Oscillation of the velvet worm slime jet by passive hydrodynamic instability

The rapid squirt of a proteinaceous slime jet endows velvet worms (Onychophora) with a unique mechanism for defence from predators and for capturing prey by entangling them in a disordered web that immobilizes their target. However, to date, neither qualitative nor quantitative descriptions have been provided for this unique adaptation. Here we investigate the fast oscillatory motion of the oral papillae and the exiting liquid jet that oscillates with frequencies f~30–60 Hz. Using anatomical images, high-speed videography, theoretical analysis and a physical simulacrum, we show that this fast oscillatory motion is the result of an elastohydrodynamic instability driven by the interplay between the elasticity of oral papillae and the fast unsteady flow during squirting. Our results demonstrate how passive strategies can be cleverly harnessed by organisms, while suggesting future oscillating microfluidic devices, as well as novel ways for micro and nanofibre production using bioinspired strategies.

Indeed, just after dissection the reservoir length was L = 30 ± 1 mm. c, Solidified slime obtained from a different specimen. In this case D = 1.9 mm in the wider part. This 3D template clearly shows the difference between the narrow canal (leftmost region) and the center of the reservoir.       We have used optical microscopy to obtain the geometry of the reservoir and papilla system, as well as to analyze the tissue structure at papilla and reservoir level.
In Supplementary Fig.1 a − b, a longitudinal cut at middle reservoir level clearly shows that the specimen anatomy has a large reservoir (Re) that ends in a narrow canal (black arrow). That narrow canal leads to the opening of the oral papilla (See Fig. 1 main text). This geometry is in contrast to previously reported anatomical studies 4 where no mention to the sudden cross section change has been reported. The inner dimensions and 3-D structure of the reservoirs were directly confirmed as a solidified template of it was found during dissection, Supplementary Fig.1 c. The Onychophora slime was found in a solid state confirming the dimensions obtained by microscopy. The ratio between D and d is ∼ 20. This provides a simple, syringe-like mechanism that permits high squirting velocities at oral papilla level.
Supplementary Note 2: Average squirt times.
In Supplementary Fig.2 examples of Onychophora and the attack process are shown.
Analyzing videos of the squirting process for several specimens we counted the number of frames (fps: frames per second) at which the liquid jet was being expelled from the oral papilla. Our results are summarized in Table I. From this data we get an average of ∆t ave = 0.064 ± 0.005 s. As described in Table II, and consistent with field observations we have seen that right and left oral papillae may squirt at slightly different times, leaving the full body contraction hypothesis on weak grounds.
Supplementary Note 3: Slow muscular contraction for fast squirt.
As reported in Supplementary Note 1, the squirting mechanism is composed of a large reservoir that has strong muscle fibers in a configuration that makes the wall a contractile object 1 . This reservoir is connected to a narrow duct through which the slime is expelled.
Based on this information the basic model of the squirting system is much alike a simple syringe (See Supplementary Fig.3). The slime is stored in a large cylinder (reservoir) of radius R re and length L re that radially contracts and pushes its content all the way through the narrow canal until it reaches the external opening located at the oral papilla (Supplementary Fig.1 a-b and Supplementary Fig. 3). As shown in Supplementary Fig. 3 (Geometry supported from our own observations and studied by Baer et al.) the whole mechanism is composed by the reservoir that contracts radially, and glands that produce the slime which transits to the reservoir. For the specimens used, the reservoir length L re ∼ 3 · 10 −2 m, and its radius has a typical value R re ∼ 2 · 10 −3 m. According to our measurements of the dry fiber and the organ opening (See Supplementary Figs. 1,4), the average fiber diameter is d ∼ 150 µm. Using a high speed camera (240-480 fps) we have also measured the fiber propagation speed which is not constant but of order V ∼ 3 − 5 m s −1 .
Using the above mentioned information we can ask how large the reservoir radial contraction should be in order to produce such a fast liquid squirting. This speed allows this slow moving worm to capture fast moving preys such as fireflies, and crickets between others. A cricket has a typical scape speed 5 of order 2 m s −1 .
We proceed to provide simple formulae that relate the vesicle geometry and jet dynamics considering an open geometry. The vesicle volume before and after contraction are vol i = πR 2 re L re and vol f = π (R re − ∆R re ) 2 L re , where ∆R re is the change in the reservoir radius due to muscular contraction. Considering that ∆R re /R re 1 the volume of liquid expelled is vol i − vol f ≈ 2πR re ∆R re L re and that volume should be equal to the one of the squirted slime. In our experiments fiber length C ∼ 60 ∼ 10 −2 m, and its mean radius r ∼ 75µm.
The previous estimate is a consequence of overall volume conservation. However, the liquid slime is at a good approximation incompressible. Therefore, flow conservation holds.
Balancing injected flow from the vesicle and squirting flow from the papilla we obtain: using Eq. S1 we find that an estimate for the squirting time is given by: where τ is the typical time scale in which the liquid is actively expelled. Given that V out has been measured from high speed movies (Supplementary Movies 1-2) together with a standard videocamera for triangulation (Supplementary Movie 3), and ∆R re was obtained from volume conservation (Eq. S1) we can extract the typical time scale for the vesicle contraction that allows the fast squirt .
which is well within the reach of the muscles found in Onychophora, and consistent with experimental observations (Supplementary Movies 1 and 2). The radial contraction speed is quite slow ∆R re /τ ∼ 3 · 10 −4 m s −1 .

Supplementary Note 4: Elastic properties measurements.
We determine the bending stiffness of the artificial papilla B = EI, and from that determine the Young's modulus, E, of the PDMS produced. We used a dynamical way to measure B. Free oscillation frequency, Ω, of a PDMS cantilever was measured recording at 8000 fps. Considering only the first eigenvalue λ 0 for the cantilever problem, we can find B using where µ is the linear mass density, and L is the beam length. From the beam geometry we know that the moment of inertia is I = wh 3 /12. We use this information and its relation with B to compute E of the PDMS sample. Experimental damped oscillations, and the best fit to A(t) = A max cos(Ωt + δ) exp(−νt) are shown in Supplementary Fig. 6 from where E = 288 kPa, and ν = 5.35 s −1 is a damping factor.
In order to determine Papilla Young's Modulus (after dissection), we attached a small steel bearing ball (1mm in diameter) to the tip of an oral papilla, and used a magnet that was perpendicular to the sample to pull it. Thus, we deformed an oral papilla using magnetic forcing 3 in a free end cantilever configuration. In this case F = 3δEIL −3 , where F is the magnetic force, and L = 3.0 mm, the papilla length. We obtained E ∼ 40 kPa (Supplementary Table II). This large value could be due to post-mortem rigidity combined with dry slime at the inner part of the tube.
Supplementary Note 5: Estimates of physical parameters.
The parameters that naturally emerge when obtaining the dimensionless Eq. 4 are: β, and γ corresponding to the ratio between the masses of liquid versus the total mass of the system, and the ratio between elastic forces and weight, respectively. Furthermore, typical speed u 0 and time τ 0 scales are needed to describe the dynamics of the system. We determine the values of these quantities to define the relevant parameter space in our experiments.
The mass parameter for a tube of circular cross section is: comparable to the one found for the real specimen.
Using these parameters obtained from our experimental data we obtained a spatiotemporal plot shown in Supplementary Fig.8.
To further characterize the natural and artificial systems we introduce an elasto-fluidic Reynolds number as: where ρ, and µ are the liquid density and the dynamical viscosity respectively. The dynamical viscosity of the Onychophora slime is unknown up to date. However, it should not be far from the value for water as before to enter in contact with air it is composed of 90 percent of water, and the remnant 10 percent is made out of proteins, sugars, lipids and nonylphenol 9 .
In order to analyze the effect of local changes in the bending stiffness EI we built micro tubes with rectangular cross section, but with variable width, w ( Supplementary Fig. 7).
The thickness was constant h = 2.2 mm. The modulation was a sinusoidal pattern where the maximum always reached the same point. That is, the maximum width of all samples was constant (w = 2.0 mm ), but the local width of the pipe wall varied as shown in Supplementary Fig. 9.
In order to find V c for these samples we repeated the procedure shown and described in Fig. 3 (Main text). The only difference is that images were acquired at 3200 fps for better image resolution.
The control parameter in Supplementary Fig. 9 is the ratio (I/I min ) 1/2 , where I min is the smallest local moment of the inertia for pipes used in this experiment (See top left image Supplementary Fig. 9). We used this dimensionless parameter as we know that the typical speed in this system is: Therefore, when keeping M , E, and L constant, V c ∼ I 1/2 . Therefore, V c /V min ∼ (I c /I min ) 1/2 . We have used as reference V min = 4.7 m s −1 corresponding to the lowest critical speed measured in the micro pipe with the largest amplitude of the modulation (See leftmost inset Supplementary Fig. 9). Our data shows the onset of the instability occurs at lower fluid speeds when weak points are present in the micro pipes.