The hydrodynamics of locomotion at intermediate Reynolds numbers:undulatory swimming in ascidian larvae (Botrylloides sp.)

Understanding how the shape and motion of an aquatic animal affects the performance of swimming requires knowledge of the fluid forces that generate thrust and drag. Despite recent advances towards understanding the biomechanics of locomotion (see Dickinson et al., 2000 for a review), these forces are poorly understood in swimming animals that are a few millimeters in length. The large diversity of larval fish and marine invertebrates at this scale generate hydrodynamic force that is dependent on both the viscosity and the inertia of the surrounding water. To understand the relative contribution of inertial and viscous forces to the generation of thrust and drag, theoretical models have been developed for the hydrodynamics of swimming at this scale (e.g. Jordan, 1992; Vlyman, 1974; Weihs, 1980). However, little experimental work has attempted to test or refine these theories (exceptions include Fuiman and Batty,1997; Jordan,1992). The goal of the present study was to test hydrodynamic theory by comparing the predictions of theoretical models with measurements of the speed of freely swimming animals and the forces generated by tethered animals.

Swimmers that are millimeters in length generally operate in a hydrodynamic regime characterized by Reynolds numbers (Re) between 10⁰and 10³, which is a range referred to as the intermediate Re in the biological literature (e.g. Daniel et al., 1992). Re (Re=ρūL/μ, where ūis mean swimming speed, L is body length, ρ is density of water,and μ is dynamic viscosity of water) approximates the ratio of inertial to viscous forces and suggests how much different fluid forces contribute to propulsion. At intermediate Re, a swimming body may experience three types of fluid force: skin friction, form force and the acceleration reaction. Skin friction and form force are quasi-steady and therefore vary with the speed of flow. In previous studies on intermediate Re swimming, these forces have collectively been referred to as the `resistive force' (e.g. Jordan, 1992). However, we will consider these forces separately because the present study is concerned with how they individually contribute to the generation of thrust and drag.

Skin friction is generated by the resistance of fluid to shearing. This is a viscous force, which means that it increases in proportion to the speed of flow. Skin friction (also called the `resistive force' by Gray and Hancock, 1955)dominates the undulatory swimming of spermatozoa(Re≪10⁰; Gray and Hancock, 1955) and nematodes(Gray and Lissmann, 1964) and has been hypothesized to contribute to thrust and drag in the intermediate Re swimming of larval fish(Vlyman, 1974; Weihs, 1980) and chaetognaths(Jordan, 1992).

The form force is generated by differences in pressure on the surface of the body and it varies with the square of flow speed(Granger, 1995). This inviscid force is equivalent to the resultant of steady-state lift and drag acting on a body at Re>10³. The form force is thought to contribute to the generation of thrust and drag forces at the intermediate Reswimming of larval fish (Vlyman,1974; Weihs, 1980)and may dominate force generation by the fins of adult fish(Dickinson, 1996).

The acceleration reaction [also referred to as the `reactive force'(Lighthill, 1975), the `added mass' (Nauen and Shadwick,1999) and the `added mass inertia'(Sane and Dickinson, 2001)] is generated by accelerating a mass of water around the body and is therefore an unsteady force (Daniel, 1984). This force plays a negligible role in the hydrodynamics of swimming by paired appendages at Re<10¹(Williams, 1994) but is considered to be important to undulatory swimming at intermediate Re(Brackenbury, 2002; Jordan, 1992; Vlyman, 1974) and dominant in some forms of undulatory swimming at Re>10³(Lighthill, 1975; Wu, 1971). Although it is assumed that the acceleration reaction does not play a role in undulatory swimming at Re<10⁰(Gray and Hancock, 1955), it is not understood how the magnitude of the acceleration reaction varies across intermediate Re.

Weihs (1980) proposed a hydrodynamic model that predicted differences in the hydrodynamics of undulatory swimming in larval fish at different intermediate Re. He proposed a viscous regime at Re<10¹, where viscous skin friction dominates propulsion, and an inertial regime at Re>2×10², where inertial form force and the acceleration reaction are dominant (also see Weihs, 1974). For the range of Re between these domains, thrust and drag were hypothesized to be generated by a combination of skin friction, form force and the acceleration reaction. Although frequently cited in research on ontogenetic changes in the form and function of larval fish (e.g. Muller and Videler, 1996; Webb and Weihs, 1986), it remains unclear whether Weihs'(1980) theory, which is founded on measurements of force on rigid physical models, accurately characterizes the forces that act on an undulating body(Fuiman and Batty, 1997).

The present study used a combination of empirical measurements and mathematical modeling of the larvae of the ascidian Botrylloides sp. to test whether the hydrodynamics of swimming in these animals is better characterized by a quasi-steady or an unsteady model. By taking into account the acceleration reaction, skin friction and form force generated during swimming, models were used to formulate predictions in terms of the speed of freely swimming larvae and force generation. By comparing these predictions with measurements of force and speed, we were able to determine whether larvae generate thrust and drag by acceleration reaction (the unsteady model) or strictly by form force and skin friction (the quasi-steady model). Ascidians are an ideal group for exploring these hydrodynamics because the larvae of different species span nearly two orders of magnitude in Re [e.g.≈5×10⁰ in Ciona intestinalis(Bone, 1992); Re≈10² in Distaplia occidentalis(McHenry, 2001)].

Colonies of Botrylloides sp. were collected in the months of August and September from floating docks (Spud Point Marina, Bodega Bay, CA,USA) in water that was between 14°C and 17°C. Colonies were transported in coolers and placed in a recirculating seawater tank at 16°C within 2 h of collection. To stimulate release of larvae, colonies were exposed to bright incandescent light after being kept in darkness overnight(Cloney, 1987). Released larvae were used in either force measurement experiments, free-swimming experiments or for morphometric analysis. In all cases, observation tanks were equipped with a separate outer chamber into which chilled water flowed from a water bath equipped with a thermostat (1166, VWR Scientific) that kept larvae at 16°C.

To calibrate the tether, we measured its stiffness and damping constants in a dynamic mechanical test. This test consisted of pulling and releasing the tether and then recording its passive movement over time(Fig. 2A). The tether oscillated like an underdamped pendulum(Meriam and Kraige, 1997a)with a natural frequency (101 Hz) well outside the range of tail-beat frequencies expected for ascidian larvae(McHenry, 2001). Using the equation of motion for the tether (equation 2, with F=0), its oscillations were predictable if the mass and the stiffness and damping coefficients were known. Conversely, we solved for the stiffness and damping coefficients from recordings of position and a measurement of the mass of the tether (see Appendix for details).

We examined how errors in our measurement of stiffness and damping coefficients were predicted to affect calculations of the force generated by larvae (Fig. 2C—H). By simulating the input force generated by a larva as a sine wave with an amplitude of 20 μN, we numerically solved equation 2 (using MATLAB, version 6.0, Mathworks) for the position of the tether over time at 1000 Hz (the sampling rate of our recordings). From these simulated recordings of tether position, we then solved equation 2 for F, the force generated by the larva. This circular series of calculations demonstrated that our sampling rate was sufficient to follow rapid changes in input force(Fig. 2C). Furthermore, we found that a minimum of 92% of the instantaneous moments resisting the input force were generated by the stiffness of the tether (i.e. the weight and damping of the tether provided a maximal 8% of the resistance to input force). If the values of stiffness and damping coefficients used in force measurements differed from those used to simulate tether deflections, then measured force did not accurately reflect the timing or magnitude of simulated force(Fig. 2D). This situation is comparable with using inaccurate values of stiffness and damping coefficients for measurements of force in an experiment.

By varying the difference between the stiffness and damping coefficients used to simulate changes in tether position over time (i.e. the actual coefficients) and those used for force measurements (i.e. the measured coefficients), we explored how inaccuracy in measured coefficients was predicted to alter the timing and magnitude of measured force(Fig. 2E—H). We simulated changes in force at 18 Hz, to mimic oscillations in force at the tail-beat frequency (McHenry, 2001), and at 180 Hz, to simulate rapid changes in force. Within the level of precision(i.e. ±2 S.D.) of our measurements of stiffness and damping coefficients, measured force was not predicted to precede or lag behind simulated force by more than 1 ms, which is just 1.8% of an 18 Hz tail-beat period (Fig. 2E,F). Error in the damping coefficient may have caused measurements to overestimate rapidly changing force by as much as 7.5% (Fig. 2G). Within the precision of measured stiffness coefficients,measured forces may have differed from actual values by as much as 2.0%(Fig. 2H). These findings suggest that our measurements accurately reflect the timing of force generated by larvae, but the magnitude of force may be inaccurate by as much as 7.5%.

The ventral surface of the body was recorded during tethered swimming(Fig. 1A) with a high-speed video camera (Redlake Imaging PCI Mono/1000S Motionscope, 320 pixels×280 pixels, 500 frames s^-1) mounted to a dissecting microscope (Wild,M5A) beneath the glass tank containing the tethered larva. The video signal from this camera was recorded by the same computer (Dell Precision 410, with Motionscope 2.14 software, Redlake Imaging) as was used to record micropipette deflections, which allowed the recordings to be synchronized.

We measured the shape of the body to provide parameter values for our calculations of fluid forces and to estimate the body mass, center of mass and its moment of inertia. The peripheral shape of the body was measured (with NIH Image version 1.62 on an Apple PowerMac G3) using digital still images of larvae from dorsal and lateral views that were captured on computer (7100/80 PowerPC Macintosh with Rasterops 24XLTV frame grabber) using a video camera(Sony, DXC-151A) mounted on a dissecting microscope (Nikon, SMZ-10A). These images had a spatial resolution of 640 pixels×480 pixels, with each pixel representing approximately a 6 μm square with an 8-bit grayscale intensity value. Coordinates along the peripheral shape of the body were isolated by thresholding the image (i.e. converting from grayscale to binary; Russ, 1999). We found coordinates at 50 points evenly spaced along the length of the trunk and 50 points evenly spaced along the length of the tail (using MATLAB). From images of the lateral view, we used the same method to measure the dorso-ventral margins of the trunk, cellular tail and tail element. By the same method, we measured the width of the trunk from the dorsal view.

In order to test the effect of tissue density, we ran simulations (see`Modeling free swimming' below) with the mean kinematics and morphometrics at high tissue density (ρ_body=1.250 g ml^-1, the density of an echinopluteus larva of an echinoid with calcareous spicules;(Pennington and Emlet, 1986)and low tissue density (ρ_body=1.024 g ml^-1, the density of seawater at 20°C; Vogel,1981). All other simulations were run with a tissue density typical of marine invertebrate larvae not possessing a rigid skeleton(ρ_body=1.100 g ml^-1; Pennington and Emlet,1986).

Freely swimming larvae were filmed simultaneously with two digital high-speed video cameras (recording at 500 frames s^-1) using the methodology described by McHenry and Strother(in press). These cameras(Redlake PCI Mono/100S Motionscope, 320 pixels×280 pixels per camera,each equipped with a 50 mm macro lens, Sigma) were directed orthogonally and both were focused on a small volume (1 cm³) of water in the center of an aquarium (with inner dimensions of 3 cm width × 3 cm depth ×6 cm height). Larvae were illuminated from the side with two fiberoptic lamps(Cole Parmer 9741-50).

We modeled the hydrodynamics of the tail using a blade-element approach that divided the length of the tail into 50 tail elements and calculated the force generated by each of these elements. Each element was dorso-ventrally oriented, meaning that the length of each element ran from the dorsal to the ventral margins of the fin. For each instant of time in a swimming sequence,the force acting on each element (E_j, where j is the tail element number) was calculated by assuming that it generated the same force as a comparably sized flat plate moving with the same kinematics. Our models assume that each tail element generates force that is independent of neighboring elements. This neglects any influence that flow generated along the length of the body may have on force generation. The total force generated by such a plate is the sum of as many as three forces: the acceleration reaction (E_ja), skin friction(E_js) and the form force(E_jf; Fig. 3). The contribution of each of these forces to the total force and moment instantaneously generated by the tail was calculated by taking the sum of forces and moments generated by all elements (see Appendix). Dividing the tail into 75 and 100 tail elements did not generate predictions of forces or moments that were noticeably different from predictions generated with 50 tail elements, but models with 25 tail elements did generate predictions different from models with 50 elements. Therefore, we ran all simulations with 50 tail elements.

We modeled the swimming of larvae with both quasi-steady and unsteady models. In the quasi-steady model, the force generated by the tail (F)was calculated as the sum of skin friction (F_s) and the form force (F_f; F=F_f+F_s), and the total moment(M) was calculated as the sum of moments generated by skin friction(M_s) and the form force (M_f; M=M_f+M_s). According to this model,the force acting on a tail element is equal to the sum of the form force and skin friction acting on the element(E_j=E_jf+E_js; Fig. 3B). In the unsteady model, the force generated by the tail was calculated as the sum of all three forces(F=F_f+F_s+F_a,where F_a is the acceleration reaction generated by the tail), and the total moment was calculated as the sum of moments generated by all three forces(M=M_f+M_s+M_a,where M_a is the moment generated by the acceleration reaction). According to the unsteady model, the force acting on a tail element is equal to the sum of the form force, skin friction and acceleration reaction(E_j=E_jf+E_js+E_ja; Fig. 3C).

In order to examine how the relative magnitude of form force and skin friction changes with the Re of the body, we ran a series of simulations using model larvae of different body lengths. Each simulation used the mean morphometrics and kinematic parameter values. The non-dimensional morphometrics and kinematics were scaled to the mean measured tail-beat period and the body length used in the simulation. This means that animations of the body movements in the model appeared identical for all simulations (i.e. models were kinematically and geometrically similar), despite being different sizes.

We tested our mathematical models by comparing the measured forces and swimming speeds of larvae with model predictions. We measured the mean thrust(force directed towards the anterior) and lateral force generated by a tethered larva and used our model to predict those forces using the same kinematics as measured for the tethered larva and the mean body dimensions. Such measurements and model predictions were made for a number of larvae, and a paired Student's t-test (Sokal and Rohlf, 1995) was used to compare measured and predicted forces. Such comparisons were made with predictions from both the quasi-steady model and the unsteady model.

Predictions of mean swimming speeds from both models were compared with measurements of speed. Model predictions of swimming speed were generated using the mean body dimensions and the tail kinematics of individual larvae measured during tethered swimming. This assumes that the midline kinematics of freely swimming larvae were not dramatically different from that of tethered larvae. Mean swimming speeds were measured on a different sample of freely swimming larvae, and an unpaired t-test(Sokal and Rohlf, 1995) was used to compare predictions of swimming speed with measurements. We verified that samples did not violate the assumption of a normal distribution by testing samples with a Kolomogorov—Smirnov test (samples with P>0.05 were considered to be normally distributed).

Using measured kinematics (Fig. 4, Table 1), we tested the ability of hydrodynamic models to predict both the timing and mean values of forces generated by larvae. The magnitude of predictions of form force and the acceleration reaction (Fig. 5A-C) were approximately two orders of magnitude greater than the predictions for the tail inertia and skin friction forces(Fig. 5D,E). Due to the low magnitude of skin friction, the tail force predicted in the lateral direction by the quasi-steady model (F=F_f+F_s)was qualitatively indistinguishable from the prediction of form force(Fig. 5F). The prediction for the tail force in the lateral direction by the unsteady model had the addition of the acceleration reaction(F=F_f+F_s+F_a),which generated peaks of force when the form force was low in magnitude(Fig. 5F). These force peaks were not reflected in the measurements of lateral force(Fig. 5G). This measured force oscillated in phase with trunk angle (θ; phase lag mean ± 1 S.D.=0.03±0.02 tail-beat periods, P=0.230, N=11; Figs 5H, 6), unlike the acceleration reaction, which was predicted to be out of phase with trunk angle. Both quasi-steady and unsteady models predicted mean thrust and mean lateral force that was statistically indistinguishable from measurements(Table 2).

The force predictions by the quasi-steady model more closely matched the timing of measurements than those of the unsteady model(Fig. 7). The force predicted by the quasi-steady model (F=F_f+F_s)oscillated in phase with measured lateral forces. However, the unsteady model(F=F_f+F_s+F_a)predicted peaks of force generation by the acceleration reaction acting in the direction opposite to the measured force (Figs 5, 7). At instants of high tail speed, the form force was large and was followed by the acceleration reaction acting in the opposite direction as the tail decelerated and reversed direction. Although both models accurately predicted mean forces(Table 2), the timing of force production suggests that the acceleration reaction does not generate propulsive force in the swimming of ascidian larvae.

Simulations of free swimming allowed the body of larvae to rotate and translate in response to the hydrodynamic forces generated by the body. As such movement could contribute to the flow encountered by a swimming larva,the forces generated by freely swimming larvae were not assumed to be the same as those generated by tethered larvae. Therefore, simulations of free swimming were a closer approximation of the dynamics of freely swimming larvae and provided a test for whether the results of tethering experiments apply to freely swimming larvae.

The results of these simulations support the result from tethering experiments that the acceleration reaction does not play a role in the hydrodynamics of swimming. The quasi-steady model(F=F_f+F_s) predicted a mean swimming speed that was statistically indistinguishable from measured mean swimming speed. By contrast, the unsteady model(F=F_f+F_s+F_a) predicted a mean swimming speed that was significantly different from measurements(Table 2). We found small(<4%) differences in predicted mean speed between models using a high(ρ_body=1.250 g ml^-1) and low(ρ_body=1.024 g ml^-1) tissue density, suggesting that any inaccuracy in the tissue density used for simulations(ρ_body=1.100 g ml^-1) had a negligible effect on predictions.

Reynolds number values varied among different regions of the body(Table 3). The mean Reynolds number for the whole body (Re=7.7×10¹) was larger than the Reynolds number for the trunk(Re_a=2.8×10¹) because the whole body is greater in length than the length of just the trunk. The mean height-specific Reynolds number (Re_jl) and the mean position-specific Reynolds number (Re_js) were larger towards the posterior (Table 3).

Predictions by the quasi-steady model showed how thrust and drag may be generated differently by form force and skin friction at different Re. At Re≈10⁰, both thrust and drag were predicted to be dominated (>95%) by skin friction acting on the trunk and tail (Fig. 8A,B). At Re≈10¹, most drag (63%) was generated by skin friction acting on the trunk, and most thrust (69%) was generated by skin friction acting on the tail (Fig. 8C,D). At Re≈10², drag was generated by a combination of skin friction and form force, but thrust was generated almost entirely by form force acting on the tail (Fig. 8E,F). By running simulations throughout the intermediate Re range (10⁰<Re<10²), we found that form force gradually dominates thrust generation (up to 98%) with increasing Re. Although the proportion of drag generated by form force increases with Re, skin friction generates a greater proportion of drag (>62%) than does form force, even at Re≈10².

It is surprising that our results suggest that the acceleration reaction does not contribute to thrust and drag in the steady undulatory swimming of Botrylloides sp. larvae. Vyman's (1974) model for the energetics of steady swimming in fish larvae assumes that the acceleration reaction should operate at the Reynolds number at which these larvae swim(Re≈10²). Although the energetic costs of locomotion predicted by Vyman (1974) show good agreement with measurements, these predictions from an unsteady model have not been compared with the predictions of a quasi-steady model. Furthermore, the hydrodynamics assumed by Vyman(1974) have yet to be experimentally tested. By contrast, Jordan(1992) did compare quasi-steady and unsteady predictions with measurements of the startle response behavior of the chaetognath Sagitta elegans. This study found that the unsteady model better predicted the trajectory of swimming than did the quasi-steady model, which suggests that the acceleration reaction is important to undulatory swimming at intermediate Re.

This discrepancy between our results and Jordan(1992) on the relative importance of the acceleration reaction may be reconciled if the acceleration reaction coefficient varies with Re. The acceleration reaction is the product of the acceleration reaction coefficient (which depends on the height of the tail element), the density of water and the acceleration of a tail element (equation 12). Both Jordan(1992) and the present study used the standard inviscid approximation (equation 13) for the acceleration reaction coefficient (used in elongated body theory; Lighthill, 1975). However,chaetognaths attain Re≈10³ and more rapid tail accelerations than ascidian larvae. If the actual acceleration reaction coefficient is lower than the inviscid approximation at the Re of ascidian larvae (Re≈10²), then predictions of the acceleration reaction would be smaller in magnitude. The chaetognath may still generate sizeable acceleration reaction in this regime by beating its tail with relatively high accelerations.

Although swimming at Re>10² has not been reported among ascidian larvae, numerous vertebrate and invertebrate species do swim in this regime. We predict that as Re approaches 10³, the acceleration reaction contributes more to the generation of thrust in undulatory swimming. Although it remains unclear how the magnitude of the acceleration reaction changes with Re, the unsteady models proposed here (F=F_f+F_s+F_a)and elsewhere (Jordan, 1992; Vlyman, 1974) should approximate the hydrodynamics of undulatory swimming at Re≈10³.

In support of prior work (e.g. Fuiman and Batty, 1997; Jordan,1992; Vlyman,1974; Webb and Weihs,1986; Weihs,1980), our quasi-steady model(F=F_f+F_s) predicted that the relative magnitude of inertial and viscous forces is different at different Re. At Re≈10°, skin friction (acting on both the trunk and tail; Fig. 8)dominated the generation of thrust and drag(Fig. 9). This result is consistent with the viscous regime proposed by Weihs(1980) for swimming at Re<10¹. Also in accordance with Weihs(1980) are the findings that form force contributes more to thrust and drag at high Re than at low Re (Fig. 9) and that thrust (Fig. 8) is dominated by form force at Re≈10². However, it is surprising that drag was generated more by skin friction than form force at Re≈10² (Figs 8, 9). Contrary to Weihs'(1980) proposal for an inertial regime at Re>2×10², this result suggests that the fluid forces that contribute to thrust are not necessarily the same forces that generate drag. This is unlike swimming in spermatozoa (at Re≪10⁰), where both thrust and drag are dominated by skin friction acting on both the trunk and flagellum(Gray and Hancock, 1955), or some adult fish (at Re≫10²), where thrust and drag are both dominated by the acceleration reaction(Lighthill, 1975; Wu, 1971).

Our results suggest that ontogenetic or behavioral changes in Recause gradual changes in the relative contribution of skin friction and form force to thrust and drag. As pointed out by Weihs(1980), differences in intermediate Re within an order of magnitude generally do not suggest large hydrodynamic differences. Although it has been heuristically useful to consider the differences between viscous and inertial regimes (e.g. Webb and Weihs, 1986), it is valuable to recognize that these domains are at opposite ends of a continuum spanning three orders of magnitude in Re. This distinction makes it unlikely that larval fish grow through a hydrodynamic `threshold' where inertial forces come to dominate the hydrodynamics of swimming in an abrupt transition with changing Re (e.g. Muller and Videler, 1996).

In summary, our results suggest that the acceleration reaction does not play a large role in the hydrodynamics of steady undulatory swimming at intermediate Re (10⁰<Re<10²). Our quasi-steady model predicted that thrust and drag are generated primarily by skin friction at low Re (Re≈10⁰) and that form force generates a greater proportion of thrust and drag at high Re than at low Re. Although thrust is generated primarily by form force at Re≈10², drag is generated more by skin friction than form force in this regime. Unlike swimming at Re>10² and Re<10⁰, the fluid forces that generate thrust cannot be assumed to be the same as those that generate drag at intermediate Reynolds numbers.

List of symbols
 
• a

  length of the trunk

 
• A

  acceleration of the body

 
• B

  position of the center of mass

 
• c_ja

  added mass coefficient

 
• c_jf

  coefficient of force on tail element due to form force

 
• c_jnorm

  coefficient of total force on tail element in the normal direction

 
• c_js

  coefficient of force on tail element due to skin friction

 
• c_{js+f norm}

  coefficient of force on tail element in the normal direction due to form force and skin friction

 
• D

  position of the center of volume of the trunk

 
• E_j

  total force acting on a tail element

 
• E_ja

  acceleration reaction on a tail element

 
• E_jf

  form force on a tail element

 
• E_js

  skin friction on a tail element

 
• F

  total force generated by the tail

 
• F_a

  tail force generated by acceleration reaction

 
• F_f

  tail force generated by form force

 
• F′_f

  tail force generated by form force in the direction of thrust

 
• F_inertia

  tail inertia force

 
• F_norm

  force in the normal direction measured on a plate

 
• F_s

  tail force generated by skin friction

 
• F′_s

  tail force generated by skin friction in the direction of thrust

 
• g

  acceleration due to gravity

 
• h_cm

  distance from the tether pivot to the center of mass of the tether

 
• H_{f tail}

  percentage of thrust generated by form force on the tail

 
• h_objective

  distance from the tether pivot to the objective

 
• h_tip

  distance from the tether pivot to the tip of the pipette

 
• i

  volumetric element number

 
• I

  inertia tensor for the body of a larva

 
• I^B

  inertia tensor for the body in the body's coordinate system

 
• I_tether

  moment of inertia of the tether

 
• j

  tail element number

 
• k_damp

  damping coefficient

 
• k_f

  coefficient of the form force on the trunk

 
• k_spring

  spring coefficient

 
• l

  height of a tail element

 
• L

  body length

 
• m_body

  mass of the body of a larva

 
• M

  total moment

 
• M_a

  moment generated by the acceleration reaction

 
• M_f

  moment generated by form force

 
• M_s

  moment generated by skin friction

 
• m_tether

  mass of the tether

 
• O

  moment generated by force on the trunk

 
• p

  speed of the trunk

 
• P

  tail-beat period

 
• P

  velocity of the trunk

 
• q

  total number of volumetric elements

 
• R_j

  position of the tail element with respect to the center of mass

 
• Re

  Reynolds number for whole body

 
• Re_a

  Reynolds number of the trunk

 
• Re_jl

  height-specific Reynolds number of a tail element

 
• Re_js

  position-specific Reynolds number for a tail element

 
• r_tether

  inner radius of the micropipette

 
• s

  distance along the tail from the tail base to the element

 
• s_j

  position of the element down the length of a tail

 
• S

  projected area of the trunk

 
• t

  time

 
• T

  force acting on the trunk

 
• T_f

  form force acting on the trunk

 
• T_s

  skin friction acting on the trunk

 
• ū

  mean swimming speed

 
• \({\bar{v}}\)

  mean tail element speed

 
• V_i

  velocity of a tail element

 
• νj norm

  speed of the normal component of the velocity of a tail element

 
• νj tan

  speed of the tangent component of the velocity of a tail element

 
• V_{j norm}

  normal component of the velocity of a tail element

 
• V_{j tan}

  tangent component of the velocity of a tail element

 
• x_i

  x-coordinate of volumetric element

 
• y_i

  y-coordinate of volumetric element

 
• z

  position of inflection point along the length of the tail

 
• α

  amplitude of change in curvature

 
• χ

  amplitude of change in trunk angle

 
• δ

  linear deflection of the tether

 
• Δs

  width of a tail element

 
• Δw_i

  volume of a volumetric element

 
• ϵ

  wave speed of inflection point

 
• φ

  radial deflection of the tether at its pivot

 
• φe

  measurement of tether deflection

 
• γ

  period of change in curvature

 
• κ

  tail curvature

 
• μ

  dynamic viscosity of water

 
• θ

  trunk angle

 
• ρ

  density of water

 
• ρbody

  density of tissue

 
• Ω

  rate of rotation about the center of mass

 
• ζ

  phase lag of inflection point relative to trunk angle

 
• *

  non-dimensional quantity

We thank M. Koehl for her guidance and advice, S. Sane for his wisdom on hydrodynamics, and A. Summers, W. Korff and W. Getz for their suggestions on the manuscript. This work was supported with an NSF predoctoral fellowship and grants-in-aid of research from the American Society of Biomechanics, Sigma Xi,the Department of Integrative Biology (U.C. Berkeley) and the Society for Integrative and Comparative Biology to M. McHenry. Additional support came from grants from the National Science foundation (# OCE-9907120) and the Office of Naval Research (AASERT # N00014-97-1-0726) to M. Koehl.