70 Hz measurements (empty canal)

a'/x2

c'/x3

ampl

itude

(µm

/s)

distance to stimulus sphere centre (mm)

50

20

Figure 11: Measured and

computed longitudinal profiles

for an empty lateral line canal.

The computed longitudinal flow profiles in the empty canal, plotted in Fig. 5,are similar to those of the measurements. In both cases the shape remains relativelyconstant except for the fact that there is a slight downward shift with increasingfrequency. The cause of this can be derived from the results plotted in Fig. 4 at aheight of 0.2 mm above the canal floor. The maximum in the radial flow profileshifts towards the canal floor with increasing stimulus frequency. The decay of thevelocity at 70 Hz as a function of distance for both measurements and computationare plotted again in Fig. 11. The velocity decay of a dipole as a function of distancein 2-D is given by a/r2-bx2/r4 and in 3-D, c/r3-dx2/r5 (Tietjens, 1960), where x isthe longitudinal distance to the stimulus sphere centre. For both the measurementsand computations, x ≈ r because y is small. Therefore, 2-D gives a'/r2 dependenceand 3-D gives a c'/r3 dependence. It is clear that the measurements show a decaycloser to the c/r3 than a/r2 suggesting that the velocity decay in the lateral linecanal resembles that of free field 3-D. Surprisingly the 2-D SEPRAN results alsoshow a 3-D like decay as seen in the measurements. Apparently the canal wallforces the velocity back to zero faster than it would in free field. A similar effectwas observed by van Hengel in the rotationally symmetric 3-D geometry (vanHengel, 1996). The deviation in the 2-D plot at the edge is caused by the boundary

71

conditions at the outflow, which produces a zero velocity gradient in thelongitudinal direction. Near the bony bridge the deviation is a result of theboundary layer around the bony bridge.

Lateral line canal with cupula

The addition of the cupula to the lateral line canal, as expected, gives very differentresults from the case without cupula. The radial velocity profiles in Fig. 7 show thatthe flow profiles no longer resemble the flow in an infinitely long tube. At thelower stimulus frequencies (< 115 Hz), the flow is predominantly blocked by thecupula. This was also observed in the measurements. At stimulus frequencies above115 Hz the radial profiles become more and more like those of the empty canalwith respect to the position of the flow maxima. This is expected as the cupularmotion comes closer to that of the driving fluid with increasing stimulus frequency(van Netten, 1991). In the computations, the maximum fluid flow occurs at 115 Hz,the resonance frequency of the cupula. In the 2-D model a major part of thestimulus signal is lost through the open top wall. As a consequence the flowdirectly behind the cupula is primarily determined by the cupular motion. In thereal system (see chapter 4) much of the stimulus is conserved by the canal walls.Therefore the flow directly behind the cupula is a summation of the components ofthis stimulus with that of the cupular motion.

The computed velocity profiles in the longitudinal direction agree with whatwas seen in the radial profiles. Below 115 Hz, most of the flow is blocked by thecupula, at 115 Hz a maximum occurs, and above 115 Hz the flow resembles theflow in an empty canal. Remarkable differences can be seen between computedprofiles and measurements in the lateral line with the cupula. The only possibleexplanation for these differences is that in the real system the flow around thecupula has a profound 3-D character.

Conclusion

The 2-D model produced results which are similar to actual measurements in theempty lateral line canal. There are however limitations to a 2-D approach. This isreflected in the discrepancies between measurements and computations seen in thecomputations with the cupula present. These differences may be due to the effectsof side walls and the changing shape of the lateral line canal with distance along

72

the canal. This problem can only be solved when a true 3-D model can be producedwith the essential physical features of the lateral line canal incorporated into it.

References

Denton, E.J. and Gray, J. (1985). Mechanical factors in the excitation of thelateral lines of fishes. Sensory biology of aquatic animals (ed. by J. Atema,R.R. Fay, A.N. Popper, W.N. Tavolga), pp. 595-617. Springer-Verlag.

Hengel van, P. W. J. (1996). Emissions from cochlear modelling, PhD thesis,University of Groningen, the Netherlands.

Jakubowski, M. (1963). Cutaneous sense organs of fishes. I. The lateral-lineorgans in the stone-perch (Acerina cernua L.), Acta Biol. Cracoveniensia,Zool. 6, 59-82.

Janssen, J. and Corcoran, J. (1993). Lateral line stimuli can override vision todetermine sunfish strike trajectory. J. Exp. Biol. 176, 299-305.

Jielof, R., Spoor, A., and de Vries, H. (1952). The microphonic activity of thelateral line. J. Physiol. 116, pp. 137-157.

Kelly, J. P., and Netten, S. M. van (1991). Topography and mechanics of thecupula in the fish lateral line. I. Variation of cupular structure andcomposition in three dimensions. J. Morphol. 207, 23-36.

Meeuwissen, M. (1994). Een numeriek model van de beweging van een cupulain het zijlijnkanaal, M.Sc. thesis, Rijksuniversiteit Groningen.

Netten, S.M. van and Kroese, A. B. A. (1987). Laser interferometricmeasurements on the dynamic behaviour of the cupula in the fish lateralline. Hearing Res. 29, 55-61.

Netten, S.M. van (1991). Hydrodynamics of the excitation of the cupula in thefish canal lateral line. J. Acoust. Soc. Am. 89, 310-319.

Netten, S. M. van and Maarseveen, J. Th. P. W. van (1994).Mechanophysiological properties of the supraorbital lateral line canal inruffe (Acerina cernua L.) Proc. R. Soc. Lond. B 256, 239-246.

Netten, S. M. van (1997). Hair cell mechano-transduction: Its influence on thegross mechanical characteristics of a hair cell sense organ, BiophysicalChemistry 2274

Partridge, B. L. and Pitcher, T. J. (1980). The sensory basis of fish schools;relative roles of lateral line and vision. J. Comp. Physiol. A 135, 315-25.

Sexl, T. (1930). Über den von E.G. Richardson entdeckten "Annulareffekt".Z.Phys. 61, 349-362.

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Schlichting, H. (1987). Boundary-layer theory (McGraw-Hill PublishingCompany, New York).

Tietjens, O. (1960). Strömungslehre BD I. Springer Verlag.Tsang, P. T. S. K. and Netten, S. M. van (1997). Fluid flow profiles measured in

the supraorbital lateral line canal of the ruff. Diversity in AuditoryMechanics. World Scientific, Singapore.

Wiersinga-Post, E. C. and van Netten, S. M. (1998). Relation between functionalmechanosensitivity and morphology of neuromasts in the lateral line of theruffe. In preparation.

Womersley, J.R., (1955). Method for the calculation of velocity, rate of flow andviscous drag in arteries when the pressure gradient is known, J. Physiol.127, 553-563.

74

Summary

The ruffe, a freshwater fish common to the waters of Europe, possesses a lateralline organ for the detection of local water motion. The cephalic part of the lateralline system is located on the head of the ruffe and consists of recessed bony canalswith a series of bony plates arching over the canal, called bony bridges. Situatedunder the bony bridges are the sensory units, or neuromasts. A neuromast consistsof sensory hair cells grouped together with their stereocilia anchored in a domeshaped extracellular matrix called cupula.

The research presented in this thesis focuses on the viscosity and the flow ofthe lateral line canal fluid in the supraorbital lateral line canal organ. The first part,chapter 2, describes a novel laser viscometer developed for measuring the viscosityof biological fluids in vivo or in vitro. The viscometer consists of a laserinterferometer tracking the motion of a driven pendulum oscillating in a viscousfluid. The resonance characteristics of the driven pendulum, specifically the Q3dBvalue, is related to the viscosity of the fluid. Thus, the viscosity could be found bycomparing it with the measured Q-values of fluids with known viscosities. Thecompact dimensions of the oscillating pendulum meant that only a volume 0.04 mlof sample fluid was required. We established that the viscosity of the lateral linecanal fluid ranges from 1.8 to 1.2 times the viscosity of water for the physiologicaltemperature range (4 - 20 °C) of the ruffe. Comparing the results from the ruffewith other labyrinthine fluids, such as perilymph and endolymph from otheranimals, suggest that the all labyrinthine fluids have the same viscosity/temperaturecharacteristics. For the ruffe, the temperature of the water in which it is swimming,may be one of the factors which determines the sensitivity of its lateral line organ.

The second part of thesis (chapters 3-5) describes the investigations into theflow of the lateral line canal fluid and its interactions with the canal structure andthe cupula. A micro sense probe consisting of a fine glass fibre with a resin sphereglued onto the fibre tip is used for the purpose of visualising the flow. This probecan be placed almost anywhere within the confined space of the lateral line canal.Its motion is tracked with a modified version of the laser interferometer used for theviscosity measurements. A flow velocity profile of the local flow in the canal is thenconstructed sequentially, point by point.

The first series of flow experiments (chapter 3), investigated the influence ofthe integral structure of the lateral line canal on the stimulus. For theseexperiments, the cupula was removed from the lateral line canal. The radialprofiles measured in the close vicinity of the bony bridge are similar to the profilescharacteristic to the local flow in a closed tube. The longitudinal profiles reveal thatthe velocity decay is close that found in free field. An interesting hypothesis

75

prompted by the radial velocity profiles in the lateral line canal is that the cupulawould have to penetrate at least beyond the boundary layer of the canal fluid inorder to receive the maximally possible stimulus. This means that the height of thecupula is probably related to the boundary layer thickness of the canal fluid.

The measurements of the flow in the lateral line canal with the cupula presentis described in chapter 4. The radial and longitudinal velocity profiles differ fromthose from an empty lateral line canal. The cupula attenuates most of the flow atfrequencies below the cupular resonance frequency. At frequencies beyond thecupular resonance frequency, the flow profiles begin to resemble those found in anempty lateral line canal as the cupula moves in phase with the lateral line fluid.From the rate of the velocity decay over the longitudinal axis, it suggests that theneuromasts operate relatively independently of each other, which seems to berelated to the presence of the hydrodynamic windows.

In conjunction with the experimental work, the flow is also computed in amodel of the lateral line canal formulated in a finite element package (SEPRAN).The model is limited to 2-D due to the restrictions imposed by both SEPRAN andcomputer hardware. Computations with the model of an empty lateral line canalgives radial and longitudinal velocity profiles similar to the measurementsconducted in an empty lateral line canal. For the model with the cupula present inthe lateral line canal, there is a clear difference between the computed velocityprofiles and the measurements described in chapter 4. This indicates that the flowin the vicinity of the cupula is profoundly 3-D, meaning that it only be accuratelydescribed in 3-D. Thus, the usefulness of the 2-D models is limited, some insightinto the interactions between the lateral line canal fluid and the bony bridge couldbe gained from the empty lateral line canal model.

76

Samenvatting

De pos, een zoetwatervis die in de meeste binnenwateren van Europa voorkomt,bezit een zijlijnorgaan voor de detectie van locale waterbeweging. Het cefalischedeel van de zijlijnorgaan, dat zich vlak onder de huid bij de kop van de pos bevindt,bestaat uit benen kanalen. Over een kanaal buigen zich een reeks benen plaatjes,die beenbruggen worden genoemd. Onder de beenbruggen bevinden zich dezintuigeenheden, neuromasten genaamd. Een neuromast bestaat uit gegroepeerdezintuighaarcellen die met hun haarbundels zijn verankerd in een koepelvormigetransparante extracellulaire matrix: de cupula.

Het onderzoek dat in dit proefschrift wordt beschreven spitst zich toe op deviscositeit en de stroom van de vloeistof in het supraorbitale kanaal van hetzijlijnorgaan. Het eerste deel, hoofdstuk 2, beschrijft een nieuwe laser-viscometerdie werd ontworpen voor het meten van de viscositeit van biologische vloeistoffenin vivo en in vitro. De viscosimeter bestaat uit een laser-interferometer die debeweging volgt van een aangedreven slinger die in de visceuze vloeistof oscilleert.De resonantiekarakteristiek van de aangedreven slinger, in het bijzonder de Q3dB-waarde, is gerelateerd aan de viscositeit van de vloeistof. Zodoende kan deviscositeit worden gevonden door de gemeten waarde te vergelijken met gemetenQ-waarden van vloeistoffen met bekende viscositeiten. Door de zeer kompakteafmetingen van de oscillerende slinger kan worden volstaan met eentestvloeistofvolume van slechts 0.04 ml. De viscositeit van de zijlijnkanaalvloeistofvarieert van 1.8 tot 1.2 keer de viscositeit van water voor het fysiologischetemperatuurbereik (4-20 °C) van de pos. Als deze resultaten van de pos wordenvergeleken met andere labyrint-vloeistoffen, zoals perilymfe en endolymfe vanandere dieren, suggereert dit dat alle labyrint-vloeistoffen dezelfde viscositeit-temperatuurkarakteristieken hebben. Voor de pos kan de temperatuur van het waterwaarin hij zwemt èèn van de faktoren zijn die de gevoeligheid van het zijlijnorgaanbepaalt.

Het tweede deel van dit proefschrift (hoofdstukken 3-5) beschrijft hetonderzoek naar de stroming van de zijlijnorgaanvloeistof en de interacties met dekanaalstruktuur en de cupula. Om de stroming te visualiseren werd een microsondegebruikt bestaande uit een dunne glasfiber met een harsbolletje gelijmd aan hetuiteinde. Deze sonde kan vrijwel overal binnen de beperkte ruimte van hetzijlijnorgaan worden geplaatst. De beweging van het harsbolletje kan wordengevolgd met een aangepaste versie van de laser-interferometer die werd gebruiktvoor de viscositeitsmetingen. Een snelheidsprofiel van de plaatselijke stroming inhet kanaal kan dan punt voor punt worden geconstrueerd.

77

De eerste reeks van experimenten (hoofdstuk 3) onderzocht de invloed van detotale struktuur van het zijlijnorgaan op de stimulus. Voor deze experimenten werdde cupula verwijderd uit het zijlijnkanaal. De radiële profielen die werden gemetenop korte afstand van de beenbrug zijn vergelijkbaar met de karakteristieke profielenvan de lokale beweging in een buis. De longitudinale profielen tonen aan dat deafname in snelheid nauw overeenkomt met de vrije veld situatie. Een interessantehypothese die wordt ingegeven door de radiële snelheidsprofielen in hetzijlijnkanaal, is dat de cupula minstens tot voorbij de grenslaag van dekanaalvloeistof zou moeten penetreren om in staat te zijn de maximaal mogelijkestimulus waar te nemen. Dit betekent dat de hoogte van de cupula waarschijnlijkgerelateerd is aan de dikte van de grenslaag van de kanaalvloeistof.

De metingen van de stroming in het zijlijnkanaal waarbij de cupula aanwezigis worden beschreven in hoofdstuk 4. De radiële en longitudinalesnelheidsprofielen verschillen van die van een leeg zijlijnkanaal. De cupulaverzwakt het grootste deel van de stroming bij frequenties onder de cupularesonantiefrequentie. Bij frequenties boven de resonantiefrequentie van de cupulagaan de stromingsprofielen lijken op de profielen van een leeg zijlijnkanaal waarbijde cupula in fase beweegt met de zijlijnvloeistof. De mate van snelheidsafnamelangs de longitudinale as suggereert dat de neuromasten relatief onafhankelijk vanelkaar opereren. Dit lijkt gerelateerd te zijn aan de aanwezigheid van dehydrodynamische vensters: de alleen met huid afgedekte kanaalstukken.

Naast het experimentele werk werd de stroming numeriek berekend in eenmodel van het zijlijnorgaan met behulp van het eindige elementen pakketSEPRAN. Beperkingen opgelegd door SEPRAN en de computer hardware maaktenslechts 2-D model berekeningen mogelijk. Simulaties met het model van een leegzijlijnorgaan geven radiële en longitudinale snelheidsprofielen die vergelijkbaarzijn met de verrichtte metingen in een leeg zijlijnorgaan. Dit verschaft inzicht overde interacties tussen de zijlijnkanaalvloeistof en de beenbrug. Voor het modelwaarin de cupula aanwezig is in het zijlijnorgaan bestaat er een duidelijk verschiltussen de berekende snelheidsprofielen en de metingen beschreven in hoofdstuk 4.Dit duidt erop dat de stroming vlakbij de cupula hoofdzakelijk 3-D is, hetgeenbetekent dat deze slechts nauwkeurig beschreven kan worden in een 3-D model.Voor een nauwkeuriger beschrijving in de toekomst is verdere 3-D-analysenoodzakelijk.

78

Acknowledgements

There are many people to whom I am grateful to for helping me reach thecompletion of this project. Firstly, my promotor Diek Duifhuis for his guidance, mychief engineer Jannes Land and Edwin "Beavis" Zevenberg for making all themechanical and electronic components of my laser setup. In addition, I would liketo thank Hans Achterhof, Esther Wiersinga-Post, Eric Lee, Paul Altena, Jos vanDeemter, Hein Leertouwer, Khay Nio, Viktoria Porfireva, Wanda Ennik, FrederikeSlichter, my "Girl-Friday" Sara May, and all my friends from the Biophysicsdepartment for their support and encouragement over the last four years. Lastly, Iwould like to thank my partners in crime, Peter van Hengel and Marc van denRaadt for making it fun for me to work in my lab!

Spesjale tank oan Sietse van Netten foar syn begelieding en bydrage fantechnyske kennis; en oan Kees Schilstra foar syn grutte bydrage oan defiscositeitsmjittingen.

Mi wan taygi mi famiri, mi bun Sranan mati Reon Welch, Gerald Soesman,Steven Coutinho nanga mi bakra mati Jos van Maarseveen tangi fu ben hor mibaka stefi.

79

Stellingen

behorende bij het proefschrift

Laser interferometric flow measurements in the lateral line organ

Peter Tsang

1. Rekening houdend met de temperatuurafhankelijkheid van de viscositeit kanmen concluderen dat de vloeistof in het zijlijnkanaal van de pos mechanischgezien overeenkomt met de lymfe in de cochlea. (Zie hoofdstuk 2 van ditproefschrift).

2. De vloeistofstroming in een zijlijnkanaal zonder cupula lijkt in de buurt vande beenbrug op de stroming in een oneindig lange buis. (Zie hoofdstuk 3 vandit proefschrift).

3. Cupulae II en III functioneren grotendeels onafhankelijk van elkaar. (Ziehoofdstuk 4 van dit proefschrift, nummering van cupulae volgensJakubowski).Jakubowski, M. (1963). Cutaneous sense organs of fishes. I. The lateral-lineorgans in the stone-perch (Acerina cernua L.), Acta Biol. Cracoveniensia,Zool. 6, 59-82.

4. Het verschil tussen de gemeten viscositeit van de zijlijnkanaalvloeistof en dewaarde die gebruikt wordt in het analytische model van Van Netten kanworden verklaard door de aanwezigheid van de kanaalwand. (Zie hoofdstuk4 van dit proefschrift).Netten, S. M. van (1991) Hydrodynamics of the excitation of the cupula inthe fish canal lateral line. J. Acoust. Soc. Am. 89, pp. 310-319.

5. De vloeistofstroming in het zijlijnkanaal kan alleen correct gemodeleerdworden in 3-D. (Zie hoofdstuk 5 van dit proefschrift).

6. Vooruitgang in beschaving gaat hand in hand met vooruitgang inbewapening.

7. Een geluksvogel is iemand die het toeval naar zijn hand weet te zetten.

8. De bekwaamheid van een poolbiljartspeler is af te lezen aan de indirectheidvan de route die de pool-bal aflegt op weg naar de "pocket".

9. Soms lijkt het of de versterker van een electrische gitaar met name het egovan de gitarist versterkt.

10. Uit de praktijk van de Nederlandse wetgeving krijgt men de indruk dat zelfsde Tien Geboden met terugwerkende kracht veranderd kunnen worden.

11 Dieren en planten die het meest door de mens geconsumeerd worden, hebbende grootste kans voor uitsterving behoed te worden.