Surface tension dominates insect flight on fluid interfaces

Abstract

Flight on the 2D air-water interface, with body weight supported by surface tension, is a unique locomotion strategy well adapted for the environmental niche on the surface of water. Although previously described in aquatic insects like stoneflies, the biomechanics of interfacial flight has never been analysed. Here, we report interfacial flight as an adapted behaviour in waterlily beetles (Galerucella nymphaeae) which are also dexterous airborne fliers. We present the first quantitative biomechanical model of interfacial flight in insects, uncovering an intricate interplay of capillary, aerodynamic and neuromuscular forces. We show that waterlily beetles use their tarsal claws to attach themselves to the interface, via a fluid contact line pinned at the claw. We investigate the kinematics of interfacial flight trajectories using high-speed imaging and construct a mathematical model describing the flight dynamics. Our results show that non-linear surface tension forces make interfacial flight energetically expensive compared with airborne flight at the relatively high speeds characteristic of waterlily beetles, and cause chaotic dynamics to arise naturally in these regimes. We identify the crucial roles of capillary-gravity wave drag and oscillatory surface tension forces which dominate interfacial flight, showing that the air-water interface presents a radically modified force landscape for flapping wing flight compared with air.

Keywords: Biomechanics; Capillary waves; Capillary–gravity wave drag; Chaos; Interfacial flight.

Fig. 1.

Interfacial flight in Galerucella nymphaeae. (A) Natural habitat of G. nymphaeae in Harvard Forest, MA, USA, where the first specimens were observed and captured. Inset shows a wild specimen resting on a waterlily leaf. (B) Close-up of G. nymphaeae. Inset shows the leg resting on a drop of water, such that the tarsi are unwetted and supported on the drop while the claws at the tip are immersed in the water. Note that the drop is deformed near the claws. (C) Capillary waves are generated during interfacial flight, due to the perturbation of the interface by the insect's immersed claws. (D) Schematic illustration of interfacial flight in G. nymphaeae. (E,F) side view and rear view of G. nymphaeae posture in upstroke, midstroke and downstroke of interfacial flight. The middle pair of legs is raised above the body and the body is angled such that its weight is well supported between the four immersed legs. Scale bars 5 mm (1 mm in B inset).

Fig. 2.

Adaptations enabling interfacial flight in G. nymphaeae. (A) False colour SEM image indicating wetting and non-wetting regions on the leg. Green indicates superhydrophobic regions and blue indicates hydrophilic regions. (B) SEM images of G. nymphaeae body and legs showing hydrophobic hairy structures. (C–E) Successive magnifications of the hindleg ultrastructure (D) showing tarsi with hydrophobic setae (E), and a pair of curved, hydrophilic claws (C), which are immersed. (F–H) Similar hydrophobic–hydrophilic ultrastructural line barriers seen on hindlegs (F) and forelegs (G,H) of more beetles. (I) Schematic  diagram showing pinning of the contact line at tarsal claws during interfacial flight. (J) Meniscus formed by dipping the leg of a dead beetle into water and raising up to the maximum extent. The white curve is the computed theoretical minimal surface profile, which fits the experimentally produced meniscus well. (K) Sequence showing formation and breakage of meniscus at the claw during take-off in a live beetle. White scale bars, 100 µm; black scale bar, 5 mm.

Fig. 3.

Kinematics of interfacial flight trajectories. (A) Representative interfacial flight trajectory showing vertical oscillations. Inset shows a snapshot from the video used to generate the trajectory, with the arrow pointing to the femur–tibia joint of the insect's hindleg – a natural marker used for tracking. Error bars were calculated for tracked coordinates as the pixel resolution of the video, equal to approximately 10 µm per pixel. (B) Horizontal displacement increases rapidly, with average displacement per wingbeat varying quadratically with time. Inset shows the displacement during a single wingbeat, with a steep increase during downstroke and sigmoidal variation during upstroke. (C) Variation in horizontal velocity over time showing a linear increase in the average downstroke velocity. Velocity is maximum in the downstroke part of a wingbeat and drops to a minimum in mid-upstroke before recovering. Velocities were computed by fitting a spline to the displacement and calculating its slope. Error bars were derived from displacement errors.

Fig. 4.

Dynamic model for interfacial flight. Schematic diagram depicting the different forces acting on the beetle. Boxed insets on the right show the beetle reduced to a single particle pinned at the interface, with horizontal forces (top) and vertical forces (bottom) acting on the particle. Circled insets on the left show the direction reversal of surface tension, depending on the nature of deformation of the meniscus at the pinned contact line. A_x, horizontal air drag; C_x, horizontal capillary–gravity wave drag; T_x, horizontal thrust; W_x, horizontal water drag; G_y, vertical gravity (body weight); S_y, vertical resultant of surface tension; W_y, vertical water drag.

Fig. 5.

Experimentally derived dynamics of wing forces exerted during interfacial flight. (A) Stroke plane angle decreases linearly as body angle increases during flight. Increasing body angle leads to transitions between interfacial, airborne or backward flight modes. Error bars are estimated from pixel resolution error in extreme wing and body points used to calculate angles. (B) As an insect takes off from the interface into air, body angle steadily increases while stroke plane angle approaches zero. When stroke plane angle falls below zero, the flight direction is backwards, opposite the dorsal-to-ventral axis. (C–E) Failed take-off attempt where the legs are trapped by surface tension. In C, a sequence of images shows a failed take-off attempt, where a significant portion of the legs is wetted. Note the almost vertical posture (body angle ∼90 deg) that concentrates wing force into lift. Scale bar, 5 mm. (D) Horizontal displacement and body angle both increase initially, and then level off as thrust approaches zero (shaded region on the right.) Error bars indicate pixel resolution. (E) Mean horizontal velocity in each wingbeat is almost zero, resulting in a static equilibrium. Error bars were derived from positional errors corresponding to one pixel.

Fig. 6.

Computational modelling of capillary drag, oscillations and take-off. (A) Comparison showing that total drag experienced in interfacial flight is much larger than that in airborne flight, above a critical velocity. The additional capillary–gravity wave drag experienced in interfacial flight is shown as the shaded region. (B) Different trajectories computed for a given total wing force magnitude, by varying the wing force distribution between lift and thrust. Trajectories take off for lower thrust-to-lift ratio p and remain confined to the interface for higher p. The dashed line is the mean water level at infinity and the dotted line is the maximum vertical height of meniscus attachment (260 µm). (C–E) Phase plots for variation in take-off time for different values of lift force per unit body weight, for an insect with two legs immersed in water. The single other parameter varied in each plot is wingbeat frequency f_w in C, symmetry ratio (r) between wing force produced in upstroke and downstroke in D, and wing stroke angle at initiation of motion (φ₀) in E. White dotted lines show the maximum estimated lift and the wingbeat frequency f_w≈116 Hz.

Fig. 7.

Chaotic vertical oscillations on the interface. (A) Bifurcation diagram for trajectories at different lift-to-weight ratio q. A wide range of q shows five distinct trajectory regimes: (i) periodic and confined to interface, period equal to or half the wingbeat period; (ii) chaotic, confined to the interface; (iii) chaotic, takes off from the interface with some time delay; (iv) periodic and confined to interface, period equal to wingbeat period; (v) instantaneous take-off. The expanded section of the transition between regimes i and ii shows cascades and regions of chaos. (B) Phase plot at q=0.99 (top) shows a self-intersecting period-4 cycle. Phase plot at q=1.48 (bottom) shows a chaotic attractor also seen in other oscillators driven by surface tension (Gilet and Bush, 2009). (C) Divergence of two trajectories with closely spaced initial conditions in the chaotic regime (q=1.48), where one takes off and the other is trapped on the interface. Both trajectories are simulated with the parameters L=137 μN, f=116 Hz, r=0.15 and φ₀=π/2, but for different initial conditions which do not lie on each other's phase plot. (D) Delay plot of experimental trajectory data showing vertical displacement plotted against itself with a delay of τ=1 wingbeat≈8.67 ms. The lack of any repeated structure indicates that the vertical oscillations observed in the experimentally recorded trajectory do not have any correlation in time and are chaotic in nature.