Royal Society Publishing

Passive elastic mechanism to mimic fish-muscle action in anguilliform swimming

Sophie Ramananarivo , Ramiro Godoy-Diana , Benjamin Thiria


Swimmers in nature use body undulations to generate propulsive and manoeuvring forces. The anguilliform kinematics is driven by muscular actions all along the body, involving a complex temporal and spatial coordination of all the local actuations. Such swimming kinematics can be reproduced artificially, in a simpler way, by using the elasticity of the body passively. Here, we present experiments on self-propelled elastic swimmers at a free surface in the inertial regime. By addressing the fluid–structure interaction problem of anguilliform swimming, we show that our artificial swimmers are well described by coupling a beam theory with the potential flow model of Lighthill. In particular, we show that the propagative nature of the elastic wave producing the propulsive force is strongly dependent on the dissipation of energy along the body of the swimmer.

1. Introduction

Undulatory propulsion is a means of locomotion shared by living organisms over a wide range of scales and in many different media [1]. From snakes [2] to sandfish [3], from eels [4] to spermatozoa or motile bacteria [5], net forward motion is achieved by propagating waves along a deformable body. In fluids, the anguilliform swimming dynamics was first addressed in pioneering studies during the 1950s and 1960s by Taylor [6], Gray & Hancock [7], Machin [8] and Lighthill [9]. They established that the propulsive force originates from either viscous friction (the so-called resistive theory) or inertial momentum transfer (Lighthill's reactive model), depending on the regime of Reynolds number, which measures the importance of inertial to viscous actions in a given flow. In both cases, the characteristics of the propagating wave (phase velocity, wavelength and amplitude) are crucial in determining the swimming performance [9,10].

A vast amount of theoretical and numerical works has followed [1014], from Stokes or viscoelastic flows related to microorganism propulsion [5] to inertial regimes [15,16] or three-dimensional geometries [17]. This research shed light on various undulatory swimming modes, and not only provided the basis for a broad spectrum of applications in robotics and engineering at the macroscale, but also in the public health domain where, for instance, sperm swimming speed is significantly related to fertilization success [18].

In the inertial regime, the most relevant analytical model of fish swimming is Lighthill's reactive theory. Based on a potential flow approximation, Lighthill showed that the estimation of the thrust force requires only the knowledge of the local kinematics at the tail of the deformable body [10]. Thus, calling y, the local deflection of the slender body with respect to the axis of swimming (x-axis), and U the swimming velocity, the average total thrust force, 〈T〉, reduces, in the limit of small lateral displacement, toEmbedded Image 1.1where ρS is the added mass of fluid (with ρ the fluid density and S the swimmer cross section). The subscript r refers to the location of the tail where ∂xy and ∂ty are estimated. This expression of the force is obtained writing that the swimming power 〈TU (i.e. the product of the thrust force times the swimming velocity) corresponds to the rate of working done by the fish, 〈W〉, minus the rate of shedding of the kinetic energy associated with lateral fluid motion, say Embedded Image. The part corresponding to the energy transferred to the fluid can be seen as a source of dissipation for the mechanics of the swimming fish, as this energy is lost into the flow. In the case of passive swimmers that rely on elasticity to propagate the undulating motion, this dissipated energy will not be available to sustain the body deformation (see [19] in the context of energy harvesting). The value of the thrust force depends thus on the nature of the near- body flow field, which is extensively discussed in this work.

For living animals, the kinematics of the propagating wave is obtained by local contractions of the muscles all along the body. The phase velocity, wavelength and wave amplitude are therefore set separately in order to select a specific regime. However, in the case of artificial mimics of real swimmers, a mechanical model of fish-muscle action would require a continuous distribution of actuators [20]. Such a design is not only complicated, but also difficult to scale down in size. Another approach is to consider a local actuation and use the elasticity of the body to propagate the wave [16,2123]. The implementation of the forcing that will provide the necessary energy to create the anguilliform kinematics is therefore much simplified.

Here, we present a full description of the mechanics of such self-propelled swimmers using a careful experimental set-up and solving the complete fluid–structure problem of a forced beam in a fluid in the inertial regime. The challenge in the latter is to obtain an estimation of the tail kinematics as a function of the forcing, using a coupled fluid–elasticity approach. The average swimming velocity, along with an estimation of the swimming efficiency, is predicted afterwards by balancing the thrust from Lighthill's prediction and the drag force. We also address the more general problem of wave propagation in bounded media, using a local energy injection. The dynamics of the propagating wave that drives the anguilliform locomotion is shown, in particular, to have a crucial dependence on energy dissipation along the body of the swimmer. Furthermore, we show that in the inertial regime, self-propelled swimmers using passive elastic deformation verify Lighthill's reactive theory even at a free surface. From this work, we confirm the potential of the present experiment to be of importance in the future design of artificial swimmers.

2. Results

2.1. Swimming kinematics

The slender swimmer in the present experiment is actuated on one extremity using magnetic interactions. The induced motion of the head is a combination of heaving and pitching, and the resulting body undulations propel the filament on the surface of a water tank (see figure 1 and Material and methods for details).

Figure 1.

Experimental set-up: (a) Top-view snapshots of the swimmer propelling itself forward after the opening of the gates (that are visible at the bottom of the images). (See the electronic supplementary materials, movie S1, seeding particles have been added on the surface to visualize the flow.) (b) Swimmer at the free surface of a water tank consisting of a flexible cylindrical tail made of acrylic polymer. A magnet embedded in the head of the filament is forced to oscillate using the time-varying magnetic field produced by a Helmholtz pair. (c) Successive pictures of a filament over a period of undulation showing the forward swimming velocity, U, and the speed of the travelling wave, υφ.

The present system differs from Lighthill's configuration in that the swimmer is not fully immersed in the fluid (it is placed at the air–water interface to confine its displacements into a two-dimensional plane). In particular, surface waves generated by the moving cylinder radiate energy away from it and may represent an extra source of drag [2426]. However, this so-called wave drag can be neglected here as the ratio of the wave drag to form drag, proportional to the square of the Weber number [27], remains small (10−6 to 0.1).

We vary the length of the swimmer, L, and the frequency of the sinusoidal forcing, f, while the other parameters are kept constant for simplicity (among them, the amplitude of the prescribed displacement at the head, Af, and the flexural rigidity of the body, B). The full kinematics of the swimmer as a function of the experimental parameters is recovered from high-speed camera recordings. From this, we extract the time evolution of the local deflection y(s, t) (where s is the local curvilinear abscissa along the centreline of the body), the wave speed υφ, and the cruising velocity U (figure 1).

The results are plotted in figure 2. Figure 2a shows the evolution of the cruising velocity as a function of the beat frequency for different lengths of the swimmer. U is found to be an increasing function of the forcing frequency for all the swimmers tested. It can also be seen that ‘short’ swimmers are faster than ‘long’ swimmers in the range of parameters tested. This can be understood by considering the results displayed in figure 2b: for a given forcing amplitude and forcing frequency (Af, f), the tail amplitude, Ar, is decreasing with L. As thrust is based on the kinematics at the tail, and consequently on Ar, short swimmers will therefore swim faster. More generally, as illustrated in figure 2f, the overall shape of the swimmers tends to decrease in amplitude along the length. This characteristic shape is of course highly dependent on the forcing parameters and the length of the swimmer and is owing to the fact that the energy is not conserved along the body. A further crucial information for the swimming efficiency [9] that can be extracted from the kinematics of the undulating filament is the phase velocity, υφ, which is displayed in the inset in figure 2d. A first observation is that, as expected, the global behaviour is independent of the length of the filament.

Figure 2.

(a) Swimming velocity, (b) amplitude of the oscillations at the tail and (inset in d) phase velocity, as a function of the forcing frequency. Experimental data plotted in a non-dimensionalized form, as a function of the non-dimensionalized angular frequency Embedded Image, with Embedded Image: (c) ratio of the cruising velocity and the characteristic actuation speed Embedded Image, (d) non-dimensionalized phase velocity Embedded Image, (e) amplitude of the oscillation at the tail normalized by that of the head Embedded Image. The solid lines correspond to the theoretical predictions of the model presented in the ‘Fluid–structure model’ section. (f) Typical undamped (3 cm long filament forced at 10 Hz) and damped (4 cm long filament forced at 20 Hz) amplitude envelopes (see the electronic supplementary material, movieS2 and S3).

All the experimental observations can be collapsed by plotting the data in a compact form using the characteristic lengths and velocities of our system. We define ω0 as the natural relaxation frequency of the elastic filament: Embedded Image, with μ its mass per unit length. Figure 2ce displays, respectively, the non-dimensionalized swimming velocity Embedded Image, phase velocity Embedded Image and amplitude Embedded Image, as a function of Embedded Image, where Embedded Image denotes the angular frequency. Note that Embedded Image has the same form as the inverse of the Strouhal number, as defined in the context of self-propelled swimmers (e.g. [20,28]). Embedded Image, which compares the characteristic speed of the imposed head oscillations to the cruising speed, decreases and converges to an asymptotic value. Its decay has to be compared and correlated with the decrease of Embedded Image with Embedded Image, that is a consequence of passive elastic propagation.

2.2. Fluid–structure model

In the following section, we aim to understand the mechanisms involved in the motion of the filament, to be able to predict what would be the output from a given forcing, and conversely what would be the required forcing to achieve a given kinematics. The filament is modelled as a forced slender beam [29] of length L with circular cross section S = πd2/4 (with d its diameter), mass per unit length μ and stiffness B. It is immersed in a uniform fluid of density ρ, moving at velocity U. We neglect, in the present model, any effect of the interface by considering a filament moving in the bulk. The beam obeys the Euler–Bernouilli equationEmbedded Image 2.1with s the curvilinear coordinate, r(s, t) = (x,y) the beam position, f the fluid forces acting on the beam, T, a tension in the plate that enforces the inextensibility condition Embedded Image and the unit vector normal to the beam (figure 3a). The extremity of the beam in s = L is free, that is Embedded Image. The head of the filament (s = 0) is actuated with a prescribed periodic displacement and rotation: y(0, t) = Afcos(ωft), ∂xy(0, t) = θfcos(ωft + ϕ), with ωf the angular frequency, θf the maximal rotation angle of the head and ϕ the phase shift between pitching and heaving (see also [30]).

Figure 3.

(a) Definition of the coordinates and geometry of the beam model considered. Simulated motion of the beam when implementing equation (2.7) gradually: (b) with only the two first terms describing a classic elastic beam (c) adding the ‘flag’ terms in brackets, (d) adding the quadratic fluid term. (Online version in colour.)

In the limit of slender structures (Embedded Image), and for purely potential flow, Lighthill provided a leading order expression of the fluid force acting on the beam [31]. This so-called reactive force results from the reaction of the fluid accelerated by the body movements and can be expressed as [32,33]Embedded Image 2.2where M = ρS is the added mass per unit length owing to the fluid, κ is the beam curvature, ut and un are the longitudinal and normal components of the beam velocity, respectively, relative to the uniform water flow (so that Embedded Image, with Embedded Image the unit vector tangent to the filament). Another contribution for the fluid force is added to account for the loss of energy along the swimmer. Considering that the transverse Reynolds numbers encountered in the present experiment are Ret = Af ωf d/ν ≈ 10–140, this term is given the classic quadratic form (e.g. [6,34,35])Embedded Image 2.3with Cd the drag coefficient associated to transverse motions.

Inserting f = fr + fd into equation (2.1) and projecting the equation on the x- and y-axes gives two coupled dynamical equations for x(s, t) and y(s, t). Following Eloy et al. [33], the equations are decoupled by first using the x-projection to evaluate the tension T, and then eliminating x(s, t) and its derivatives using the inextensibility condition (see also [36]). We assume that the lateral deflections are small (Embedded Image and Embedded Image) and discard the terms of order larger than y2. The curvilinear coordinate s is approximated by its abscissa x, the first corrections coming up through this substitution being of order y3. This yields the weakly nonlinear dynamical equationEmbedded Image 2.4withEmbedded Image 2.5andEmbedded Image 2.6Equation (2.4) is non-dimensionalized using L and Embedded Image as characteristic length and time. It readsEmbedded Image 2.7

The dimensionless quantities are noted with tildes; Embedded Image is the reduced velocity, Embedded Image the mass ratio and Embedded Image the non-dimensionalized damping coefficient. Note that Embedded Image depends on L, which reflects the increasing effect of damping when the filament is longer. Equation (2.7) is solved numerically (see Material and methods) using the experimental parameters. Here, the only unknown parameter is the transverse drag coefficient Cd. It is determined using a minimization approach (see Material and methods). Figure 4 shows the experimental and computed time dynamics of a 4.5 cm long filament, using the obtained value of Cd.

Figure 4.

(a) Successive computed shapes superimposed to pictures of a 4.5 cm long swimmer forced at f = 19 Hz and (b) the corresponding experimental and theoretical amplitude envelopes. (Online version in colour.)

The first two terms of equation (2.7) are the classic ingredients of the beam equation: the inertia of the beam (with an added mass effect owing to the surrounding fluid), balanced by an elastic restoring force. The action of the fluid brings two additional effects: a quadratic dissipation term (last term) and a ‘flag’ term (in square braquets) which depends on the swimming velocity U. The flag term is known to cause the beam to flutter above a critical flow velocity (this is called the flapping-flag instability, e.g. [29,3739]). In the range of parameters of the experiments (Embedded Image, Embedded Image and Embedded Image), the dissipative part has an order of magnitude 10–30 times greater than the other dimensionless fluid term, which makes it the main fluid contribution in the dynamical balance.

Interestingly, in our system, this term is also responsible for the wave propagation along the beam, a specific feature of anguilliform swimming; this is a non-trivial question as such bounded elastic systems usually produce, by definition, standing waves. Indeed, the implementation of the first two terms describing the elastic beam (i.e. without any losses in transmission) yields a stationary wave (figure 3b). When adding the ‘flag’ term, the obtained kinematics slightly departs from a standing wave, but the velocities reached by the swimmers are not sufficient to trigger propagation (figure 3c). The flag forces related to the upstream velocity U have thus a minor effect on the mechanism of propagation, because our range of parameters lies below the onset of the flapping [40]. This assertion is supported by the fact that, experimentally, the kinematics of the filament is similar when swimming behind the gates (U = 0) and when moving freely at its cruising velocity along the water tank (see the electronic supplementary materials, movie S1). However, including the quadratic fluid force does enable propagation by drawing enough energy from the beam to prevent the build-up of a standing wave (figure 3d). This dissipative action of the surrounding fluid is thus decisive, in that it allows the undulations to develop into a travelling wave.

It is important to note that, physically, the quadratic term in equation (2.3) reflects the kinetic energy lost into the fluid through transverse flow separation at each half-stroke (e.g. [41]). In Lighthill's theory, the estimation of the thrust generated by the swimmer is based on considerations of momentum and energy losses [9]. Our system differs from this theory in that, energy is not only wasted from the tail, but also from the sides. Dissipation thus contributes to the energy balance through an additional wasted kinetic energy term that can be written in a similar way to the kinetic energy lost in the wake. The available energy to produce thrust is hence reduced, but the global mechanism for propulsion remains unchanged. At first approximation, the expression of the thrust, taking into account quadratic dissipation, has the form given by equation (1.1) but with a correction factor depending on the transverse drag coefficient Cd. For the determination of the cruising velocity and the swimming efficiency, this correction is taken into account when balancing the expression of the thrust force with the global drag of the swimmer, through the adjustable coefficient CD. Transverse flow separation impacts the performance of the swimmer mostly by damping the body oscillations along its length (owing to the fact that energy is extracted from the solid and transferred univocally to the fluid).

The results of the numerical simulation are plotted in figure 2 in solid lines. The phase velocity is computed a posteriori from the simulated motion of the beam. The measurement is performed in the same way as for experimental data, by tracking in time, the position of the points of the swimmer crossing the body midline. Comparison between experimental data and theoretical predictions shows a good agreement, with a relative deviation of 22% for Embedded Image and 29% for Embedded Image. This confirms the ability of the present model to describe our system and to predict the tail amplitude as well as the phase velocity.

One should note that, despite the possible surface effects mentioned previously, the model of a fully immersed swimmer described by equation (2.7) provides a good prediction of the time–space dynamics of our surface swimmer. This leads to the conclusion that surface effects may either be negligible or already incorporated in the adjustable quadratic fluid dissipation term (equation (2.3)). A detailed study of the surface deformations is the subject of ongoing work.

3. Discussion

3.1. Swimming speed

Using Lighthill's model (equation (1.1)), we obtain a prediction for the thrust force. The swimming speed reached in the steady-state regime is then determined by a balance between the thrust generated and the drag experienced by the filament. In the regimes encountered here, the swimmer produces a significant wake whose width is set by the amplitude of the tail displacement Ar; the drag is hence given the form [41]Embedded Image 3.1where CD is a drag coefficient and S′ = 2Ar × d the effective section. Using a simplified kinematics y(x, t) = Arcos(2π(ftx/λ)) (with λ, the wavelength), the equality between equations (1.1) and (3.1) gives us a first-order estimation of the swimming velocityEmbedded Image 3.2where Embedded Image is a function of the spatial shape characteristics of the swimmer. Although desirable, a robust measurement or prediction of CD for a slender structure whose shape is changing in time is very difficult to perform, especially as the nature of this drag is not a straightforward question.1 In the following, it has been used as an adjustable parameter and its value of CD ≈ 0.23 is in agreement with the order of magnitude of the drag coefficient of streamlined bodies [42].

All data points from the present experiments are plotted in the compact form of equation (3.2) in figure 5a. It is noteworthy that, although the theory is based on relatively simple arguments, the data collapse around a single linear relationship. A slight deviation above the master curve is observed for the longest filaments, meaning that the obtained value of CD ≈ 0.23 overestimates the actual drag. This may be related to the enhanced streamlining of the envelope of oscillation observed for long filaments where the amplitude at the tail Ar is significantly smaller than that of the head.

Figure 5.

(a) Implementation of equation (3.2) using experimental data. The line of slope unity around which the data line up is the identity defined by Lighthill's theory. A single value of CD ≈ 0.23 was used, which is obtained as an adjustable parameter giving the best fit of all points. (b) Hydromechanical efficiency, calculated from equation (3.3), plotted against γ which is a function of the kinematic characteristics of the swimmer. The dotted line corresponds to the Embedded Image function issued from Lighthill's theory.

The estimate of U provided by Lighthill's theory completes the beam model introduced previously in characterizing our swimmer.

3.2. Efficiency

Following Lighthill's steps, one can then define an efficiency (similar to the Froude efficiency of a propeller), η = UT〉/〈W〉 that illustrates the ability of the swimmer to convert the work done by its flexural movements into useful thrust. When the undulation takes the form of a travelling wave of velocity υφ, this hydromechanical efficiency can be written (see [10])Embedded Image 3.3In other words, the swimming speed U should tend toward υφ to produce thrust efficiently; U cannot be too close to υφ though, or the thrust generated will not be sufficient to overcome the drag resistance 〈D〉 (the limit case U = υφ corresponding to a swimmer slipping into the water without giving it any lateral displacement). Using equation (3.2), η can be expressed as a function of the kinematic characteristics of the swimmer:Embedded Image 3.4where γ(λ, Ar) = π3dAr/4CDλ2. The efficiency, calculated from equation (3.3), is plotted against γ using experimental data (figure 5b). The good agreement with the theoretical γ-function of equation (3.4), represented here by the dotted line, confirms the ability of the expression to describe the dependence of η on λ and Ar. As both lengths result from the elastic response of the filament, equation (3.4) thereby describes how the cost of propulsion changes with the imposed forcing.

In order to make the forcing dependency of the efficiency more explicit and to understand the observed trend in figure 5b, γ(λ, Ar) is rewritten by approximating the dispersion relation to that of a beam immersed in a still fluid without any energy losses, λ = 4π2(B/(μ + M)1/4f−1/2. It then reads γβArf, which is homogeneous with respect to the speed of the lateral displacements of the tail, with β a constant set by the characteristics of the fluid–solid system; this γ-form includes both the input (f) and the outcome (Ar) of the ‘transfer function’ represented by the elastic system. Injecting it into equation (3.4) and noticing that γ is small compared with unity, we get Embedded Image. As observed in figure 5b, swimmers thus experience an improved efficiency when increasing the forcing frequency. Note that contrary to swimmers executing standing waves, which cannot achieve efficiencies exceeding 1/2 [9], the present system yields values of η systematically higher than 1/2. This highlights the importance of developing travelling waves along the beam. As mentioned previously, however, passive propagation requires a source of dissipation. This loss of energy causes a decrease in amplitude along the swimmer that is especially pronounced for long filaments or high forcing frequencies (figure 2f). In the present experiment, the fact that an increase in frequency cannot be dissociated of a decrease in the amplitude Ar limits the efficiency that can be reached, which consequently is reflected by a saturation in the η(γ) curve. This limiting effect of the spatial damping also shows in figure 2c, where Embedded Image decays with Embedded Image. The crucial question for the present swimmers is then to remove enough energy along the passive elastic body to prevent the build-up of standing waves while maintaining a substantial amplitude of oscillation down to the tail. A key parameter is clearly the magnitude of the dissipative term in equation (2.7), which is quite large in the present experiment, but that will most likely to change for other body shapes or fully submersed swimmers.

4. Conclusions

We have presented useful results for the design of an easy-to-build model of a self-propelled swimmer, which could be adapted to other environments and scales. These swimmers differ from fishes in that they are locally actuated and use their elasticity to passively propagate a wave. We fully characterize the swimmer by first developing a model giving a complete description of its dynamics under actuation, and then showing that average swimming speed and efficiency are well predicted by a potential flow theory as introduced by Lighthill.

More precisely, we have shown that the production of anguilliform kinematics is dependent on the ability of the mechanical system to dissipate energy along the body of the swimmer in order to prevent the build-up of stationary waves. In return, this substantial amount of energy lost into the fluid leads to a decrease of the undulations along the length of the swimmer that limits its performance. This non-trivial problem is the focus of oncoming works in relation to previous studies on other passive fluid–elasticity mechanisms relevant to bioinspired propulsion in the inertial regime [43,44]. Lastly, we have showed that a precise description involves a complete understanding of the nature of the global drag acting against the forward motion. This point will also be studied in near future by taking into account the full kinematics of the swimmer's shape and incorporating viscous effects.

5. Material and methods

5.1. Experiments

The swimmers are composed of a slender cylindrical flexible tail (diameter d = 460 μm; flexural rigidity B = 5.4 × 10−10 ± 1.0 × 10−10 N m2) made of acrylic polymer (polyvynil syloxane), with a small embedded magnet constituting the head. They float through capillary forces at the free surface of a water tank (12 × 6 × 27 cm) which is placed between a pair of large coils mounted in a Helmholtz configuration (figure 1b). A spatially uniform magnetic field that actuates the head of the swimmer is generated. Applying an AC voltage to the coils produces an oscillating magnetic torque T(t) = m × B(t) (where m is the magnetic moment of the magnet and B(t) is the applied field) as the permanent magnet attempts to align with the alternating field. The rotational oscillations of the magnet generate a backward-propagating wave along the flexible tail, causing it to swim forward. On each run, the swimmer is first held still by a closed gate (figure 1a), accelerates from rest when the gate opens and reaches rapidly (in a typical time of order 0.5–1 s) a steady swimming speed U determined by a balance between the forward thrust generated by the body undulations and the drag experienced by the filament. The Reynolds number based on the cruising speed U and the length of the filament L, Re = UL/ν (ν being the kinematic viscosity of the fluid) ranges from 350 to 6100. Viscous effects are thus negligible with respect to inertia.

5.2. Numerical solution of the fluid–structure model

The motion of the swimmer is assumed to be harmonic, the angular frequency ωf being set by the forcing, Embedded Image, where Embedded Image is the dimensionless complex amplitude along the beam. We neglected the higher harmonics appearing through the nonlinear term (a time Fourier transform of the body motion showed that they were indeed negligible). Inserting this form of y into equation (2.7) yields the following amplitude equation:Embedded Image 5.1

Care needs to be taken with the complex notations for the nonlinear term. The 8/3π factor comes from the projection of Embedded Image onto the harmonic Embedded Image [32]. Equation (5.1) is re-written in a matrix form using finite differences and solved numerically in Matlab to obtain the complex amplitude Embedded Image, and subsequently the motion of the beam y(x, t). The beam is discretized over 100 points, which is sufficient to account for the spatial variations along its length. The actuation is imposed through the implementation of the boundary conditions.

5.3. Estimation of Cd

We calculated the value of Cd that minimizes the square difference between the simulated and experimental motion (denoted here yt(x, t) and ye(x, t)) over a time period T, Embedded Image. Repeating the process on 27 videos (three different frequencies and nine swimmers of different lengths) yielded mean value Cd = 2.2 ± 0.8, which is consistent with the typical values of order one obtained for bluff bodies [45].

Funding statement

Financial support from the French National Research Agency through project no. ANR-08-BLAN-0099 and of EADS Foundation through project ‘Fluids and elasticity in biomimetic propulsion’ is gratefully acknowledged.


We thank Olivier Doaré, Cyril Touzé, Sebastien Michelin, Christophe Eloy and Veronica Raspa for helpful discussions. We also thank Maxime Dana for his help with experimental data and Joseph Tavacoli for his careful reading of the manuscript.


  • 1 The drag balancing the thrust force could be a combination of form drag and skin friction. Here, form drag is assumed to be dominant, so that we define equation (3.1) with a drag coefficient CD that is expected to be fairly constant in the range of Reynolds number encountered for this ‘bluff body’ Reb = 2ArU/ν ≈ [40–103].

  • Received July 23, 2013.
  • Accepted August 2, 2013.


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